Tailoring spatial modes for sum-frequency generation

Master’s Thesis

Tailoring spatial modes for sum-frequency generation in lithium niobate waveguides

´ pez Jano Gil Lo Supervisors: Prof. C. Silberhorn and Prof. T. Zentgraf

University of Paderborn Integrated Quantum Optics May 15th , 2017

Jano Gil López

Acknowledgments This project would have never been finished without all the support that I have had during all my master’s thesis. I want to thank Vahid Ansari and Markus Allgaier for all the time spent on helping with my project, giving advice and always being available for me. I have learned a lot from you guys. Special thanks also to Michael Stefszky and John Donohue for their comments and time. No less help I received from all the Integrated Quantum Optics group every time that I needed counsel on anything. It has also mean a lot to me to find such nice people to spend time with (inside and outside the university), filling with joy the never completed heart of someone away from home and his beloved ones. Thanks! Danke! Grazie! I could never thank Natalia enough for encouraging me to complete my studies abroad. The strength and will that she has sent me from the beginning to keep working, learning and growing are invaluable. She makes every effort worthy. Thanks for your courage and conviction, te amo. I keep infinite gratitude to my parents for supporting me all this time abroad. They have provided everything I needed to follow my dreams and they shared my happiness with every success. I proudly present this thesis before them as a thanks.

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CONTENTS

Tailoring spatial modes for sum-frequency generation in lithium niobate waveguides

Contents 1 Introduction

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2 Theoretical foundation 2.1 Laser beams . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Gaussian beams . . . . . . . . . . . . . . . . 2.2 Waveguides . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Total internal reflection waveguide model . . . 2.2.2 Electromagnetic waves in waveguides . . . . . 2.2.3 Modes in slab waveguides . . . . . . . . . . . 2.2.4 Channel waveguides . . . . . . . . . . . . . . 2.3 Integrated optical devices . . . . . . . . . . . . . . . 2.3.1 Directional couplers . . . . . . . . . . . . . . 2.3.2 Adiabatic tapers . . . . . . . . . . . . . . . . 2.4 Nonlinear optics . . . . . . . . . . . . . . . . . . . . 2.4.1 Second order nonlinearities: three wave mixing 2.4.2 SFG processes in Lithium Niobate waveguides

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3 Methods for optimising two wavelength waveguide coupling 3.1 Spatial mode characterization . . . . . . . . . . . . . . . . . . . . 3.2 Approach I - Integrated solution . . . . . . . . . . . . . . . . . . . 3.2.1 The integrated two wavelength coupling device . . . . . . . 3.2.2 Tapered structure for 850 nm light . . . . . . . . . . . . . 3.2.3 Directional coupler . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Periodic poling . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Approach II - Spatial Light Modulator solution . . . . . . . . . . 3.3.1 The Digital Micromirror Device SLM . . . . . . . . . . . . 3.3.2 DMD diffraction characterization . . . . . . . . . . . . . . 3.3.3 Spatial intensity shaping . . . . . . . . . . . . . . . . . . 3.3.4 Spatial phase shaping - Holography . . . . . . . . . . . . . 3.3.5 Finding the single mode - Genetic algorithm optimization .

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CONTENTS

4 Results on the SLM coupling optimization 4.1 Spatial amplitude shaping . . . . . . . . . . 4.2 Spatial phase shaping . . . . . . . . . . . . . 4.3 Tailoring waveguide mode coupling . . . . . 4.4 Genetic algorithm . . . . . . . . . . . . . . . 4.4.1 Computer simulations . . . . . . . .

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5 Conclusions and outlook

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6 References

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1

Introduction

Over the past few decades, quantum information science has seen a fast and successful development. Principles of quantum mechanics like the uncertainty principle, the no-cloning theorem and entanglement were applied in the second half of the twentieth century in experiments to produce quantum communication protocols [1] and quantum teleportation [2, 3]. These early experiments demonstrated the potential increase in security and computational power that quantum information has to offer to communication and computation sciences [4]. Optics is one of the most promising methods for implementing these quantum technologies. Photons have outstanding performance in quantum information processing as they are the perfect quantum information carriers [5]. Furthermore, many applications can be done in existing fibre networks, thereby aiding in the real-world implementation of these technologies. Quantum networks are developed to produce systems for quantum technologies. In them, quantum operations and protocols are the nodes and the information carriers are the links between the nodes. Many properties of photons are desirable for quantum network links, as quantum bits (qubits). They exhibit low-noise properties and have many degrees of freedom to encode information [5]. One method to generate photons is the well known spontaneous down-conversion (SPDC) non-linear process [6]. Photons generated in experiments can be in entangled states, transported, measured and even stored in quantum memories [7, 8, 9]. Taking the proof of principle quantum technologies into the real world will likely require an integrated platform. Integrated devices offer smaller footprint, reduced cost, greater production capability and higher efficiencies. However, integrated technologies also come with their own set of problems that need to be overcome. Quantum states are highly sensitive to both loss and noise as they can destroy the states, entanglement and erase the information. Loss and noise in fibre networks limits the current range of the technologies. To overcome these losses and to interface the different nodes in the quantum network will likely require frequency conversion. The devices needed such as repeaters and memories usually require different light (i.e. wavelengths and spectral bandwidths) to work with. Non-linear processes such as sum-frequency and differencefrequency generation come into play at this point. With these processes, frequency converters can be built in integrated chips to interface different elements in a quantum network [10, 11]. 4

Jano Gil López

Frequency conversion involves photons and light fields at different wavelengths, the integrated devices have to be engineered to produce the desired output. Different wavelengths propagate with different spatial properties inside waveguides, leading to issues for the overlap of the interacting fields. In this thesis, we address one of this issues: the overlap of the spatial intensities of the interacting fields. The bad spatial overlap between single and multimode fields inside the integrated frequency converter, decreases the efficiency of the interaction. This is usually taken care of by the alignment of the fields into the waveguides and even then there is no way to know which modes are interacting in the process. We want to enhance the efficiency of integrated sum-frequency generation by tailoring the spatial modes coupled into the waveguides with two alignment-independent approaches. In chapter 2 the theoretical foundations needed for the experimental work are explained. The concepts of spatial intensity distribution and spatial modes are introduced. Then, the propagation of light fields in waveguides is described and continues with an introduction to integrated structures that will be used. The theory chapter is concluded with an explanation on non-linear processes to highlight the spatial overlap problem. After the theoretical introduction, the two approaches to the problem are described in chapter 3. The first approach is the design of an integrated solution, that deals with the spatial modes. The spatial modes of waveguides used are characterized for the design. Following the characterization, the different parts of the integrated structure are described. The second approach uses a Spatial Light Modulator (SLM) to shape the spatial modes before coupling them into the waveguide. The SLM is characterized for spatial intensity and phase shaping. A genetic algorithm is then introduced to optimize the spatial mode coupling. Finally, in chapter 4 results on the SLM experiments are shown. The tailoring of propagated modes through the shaping of the SLM is proved. The genetic algorithm is simulated to obtain parameters for its implementation. The thesis finishes in chapter 5 with the conclusions and an outlook on future projects motivated by this work.

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2

Theoretical foundation

Spatial modes of light describe the intensity distribution of laser beams on the plane perpendicular to the propagation direction. They are solutions to the Helmholtz’s equation that describe the properties of laser beams. The spatial modes of light are of special interest when working with processes that involve the overlap of two or more different beams in beam splitters, non-linear crystals or optical waveguides. Good overlap for efficient processes will depend on the spatial modes of the beams involved. A description of spatial modes is given so that its importance in waveguide coupling and overlapping processes like Sum Frequency Generation (SFG) can be regarded. Basic concepts from electromagnetism and ray optics will lead to equations and expressions to describe the propagation of laser beams, their spatial modes and their behavior in waveguides. The use of light and lasers in optics requires understanding of the electromagnetic (EM) field that describes them. The wavefronts of the EM field describe how the waves evolve in time and space as they propagate through a medium or the vacuum. On the wavefront, the spatial intensity distribution is what defines spatial modes of light. We will introduce the basics of nonlinear optical processes in crystals and specifically SFG. Then, a description of waveguides provides all the background needed to complete the theoretical introduction.

2.1

Laser beams

A beam is an EM wave propagating in one direction (z from now on) and usually has a high degree of spatial coherence [12]. The complex wave function completely describes an EM field as follows: U p~r, tq “ ap~rqexpriϕp~rqsexpri2πνts

(2.1)

in which: ap~rq is the amplitude of the wave, ϕp~rq the phase and ν is the frequency. It is a function of position ~r0 px, y, zq and time. It has to satisfy the wave equation ∇2 U ´

1 B2U “ 0. c2 Bt2

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(2.2)

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2.1 Laser beams

We can rewrite (2.1) as: U p~r, tq “ Ap~rqexpri2πνts

(2.3)

where U p~r, tq is called the complex amplitude and the term Ap~rq is the complex envelope. If we substitute (2.3) into the wave equation, it leads to a differential equation for the complex amplitude: ∇2 U ` k 2 U “ 0,

(2.4)

called the Helmholtz’s equation, where k “ 2πν{c is the wavenumber. Propagating EM waves, such as laser beams, must satisfy the Helmholtz’s equation. The optical intensity is obtained from the complex amplitude as Ip~rq “ |U p~rq|2 ,

(2.5)

which describes the intensity distribution. Spatial light modes are defined by this distribution in the px, yq plane, also called transversal modes. Its shape will be given by the solution of the Helmholtz’s equation. An example beam and transversal mode is depicted in figure (1). There are many beam profiles that satisfy the Helmholtz’s equation. We are interested in Gaussian beams as they provide the foundation for higher order or different spatial modes.

Figure 1: The y axis goes upward normal to the page. In a) A Gaussian beam is illustrated propagating along z. Gaussian beams are a solution of the Helmolzt equation as we will see in the next section. In b) the intensity distribution in px, y, z0 q. is illustrated 7

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2.1.1

2.1 Laser beams

Gaussian beams

Gaussian beams are easily produced by commercial lasers and they have good properties for operation in free space and in waveguides. We are going to use the paraxial approximation from now on. In the paraxial approximation, the normals of the wavefront make small angles with the direction of propagation ~z, meaning that the complex envelope of the waves varies slowly with z. That is represented by a paraxial wave U p~rq “ Ap~rqexppíkzq,

(2.6)

which is a plane wave modulated by the slowly varying complex envelope Ap~rq. The complex envelope of a paraxial wave must satisfy the Helmholtz’s equation and the paraxial approximation requires the complex envelope to vary slowly with z. This leads to the paraxial Helmholtz’s equation [13] BA “ 0, Bz is the transversal Laplacian operator. ∇2T A ´ i2k

where ∇2T “ δxx ` δyy

(2.7)

The paraboloidal wave is a solution of (2.7). Moreover, the Gaussian beam is a paraboloidal wave shifted in z by a complex number [13]. It is a new solution of the paraxial Helmholtz’s equation whose complex envelope is a Gaussian function, as we seek. Ap~rq “

A1 ρ2 exppík q, qpzq 2qpzq

qpzq “ z ` iz0 .

(2.8)

The Gaussian beam is a natural solution of the paraxial Helmholtz’s equation, with an envelope described by (2.8). qpzq is the q-parameter in which z0 is known as the Rayleigh range and A1 is a constant. To obtain the complex amplitude, we separate real and imaginary terms of 1 1 λ “ í qpzq Rpzq πW 2 pzq

1 : qpzq

(2.9)

W pzq and Rpzq are measures of the beam width and wavefront radius of curvature, respectively, as shown in figure (2). W pzq is the spot size of the beam (is the radius of the area that comprises 86% of the total power [13]). The minimun W0 is at z “ 0 8

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2.1 Laser beams

and is known as the radius ? of the waist. Finally, the Rayleigh range z0 is the distance that takes W pzq to be 2W0 . When the Rayleigh range is maximized within a given lens/initial beam constraint to keep W pzq constant the beam is said to be collimated. Substituting (2.9) in (2.8), and combining with the complex amplitude expression (2.6), the Gaussian beam complex amplitude simplifies to: U p~rq “ A0

ρ2 ρ2 W0 expp´ 2 qexppíkz ´ ik ` iζpzqq W pzq W pzq 2Rpzq πW02 λ A0 “ A1 {z0 z0 “

c z W pzq “ W0 1 ` p q2 z0 z0 2 Rpzq “ zp1 ` p q q z ´1 z ζpzq “ tan z0

(2.10)

(2.11) (2.12) (2.13) (2.14) (2.15)

The last six equations completely describe the propagation of a Gaussian beam. z0 is determined by the wavelength and W0 which are characteristics given by the source of the laser beam. The rest of the parameters are related ?to and can be calculated from z0 . We have used cylindrical coordinates with ρ “ x2 ` y 2 radial position and z the axial coordinate. With this set of equations we can give expres-

Figure 2: Characteristics of a Gaussian beam propagating in z direction. W0 is the minimun waist width or focus width, W pzq is the dependence of the width with z and describes the divergence of the beam as it propagates and Rpzq is the dependence of the wavefront radius of curvature with z. 9

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2.1 Laser beams

sions for the properties of the beam that will be used in the thesis. The intensity distribution of a wave was defined in the last section as Ip~rq “ |U p~rq|2 . With (2.10), this leads to Ipρ, zq “ I0 p

W0 2 2ρ2 q expp´ 2 q, W pzq W pzq

(2.16)

where I0 “ |A0 |2 . The intensity is a Gaussian function of the radial distance, giving the Gaussian beam its name. There are other beam solutions of the paraxial Helmholtz’s equation. The Gaussian beam is the lowest order of a more general collection of solutions known as the Hermite-Gaussian beams which are orthogonal perfectly distinguishable modes. They have different intensity distributions shown in figure (3) but they all propagate as Gaussian beams [13] with the parameters given in equations (2.11-2.15). The propagation parameters of a beam can be modified with bulk optics such as mirrors, pin holes and gratings while still fulfilling Helmholtz’s equation.

Figure 3: Six orders of the Hermite-Gaussian intensity distribution are illustrated. The first order is a Gaussian distribution. The units are arbitrary.

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2.2 Waveguides

Through this section we have described how light fields propagate in free space. The description changes when light is coupled into waveguides. The propagated modes inside waveguide depend on the coupled modes and the characteristics of the waveguides. In the next section, we solve the propagation of light fields in waveguides and describe how they change with the wavelength of the coupled field.

2.2

Waveguides

To realize optical experiments it is necessary to direct light fields onto detectors and through multiple devices. This can be done in free space, with the use of bulk optics such as lenses, mirrors, gratings and prisms. Such optical experiments need large spaces and some effort to align the light on the correct path. However, it is possible to transmit optical beams through closed conduits that we call waveguides. Waveguides allow long-distance transmission and the design of complete setups in small integrated devices [13, 14]. They are also good to overlap fields for both linear and non-linear interactions. Several optical components can be realized in chips, linked by waveguides in much the same way as wires in an electric circuit. The application of waveguides in integrated devices has open a new field in optics: integrated optics [15]. Two of the more widely used waveguides are optical fibers and strip dielectric waveguides. The working principle of waveguides can be simply explained by total internal reflection between two materials. After the simple description, we can find solutions for the spatial modes propagated inside waveguides and how they change with the characteristics of light and the structure itself. We will obtain those results solving the Maxwell’s equations in the waveguide. 2.2.1

Total internal reflection waveguide model

Ray optics draw the most simple description. We start from a slab waveguide as in figure (4). It is compose of two different materials with refractive indexes ni and nr . A ray incident with angle θ will be refracted following Snell’s law ni sin θi “ nr sin θr ,

(2.17)

where ni and θi are the refractive index and incident angle of the incident ray respectively, and nr and θr the correspondent at the other material.

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2.2 Waveguides

Figure 4: Depiction of Snell’s law. In a) the incident angle is smaller than the critical angle and the light is refracted, in b) the angle is larger than the critical angle and is reflected. Equation (2.17) yields a ray incident on the boundary between two refractive indices. It gives the refraction angle in the new material as the ray traverses the limit. However, at a certain incident angle known as the critical angle, the ray is completely reflected on the boundary. Light that is incident on the interface with an angle greater than the critical angle is always reflected. The critical angle is easily calculated setting the refracted angle as θr “ 90˝ so long as ni ą nr . When the incident angle is below the critical angle it called radiation mode [15]. On the opposite, if the angle angle, then the wave is reflected, being trapped. This is [15]. When we obtain a guided mode, the light is coupled will propagate inside. 2.2.2

will be refracted. This is is larger than the critical known as a guided mode into the waveguide and it

Electromagnetic waves in waveguides

The description given in the last section only describe how fields are guided through the waveguide. To completely describe how the light fields propagate inside waveguides and how are the guided modes we have to solve Maxwell’s equations. They will give more detailed conditions for guided modes and how are their intensity distributions. The Maxwell’s equations are

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2.2 Waveguides

~ “ íωµH ~ ∇Ê ~ “ íωµE ~ ∇ˆH ~ “0 ∇Ë ~ “ 0. ∇¨H

(2.18)

The electric and magnetic fields propagating inside the waveguide have the form ~ ~ ipωt´βzq Epx, y, z, tq “ Ee ~ ~ ipωt´βzq , Hpx, y, z, tq “ He

(2.19) (2.20)

where β is the propagation wavenumber or constant [13] inside the waveguide. The beam propagates differently in the waveguide than in the vacuum and this is described by β which depends on the higher refractive index of the medium and the geometry of the waveguide. These fields propagate along z in the waveguide. It makes sense to split them into the two transverse (x and y) directions to solve the propagation modes inside the guiding structure. We decompose the fields and the gradient operator ∇ as ~ ~ T px, yq ` zÊz px, yq Epx, yq “ xÊx px, yq ` yÊy px, yq ` zÊz px, yq “ E

(2.21)

and ∇ “ xˆBx ` yˆBy ` zˆBz “ ∇T ` zˆBz “ ∇T ´ iβ zˆ,

(2.22)

where xˆ, yˆ and zˆ are the unitary vectors on those directions. The xˆ and yˆ are the transversal components, giving ET . Then, zˆ is the longitunidal component for Ez . The change Bz into íβ is due to the dependence with the propagation direction z [16]. Together with the decomposition in equations (2.21) and (2.22) we have to note ~ T “ 0 and zˆ ¨ ∇T Ez “ 0. We also note that zˆ ˆ E ~T that: zˆ ¨ zˆ “ 1, zˆ ˆ zˆ “ 0, zˆ ¨ E ~ T have only transverse components while ∇T ˆ E ~ T is longitudinal. With all and zˆ ¨ E this we obtain a new set of Maxwell’s equations:

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2.2 Waveguides

∇T Ez ˆ zˆ ´ iβ zˆ ˆ ET “ íωµHT ∇T Hz ˆ zˆ ´ iβ zˆ ˆ HT “ íωµET ∇T ˆ ET ` iωµˆ z Hz “ 0 ∇T ˆ HT ` iωµˆ z Ez “ 0 ∇T ¨ ET ´ iβEz “ 0 ∇T ¨ HT ´ iβHz “ 0.

(2.23) (2.24) (2.25) (2.26) (2.27) (2.28)

If we perform the cross product of equations 2.23 and 2.24 with zˆ, using the vector identity B ˆ pA ˆ Cq “ BAC ´ CAB and reorganizing the terms, we obtain a linear system of two equations for ET and HT β zˆ ˆ ET ´ ωµHT “ iˆ z ˆ ∇T Ez ωµˆ z ˆ ET ´ βHT “ í∇T Hz .

(2.29)

We can rearrange this as zˆ ˆ ET “ ´ kiβ2 zˆ ˆ ∇T Ez ´ iωµ ∇T Hz kc2 c iβ iω HT “ ´ k2 zˆ ˆ ∇T Ez ´ k2 ∇T Hz , c

(2.30)

c

to isolate the transversal components. Where the cutoff wavenumber kc is defined 2 as kc2 “ ωc2 ´ β 2 . kc take different values depending on the boundary conditions (β) of the waveguide [16]. Another handy quantity is the cutoff frequency ωc “ ckc , the minimum frequency for a wave to be transmitted. It is also useful to express β in terms of ω and ωc c ω2 ω 1 ´ c2 . (2.31) β“ c ω Finally, in order to satisfy all the Maxwell’s equations, the longitudinal fields must satisfy the Helmholtz’s equations [16]: ∇2T Ez ` kc Ez “ 0 ∇2T Hz ` kc Hz “ 0.

(2.32)

Equations (2.32) have to be solved under the boundary conditions of each waveguide type. Once the longitudinal fields are known, ET and HT are calculated from (2.30). This is a hybrid solution with non-zero Ez and Hz .

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2.2 Waveguides

We can classify the solutions of this set depending on whether one, both or none of the longitudinal components are zero in: transverse electric and magnetic (TEM), transverse electric (TE) or transverse magnetic (TM) modes, naming each after the component which is non-zero. It is important to note that in the TE and TM cases, the transverse field components ET and HT depend only on the longitudinal components.

TE modes The conditions are Ez “ 0 and Hz ‰ 0. Applying these conditions to (2.30) and (2.32): ∇2T Hz ` kc2 Hz “ 0 HT “ ´ kiβ2 ∇T Hz c ET “ ηT E HT ˆ zˆ.

(2.33)

The mode impedance has been defined as ηT E “ ωµ . The third equation in (2.33) β is the same as for plane waves traveling in the z-direction but with the impedance replaced by ηT E .

TM modes The process is analogous to the treatment with TE but with Ez ‰ 0 and Hz “ 0. ∇2T Ez ` kc2 Ez “ 0 ET “ ´ kiβ2 ∇T Ez c HT “ ηT E zˆ ˆ ET . 2.2.3

(2.34)

Modes in slab waveguides

Slab waveguides are two dimensional waveguides with two clearly different regions of different refractive index. We will obtain the conditions for propagated fields, the number of modes and their intensity distribution at different wavelengths. The core has to have higher refractive index than the surrounding material to trap light. This means ni ą nr . As the cutoff wavenumber kc depends on the refractive

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2.2 Waveguides

index of the medium it is different outside and inside the waveguide. 2 “ k02 n21 ´ β 2 kc1 2 “ k02 n22 ´ β 2 kc2

(inside) (outside).

(2.35)

In order to keep the wave confined inside the waveguide, kc2 must be imaginary. In that case, the field decays in the outside region. We set kc2 “ íαc . The second equation in (2.35) becomes αc2 “ β 2 ´ k02 n22 .

(2.36)

Inserting this into the Helmholtz’s equation for the TM modes (first equation in (2.34)), we arrive at a system for the electric field in the two regions: Bx2 Ez pxq ` kc2 Ez pxq “ 0 for |x| ď a Bx2 Ez pxq ` αc2 Ez pxq “ 0 for |x| ě a.

(2.37)

The solutions of equations (2.37) are of the kind sinpkc xq and cospkc xq. They can also be written as e˘ikc x for the even and odd modes respectively. Even and odd refer to the symmetry of the magnetic field. For even modes we obtain $ ’ ’ & E1 sinpkc xq if |x| ď a Ez pxq “ (2.38) E2 e´αc x if x ě a ’ ’ % E3 eαc x if x ď á. It follows from the last two equations in (2.38), that the solution outside the waveguide decays exponentially on both sides. The x-components Ex are obtained by applying [16] Ex “ ´

iβ Bx Ez kc2

,

(2.39)

leading to: $ iβ ’ ’ & ´ kc E1 cospkc xq if |x| ď a Ex pxq “ ´ αiβc E2 e´αc x if x ě a ’ ’ % iβ E3 eαc x if x ď á. αc

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(2.40)

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2.2 Waveguides

Finally the magnetic fields arise $ ’ ’ & Hy pxq “ ’ ’ %

from Hy “

1 E ηT M x

([16]) as:

H1 sinpkc xq if |x| ď a H2 e´αc x if x ě a H3 eαc x if x ď á.

(2.41)

Here we have defined: H1 “

iβ E1 , kc ηT M

H2 “ ´

iβ E2 , αc ηT M

H3 “

iβ E3 αc ηT M

(2.42)

. For the odd TM modes, the trigonometric functions on Ez pxq and Hy pxq are interchanged. This sets of equations describe the possible light fields inside the slab waveguides. Light coupled into the waveguide will excite and propagate one or several of the possible fields depending on the characteristics of the waveguide and the wavelength of the input field. The expressions for the electric and magnetic fields depend on kc and αc , both coefficients depend on β. As we obtained in (2.31), β depends on the frequency of the field and the cut-off frequency, therefore different input frequencies will propagate different modes. The last step is to apply the proper boundary conditions. The tangential components of the electric and magnetic fields (Ez and Hy ) have to be continuous across the slab interfaces with the surrounding material at x “ á and x “ a. Also the normal components of the electric field Ex must be continuous. Since Hy “ ηT1M Ex , the continuity of Hy follows from the continuity of Ex . The constraints for the fields that yield the boundary conditions are [16]: αc “ kc tanpkc aq (even modes) αc “ ´kc cotpkc aq (odd modes).

(2.43)

Equations (2.35) and (2.43) relate the three paramters of the waveguide (kc , αc and β) with the frequency of the wave ω. We eliminate β combining the two equations in (2.35): αc2 ` kc2 “

ω2 2 pn1 ´ n22 q. 2 c0

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(2.44)

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2.2 Waveguides

Finally, we obtain the numerical solution of equations (2.43) and (2.44). In order to do that, we can define the dimensionless quantities u “ kc a and v “ αc a, giving: # even modes

v “ u tanpuq , v 2 ` u2 “ R 2

# odd modes

v “ ú cotpuq v 2 ` u2 “ R 2

(2.45)

Where R is called the normalized frequency variable [16], R“

2πa NA λ

(2.46)

a and NA “ n21 ´ n22 is the numerical aperture of the slab. There are many solutions that are the intersection of the tangent and cotangent with the circles of radius R. The graphic solutions are shown in figure (5). For R ď π{2 there is only one even solution for the system. Bigger values of R have more than one solution. Each solution has a different spatial intensity distribution that can be calculated from (2.45). Through this section, we have stated how light propagates inside waveguides and which spatial modes are allowed by the boundary conditions of the system. With

Figure 5: The functions for even and odd modes cut the different radius circles. Each intersection in a circle is a solution, i.e. a propagating mode inside the slab waveguide. 18

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2.2 Waveguides

equation (2.46) we can calculate how many solutions (i.e. modes) will a field with wavelength λ propagate for a certain waveguide with width a and aperture NA . We have learned that each propagated mode will have a different spatial intensity distribution because of the dependence on the frequency of the EM fields inside the waveguide. a For example, a 1550 nm field in a 7 µm wide waveguide with n21 ´ n22 “ 0.148 (for Titanium diffused Lithium Niobate waveguides), will have only one propagated mode as R “ 1.33 ă π{2. However, a 888 nm field in the same waveguide will have two propagated modes as R “ 2.33 ą π{2. The propagation modes for both wavelengths are completely different and this is a problem when we want to overlap both fields in a waveguide to produce non-linear interaction. From section 3, we will describe two approaches to couple the multimode field only into one propagated mode. 2.2.4

Channel waveguides

In last section, we outlined slab waveguides because is easy to obtain analytical solutions for the propagated fields in them. However, in the experiments we used channel waveguides. The difference with slab waveguides are that channel waveguides are three dimensional structures with a refractive index distribution that varies continuously along the transversal direction from ni to nr . The analytical soluion of Maxwell’s equations in channel waveguides are very hard to obtain and usually only nummerical solutions are available through simulations [17]. However, the wavelength dependence of the mode number has similar properties between slab and channel waveguides.

Figure 6: Front and top view of a channel waveguide. The gray section has a higher refraction index ni ą nr so that light is guided through it.

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2.3

2.3 Integrated optical devices

Integrated optical devices

The propagation processes described in section (2.2) allow the design of more complicated structures than waveguides. Many bulk optics components can be reproduced with integrated components at the scale of the micro or nano meters. Since the first ideas for design and production of integrated structures in 1969 [18] the field has grown with great success. Along with new and improved production techniques such as epitaxy [19], lithography [20], material exchange [21] and diffusion [22], new materials like Silicon, Lithium Niobate, Potassium Titanyl Phosphate and Galium among others have been researched for the devices. Different materials have different characteristics for the refractive indices contrast, propagation constants, damage threshold, reflectivity and non-linearity. Many technologies have been improved by means of the integrated devices: optical resonators, photon generators, lasing systems, photonic circuits, computing. The development of the field has lead to powerful applications in photonics, biological imaging, medicine and communications. All this technologies relay on well known and widely extended integrated structures available. In this section, we are going to describe and summarize some of them that will be use in this thesis. 2.3.1

Directional couplers

Beam splitters and polarized beam splitters have a wide variety of applications in optics: beam switches, intereferometry experiments and power filters. Directional couplers are an integrated waveguide structure that allows the transfer of an optical field from one waveguide to another [13, 17]. They are the integrated equivalent to beam splitters in bulk optics. Directional couplers are made of two waveguides that are physically close enough in the direction transversal to the propagation direction such that the evanescent part of the fields reaches the other waveguide before it completely decays, as in figure (7). For weak coupling (defined by the proximity of both channels), coupling-mode theory can be calculated to describe the propagation of fields in a directional coupler [13]. For this prof we assume single mode behavior in both waveguides and lossless mediums. In coupling-mode theory , the modes of each waveguide are not affected by the presence of the other waveguide. Using the description developed in the chapters before, we can write our modes for the first and second waveguides as u1 pyqeíβz and u2 pyqeíβz respectively. Coupling is assumed to modify only the amplitudes of 20

Jano Gil López


the modes as they propagate and couple into the other waveguide. To describe that, we calculate the amplitudes of the fields as functions of the propagation direction z: a1 pzq and a2 pzq. Coupling can be seen as a scattering effect. The field on the first waveguide is scattered in the second waveguide, creating a source of light and changing the amplitude of the field in the second waveguide. This can be written as two coupled differential equations for the amplitudes in both waveguides [13]: da1 dz da2 dz

“ íC21 ei∆βz a2 pzq “ íC12 ei∆βz a1 pzq,

(2.47)

where ∆β “ β1 ´ β2 is the phase mismatch per unit length and C21 and C12 are coupling coefficients that depend on the refraction indices, the propagation constants, the waveguide widths and the distance between the waveguides. They can be different for each waveguide. Different coupling coefficients represents the same conditions than different reflectance and transmittance in a beam splitter. Solving the differential equations in (2.47) we obtain: a1 pzq “ Apzqa1 p0q ` Bpzqa2 p0q a2 pzq “ Cpzqa1 p0q ` Dpzqa2 p0q,

(2.48)

Figure 7: A directional coupler between two waveguides of width a separated by a distance d, close enough to allow power transfer. The red lines show the spatial intensity of a field being transferred. The wave is coupled from the first waveguide into the second one in a length Lc .

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where Apzq “ D˚ pzq “ expp i∆βz qpcospγzq ´ 2 i∆βz C21 Bpzq “ iγ expp 2 qsinpγzq qsinpγzq Cpzq “ Ciγ12 expp i∆βz 2

i∆β sinpγzqq 2γ

(2.49)

with a ∆β 2 q ` C 2 , C “ C12 C21 . (2.50) 2 If no field enters the second waveguide, a2 p0q “ 0. Then, the intensities I1 pzq and I2 pzq that depend on a1 pzq and a2 pzq are: γ2 “ p

q2 sin2 pγzqq I1 pzq “ I1 p0qpcos2 pγzq ` p ∆β 2γ 2

I2 pzq “ I1 p0q|Cγ212| sin2 pγzq

(2.51)

With identical waveguides (n1 “ n2 , β1 “ β2 ), the phase mismatch ∆β is always zero, the waves are said to be phase matched and the equations are simplified as: I1 pzq “ I1 p0qcos2 pCzq I2 pzq “ I1 p0qsin2 pCzq

(2.52)

Figure 8: The graphs show the power in both channels of the coupler described by equations 2.51 as a function of the propagation distance z. a) Uncoupled case with a phase mismatch ∆β ‰ 0. With a distance d large enough, there would be no transfer at all. b) Coupled case with ∆β “ 0 where complete transfer is possible. 22

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The two waveguides exchange power periodically as we see in figure (8). However, the situation is very different for ∆β ‰ 0 and ∆β “ 0. For the mismatched case there is no complete power transfer between the waveguide channels, only a small fraction of the power is transferred because of the faint overlap. In the matched case we see how as the power in channel 1 decreases, power in channel 2 increases at the same rate. At a certain distance Lc , the power has been completely transfered to channel 2. If the directional coupler is longer, the process will start to reverse, transferring power back to channel 1. It is also important to note that, depending on the interaction length L in z, the power transfer will be complete or a percentage of the initial power. One can use this principle to construct switches and beam splitters. 2.3.2

Adiabatic tapers

In section (2.2.3) we saw that in the same waveguide, different wavelengths may propagate a different number of modes with different spatial intensity distributions. Single (fundamental) mode operation is desirable for processes that involve the interaction of different fields in integrated devices. Both interacting waves have to overlap in every degree of freedom to have an efficient nonlinear process. A structure to keep two different wavelengths propagating in the fundamental mode in the same waveguide is needed. Adiabatic tapers provides a means of transferring a single mode to a larger or

Figure 9: Example of a taper structure. The grey area is the waveguide region. The taper has a length Lt along which the width of the waveguide increases with an angle θ. The waveguide increases its width from di to df after the taper. An ideal taper would keep the wave propagating in the fundamental mode while increasing its size, without any power leaking into higher modes. 23

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2.4 Nonlinear optics

smaller waveguide [14, 23, 24]. The taper increases the width of a waveguide that is originally single mode for a wavelength with an small angle so that the field doesn’t couple into the higher modes as the waveguide grows in width [25, 26].Figure (9) shows the concept of a tapered waveguide. The propagating field grows through the taper length to the new size keeping its spatial mode.

2.4

Nonlinear optics

Waveguides and other integrated optical structures can be built in linear dielectric materials. These are characterized by a linear relation between the polarization density (dipole moment per unit volume induced by the electric field) P and the electric field E in the material: P “ 0 χE. Where 0 is the electrical permittivity of free space and χ the electric susceptibility of the medium. However, many interesting processes such as Second Harmonic Generation (SHG) or Parametric Down-Conversion (PDC) in optics are nonlinear processes [17]. They need a nonlinear dielectric medium characterized by a nonlinear relation between P and E. The polarization density can be expressed in terms of the individual dipole moment p induced by the electric field and the number density of dipole moments N : P “ pN . The nonlinearity can be in p or in N . The relation between p and E becomes nonlinear when the electrical field is comparable in power to interatomic electric fields [13]. In that situation, the relation between P and p with E is nonlinear. It can also be that the number density N depends on the optical field. Electric fields are typically smaller than the interaction fields withing the crystalline structure, so that the nonlinearity is small too. Therefore, the relation between P and E is approximately linear, deviating from linearity as the electrical field increases. We can then expand P in a Taylor series about E “ 0 P “ 0 pχE ` χp2q E 2 ` χp3q E 3 q,

(2.53)

where χpnq are the coefficients describing the strength of the n-order nonlinearity. As an approximation, this equation is the most basic description of a nonlinear optical medium, enough for the scope of this thesis. A complete description has to describe the interaction between the fields and the electric fields of the crystalline structure. It is useful to split equation (2.53) into linear and nonlinear terms like P “ 0 χE ` PN L , where PN L contains all nonlinear terms. 24

(2.54)

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2.4.1


Second order nonlinearities: three wave mixing

We consider optical mediums in which nonlinearities of order higher than 2 are negligible. The input are two optical fields at different frequencies, ωsignal and ωpump Eptq “ RerEpωsignal qexppiωsignal tq ` Epωpump qexppiωpump tqs

(2.55)

Then the nonlinear component PN L “ χp2q E 2 has the following five terms, 2

2

PN L p0q “ χ{2r Epωsignal q ` Epωpump q s PN L p2ωsignal q “ χ{2Epωsignal qEpωsignal q PN L p2ωpump q “ χ{2Epωpump qEpωpump q PN L pω` q “ χEpωsignal qEpωpump q PN L pω´ q “ χEpωsignal qE ˚ pωpump q

(2.56)

They correspond to the five different frequencies produced in the nonlinear process: 0, 2ωsignal , 2ωpump , ω` “ ωsignal ` ωpump and ω´ “ ωsignal ´ ωpump . Second order nonlinear media can be used to mix two optical waves at different frequencies to obtain a third wave at a different frequency ωout . The second and third equations in (2.56) are the Second Harmonic Generation (SHG) frequencies, the last two equations are sum-frequency generation and difference-frequency generation respectively. The incident waves at frequencies ωsignal and ωpump are able to produce polarization frequencies at all the five possible in (2.56), although not all of them are always generated. Conditions known as phase and frequency matching have to be met [13]. Although a detailed calculation of the conditions is not the focus of this thesis, a short description is needed to understand the purpose of this project. We will understand how different frequency fields interact in the non-linear medium and why their spatial characteristics are important to the efficiency of the process. We focus now only on SFG processes. For the sake of simplicity, we consider two plane waves with wave vectors ~ksignal and ~kpump . Therefore, the complex amplitudes are Asignal exppí~ksignal ¨ ~rq and Apump exppí~kpump ¨ ~rq. The third generated field for a sum-frequency generation would be (from third equation in (2.56)) PN L pωidler q “ χp2q Asignal Apump exppí~kidler ¨ ~rq where: ωidler “ ωsignal ` ωpump

Frequency matching condition (2.57)

~kidler “ ~ksignal ` ~kpump

Phase matching condition 25

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Figure 10: The input fields frequencies are ωsignal and ωpump . The generated frequency is ωidler . These two equations allow the calculation of the correct characteristics for the three interacting field to be temporal and spatial phase matched for their interaction through the process in the nonlinear medium. Only waves yielding the phase matching condition will be generated. An SFG process is depicted in figure (10). The signal and idler input enter the nonlinear medium and overlap spatially for the interaction. They fulfill phase and frequency matching conditions so that at the other end of the crystal a new idler field is obtained with frequency ωidler “ ωsignal `ωpump . The power efficiency of non-linear processes depends on the characteristics of the waveguide, the frequencies of the interacting fields, the power of the fields and the normalized overlap between them. The overlap is defined as 8 ĳ

O9

ωpump px, yqωsignal px, yq dxdy

(2.58)

´8

where pump px, yq and signal px, yq are the spatial intensities of the pump and signal fields respectively. Improving the overlap is a way to increase the efficiency of the SFG process. Tailoring of the spatial modes of the fields in the waveguides can help increase the overlap. 2.4.2

SFG processes in Lithium Niobate waveguides

Lithium Niobate is a ferroelectric material with great electro-optical, acousto-optical and nonlinear characteristics. Very low-loss waveguides are fabricated with the Titanium indiffusion process [14]. The Ti diffused into the bulk material slightly increases the refraction index (∆n « 0.005 [27]) creating the refraction index difference needed to guide the fields.

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From section (2.2) we know that the characteristics of the propagated field depend upon many parameters: the refraction index difference, the cut-off frequency of the waveguide, the waveguide width, the wavelength of the input wave and the coupling with the propagated modes. For this reasons, waveguides are fabricated so that they optimally propagate one wavelength. This means that the waveguides will be single mode for a frequency range so that the normalized frequency of the waveguide (see equation (2.46)) is R ď π{2. As a consequence, longer wavelengths will not be propagated, and smaller wavelengths will be multi-mode. This is an issue if we want to implement SFG in an LiNbO3 integrated device. The process requires the frequency and phase matching conditions and overlap in every degree of freedom. If the device is optimized for the 1550 nm signal wavelength, it will be multimode for the pump field at around 850 nm, leading to a bad overlap and an inefficient SFG process. One way of addressing this issues is tailoring the spatial modes through the coupling properties of the waveguides. To produce SFG in the waveguides, the phase matching condition has to be achieved between the two interacting fields. In Lithium Niobate this can be done through a treatment of the material called periodic poling. Phase matching and periodic poling are not the focus of this thesis, although a short explanation is included for the better understanding of the problem. Periodic poling Perfect phase matching is difficult to achieve and would constrain the choice of the crystal material to find the correct nonlinear coefficient. Instead, a phase mismatch is allowed and compensated by using a medium with position-dependent periodic nonlinearity [13]. In waveguides, this can be achieved through the periodic poling of the material. The periodic poling introduces an opposite phase that completes the phase matching condition. The second equation in (2.57) would turn into ~k1 ` ~k2 ` 2π “ ~k3 Λ where Λ is the period of the poling in the material.

(2.59)

This periodic poling of the material can be achieved with the application of an electric field on specially and periodic masked areas of the sample. The internal polarization of the material interacts with the electric field, locally changing the 27

Jano Gil López


spontaneous polarization of the material by moving the ion structure. Therefore, the sign of the nonlinear coefficient is periodically changed, compensating the accumulated phase difference [28, 29].

Figure 11: Periodically polled waveguide. Λ is the period of the poling. The arrows on the side view indicate the sign change of the nonlinear coefficient.

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3

Methods for optimising two wavelength waveguide coupling

In the current state of the art, integrated devices to perform SFG processes suffer as they cannot be designed to be single mode for all wavelengths involved simultaneously in the processes. This optimized field is usually chosen as the one that will have the photons with the information (i.e. the signal field), to reduce losses in the photon number. When both fields are coupled into the device, there is no more way of controlling the coupling of the pump field into higher order spatial modes than through the alignment of the setup. The SFG process efficiency is enhanced only by alignment. However, there is no way to know which modes are interacting in the SFG process. If both fields were propagating in the same spatial mode, the overlap would improve and therefore enhancing the efficiency of the non-linear process. The aim of this master’s thesis project is to achieve single mode propagation for two different wavelengths in waveguides optimized for a signal field at 1550 nm. We want to achieve single mode propagation for the pump wavelength (« 850 nm) in these signal optimized waveguides. By doing so, the improved spatial overlap between the two wavelengths will enhance SFG processes. The final goal is to design a completely integrated structure in LiNbO3 that controls the spatial modes, couples the two fields together and performs the nonlinear process. The idea is to find a pump field intensity distribution that spatially overlaps only the fundamental mode of the waveguide. In this way, we obtain effective single mode propagation for a wavelength that would normally be multimode. First, we will characterize the modes at different wavelengths to identify the issues. For that, we use a sample with many straight waveguides with different widths. With these results, we will describe the integrated coupling approach to the problem. A completely integrated structure is designed to take care of the coupling modes. With it, we will control the spatial modes of the signal and the pump to enhance the overlap for the SFG process. It will have the advantages of being a small and stable solution, easily adaptable to different experiments. This integrated solution will be fabricated in different materials like Lithium-Niobate or Potassium titanyl phosphate. Another way to address the problem is to tailor the pump spatial intensity before the waveguide coupling. For that, we use a Spatial Light Modulator (SLM) to shape 29

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3.1 Spatial mode characterization

the spatial modes of input light for the waveguide. A genetic algorithm will be used to optimize the pump field single mode in the waveguide. This second approach will give another non integrated solution for the single mode problem but useful information about the propagated modes to apply on the design of the integrated solution. The drawback is that this solution relies on bulk optics before the waveguide coupling, making it a large setup that needs a lot of alignment. Once working, optimization algorithms can be used to find the correct coupling modes.

3.1

Spatial mode characterization

We characterize the spatial modes of the pump at 888 nm in a sample of Titanium indiffused LiNbO3 waveguides with different widths and compare them with the signal modes at 1550 nm. These waveguides are optimized for the signal field at λ « 1550 nm. The optimized waveguides have widths about 6 and 7 µm that make them single mode for the signal wavelength. However, if we calculate the number of modes for 888 nm in a 6 µm waveguide we obtain R “ 2.33. Taking a look at figure (5), we see that ideally two modes are propagated. The sample has straight waveguides with widths from 1 to 7 µm. With a simple setup like in figure (12) to couple the 888 nm light into the waveguide and out couple the output to a camera, we are able to characterize the propagated modes. With a half wave plate we are able to change the polarization from vertical to horizontal to characterize TM and TE modes respectively. Results from the characterization measurements are shown in figure (13) a). TM and TE behave differently, as TM is single mode for bigger waveguides than TE.

Figure 12: Spatial modes characterization setup. The half wave plate is used to change from vertical to horizontal polarization. The laser is coupled into the waveguides with an incoupling lens and the output modes are outcoupled and imaged on a camera with the outcoupling lens. 30

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3.1 Spatial mode characterization

We measure six waveguides of each width to obtain contrastable resutls. From the measurements we can state that TM is single mode up to 3 µm and TE is single mode below 2.5 µm. Figure (13) b) shows the overlap issue between the 888 nm and 1550 nm modes in the waveguides. The 1550 nm mode shown is a simulation of the fundamental mode at the signal wavelength in the 7 µm wide waveguide. The spatial intensity overlap between the 888 nm modes and the fundamental 1550 nm mode is poor and reduces the efficiency of the SFG process. Usually this problem can only be partially addressed by the alignment of the fields for a higher efficiency.

Figure 13: a) Pictures of the output modes at 888 nm for TM and TE polarization and different waveguide widths. The different positions are due to changes in the setup alignment, not the output of the waveguide. b) A comparison of 888 nm and simulated 1550 nm modes in a 7 µm to show the poor spatial overlap. 31

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3.2 Approach I - Integrated solution

For the rest of the chapter we outline two new approaches to couple both fields to the fundamental mode. Working with the fundamental is easier for coupling in and out of the waveguide and working with single mode fibers. We aim for an alignment independent solution that will couple both fields to the fundamental mode in a stable configuration for continuous operation.

3.2

Approach I - Integrated solution

A integrated coupling structure is presented to deal with the overlap problem between a 1550 nm signal and a « 850 nm pump fields in SFG processes. First, the whole structure is described. The device is designed to be produced on Titanium indiffused LiNbO3 . The different structures needed are introduced for their future manufacture. 3.2.1

The integrated two wavelength coupling device

This approach is the most complete. The presented structure will take care of the incoupling modes, the propagation of the fundamental modes for their overlap and the transfer of both fields to the same waveguide for the SFG process. The device will keep both fields in the fundamental mode through all the process, from the coupling to the nonlinear process at the end. The design scheme is shown in figure (14). The fields are incoupled from the left side. The device starts with two incoupling channels that are single mode for each

Figure 14: Integrated SFG device. Li is the incoupling region with widths that are single mode for each wavelength, Lt is the tapered region for the pump, Lc is the directional coupler length and Lp is the poled region. 32

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of the two wavelengths, this part has a length Li . The pump wavelength is around 850 nm and initially, the waveguide has width w850 ă w that is single mode. Then, the pump channel has to be widened to have a width w, the signal optimized waveguides width. It is done by tapering the initial width w850 to the final width w along Lt as described in section (2.3.2). The signal channel remains the same. After the taper, both channels are propagating the fundamental mode in w wide waveguides. The next step is to bring both fields into the same channel. The directional coupler comes into play at this point. Along the distance Lc , the signal field will be completely transfered into the pump channel following the description in section (2.3.1). The distance between the waveguides in the coupler is d. As both waveguides start to approach in the bends, energy starts to be slowly transferred even before the distance is d. This has to be taken into account when fabricating the device, as the coupling distance Lc will be shorter than in the simulation. No energy should remain in the signal channel after the coupler. Since the pump channel has a width w it is single-mode for the signal 1550 nm field. The pump 850 nm field has been adiabatically transferred to the w width waveguide and it will remain principally in the fundamental mode. Therefore, with this structure, the overlap between the two fields has been enhanced. Both fields propagate the fundamental mode in the same waveguide. 3.2.2

Tapered structure for 850 nm light

For the design of the taper, we use established design procedures and device specifications. It will be addressed in future projects. Linear adiabatic tapers have been fabricated within other experiments in the research group and those designs will be used for this work [14]. The design is non-trivial and will need experimenting with the different parameters. 3.2.3

Directional coupler

After the taper for the pump, the two fields that we want for SFG are both single mode propagated in waveguides of the same width. The next step is to overlap them in the same waveguide. To to that, we include a directional coupler that will transfer one of the fields into the others field waveguide, while not transferring the later. In order to know the behavior of the different fields in a directional coupler, we run a simulation in Rsoft, which is a software to simulate waveguide structures and 33

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Figure 15: Power in the input channel for the two wavelengths. dpµmq is the distance between the two channels, Zpµmq is the distance in the propagation direction. The sinusoidal behavior can be seen following horizontal constant d lines. The loss of power in the input channel is transferred to the second channel. field propagation within them. We want only one of the wavelengths to be completely transferred. The simulations are done for 1550 nm and 850 nm separately, in 6 µm waveguides with a distance d between the two coupled waveguides and with a single mode field being propagated. The results are shown in figure (15). The situation is very different for both wavelengths. The signal field is transfered much quicker than the pump. If we choose a coupling distance Lc “ Z “ 2 cm with a distance between the waveguides d “ 5 µm, the 1550 nm will transfer almost 100% transferred while only 2 ´ 4% power from the 800 nm field is transferred. With this coupler after the taper, we have our two single mode fields in the same waveguide, ready to be sent to the nonlinear process. 3.2.4

Periodic poling

The periodic poling has been broadly investigated and experimented within the research group. Results from earlier studies will be used, as in [30]. For our range of signal and pump wavelengths we need a poling period of Λ « 4.28 µm to fulfill quasi-phase matching conditions.

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3.3

3.3 Approach II - Spatial Light Modulator solution

Approach II - Spatial Light Modulator solution

The spatial modes of the pump can be tailored before coupling into the waveguide. The waveguide will only propagate modes that overlap with the mode of the incoupling laser in free-space. Therefore, by tailoring the spatial intensity distribution of the pump in free space, we can couple only the fundamental mode in the waveguide. In this section we describe a non-integrated solution for the single mode coupling. We need a way to shape the pump spatial intensity distribution in free space. With the shaped pump, we need a fitness function to analyse the coupled modes in the waveguide and compare them with the desired single mode behavior. Finally, a feedback loop from the fitness to the shaping device takes advantage from a optimization algorithm to find the fundamental mode coupling intensity distribution for the pump field. This approach will give useful information about mode propagation in nonsingle mode waveguides for a certain wavelength as well as provide a bulk optics solution for the problem. Spatial light modulators are devices that allow the shaping of laser spatial intensity distributions and/or phase at will. First, we will explain which kind of SLM we will be using and how it works. Following a characterization of the SLM, we will explain which techniques allow spatial mode and phase shaping. Thereafter, we will build the shaping setup and send the shaped beams to straight waveguides. Through the use of a genetical algorithm, we will try to find the input pump field that will propagate only the fundamental mode in the waveguide, virtually making it single mode. 3.3.1

The Digital Micromirror Device SLM

Digital Micromirror Devices (DMD) are amplitude-only SLMs. We use the DLPC4500 DMD from Texas Instruments. The screen of the SLM is an array of 1140 ˆ 912 effective mirrors that are individually addressable. Each mirror can be tilted 12 or ´12 degrees. The two tilting angle define two different states for each mirror that change the direction of the reflected light. The device is represented in figure (16). To shape a beam, we build black and white 1140 ˆ 912 pixels pictures in the computer in which the binary colors correspond to the two possible states of a mirror. This way we individually tilt each mirror. ”Black” state mirrors will reflect light into a different direction than ”white” mirrors. We call these pictures masks. The mask is then send to the DMD screen and the beam is directed onto it. If we only look at one of the two reflected directions, we are shaping the input intensity distribution in 35

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Figure 16: The DMD is a screen of 1140 ˆ 912 mirrors. Each px in a) is a mirror. The units in b) are in µm. a) drawing of the DMD screen with the mirrors/pixels dimensions. b)small part of the DMD it zoomed in. We can see the mirrors in the diamond configuration. Columns effectively take two actual columns of mirrors. c) A side view of two mirrors tilted in the two different angles, reflecting light on different directions. Yellow arrows represent the incident light, black arrows are the reflected light. The pitch between the mirrors can be seen. a binary basis. We are multiplying the original intensity distribution by a discrete transfer function of 0 and 1. Through this process the intensity distribution can be shape at will. The pitch between the mirrors and the tilting angles (as in figure (16) b) ) give to the surface of the DMD the tooth configuration of a ruled diffraction grating. The light is then not only reflected by the mirrors but also diffracted into many diffraction orders. We will characterize this behavior and find the most efficient diffraction order. When working with an ultrafast laser, it is also important to compensate the angular dispersion that gratings introduce in ultrafast pulses.

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3.3.2


DMD diffraction characterization

Because of the diffraction grating effect from the tilted mirrors, light reflected from the DMD will be distributed into different diffraction orders following the equation dpsinpθm q ` sinpθi qq “ mλ,

(3.1)

where d is the size of the grating slit (the size of the mirrors for the DMD), θm is the diffracted angle for the diffraction order m, θi is the incidence angle, m is the diffraction order and λ is the wavelength of the incident light. The angles are measured from the normal to the surface of the DMD as in figure (17). The pump shaped by the SLM will interact with quantum light in the SFG process. Losses are a major issue in quantum applications. Therefore, we have to find the most efficient diffraction order to reduce power losses. We find it using the setup in figure (17). We will measure the diffraction efficiency for a completely blank (all pixels white) and for a completely black mask to see if there are also differences between them. From this measurement we will obtain the angles configuration θi and θm for the highest efficiency diffraction order from the SLM. With a fixed L distance, we measure dm for increasing θi angles. The angle θi is also the angle of the reflected light. By simple geometry of the triangle rectangle, we can calculate each θm with θi , L and dm . All angles are measured from the normal of the DMD.

Figure 17: Diffraction efficiency measurement setup. L is the distance from the screen to the DMD, dm is the distance from the incident laser beam to the diffraction order m on the screen and θi is the angle of rotation of the DMD, starting with its normal parallel to the incident beam. The numbers on the left of the screen are different diffraction orders.

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Figure 18: Detailed scheme of the measured distances and angles. The dashed line is the normal of the DMD. Angles have opposite angle to the left and the right of the normal.

φ “ tan´1 p dLm q ϕ “ φ ´ θi

(3.2)

A more detailed scheme of the geometry is illustrated in figure (18). With this measurement we characterize the diffraction behavior of the DMD and the most efficient diffraction angle. We saw different results for the two different states (tilt angles) of the mirrors. Blank and black masks represent the two different states. We can see the results in the graphs in figure (19). We measured only the reflected light and the brightest diffraction order. It can be seen on the graphs that only a small percentage of the incident power remains on the reflected 0th order. We learn that black state mirrors are more efficient on the diffraction and the most efficient ”black” diffraction order gives almost 50% efficiency, while the ”blank” most efficient order gives around 45% efficiency. We can calculate which order is the most efficient with (3.1). For the black mask it is the 4th and for the blank mask is the ´3rd order. To achieve the highest efficiency with the DMD we will use the 4th diffracted order of black masks, with an incident angle of 24˝ and diffracted angle of 48˝ .

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Figure 19: Diffraction efficiency measurements for blank and black masks at different angles. The angle is θ from figure (18). The 0th and brightest diffracted mth orders are measured. Angular dispersion For the diffraction characterization we have used monochromatic light. The final applications of these experiments will work with ultrafast pulses. Ultrafast pulses have a really short time length and a broad spectrum. We have to account for the angular dispersion that the grating introduces to the pulses. This wide spectrum is affected by the diffraction. A wide spectrum ∆λ in (3.1), results in wide ∆θ angles for each order. As the diffracted light propagates, it will grow in width, broadening the spectral properties of the field. We have to compensate this angular dispersion to keep the spectral and time properties of the pulses unchanged. We can achieve spectral dispersion compensation following a design from [31]. With a grating, two lenses and a mirror we compensate the spectral dispersion in a setup like in figure (20). The grating, the second lens and the mirror will introduce a spectral dispersion on the pulses with the opposite sign from the introduced by the SLM. The first lens in combination with the second one is used as a telescope to enlarge the width of the incident beam up to the size of the DMD screen (6 ˆ 8 mm). By covering the whole screen we will have the maximum resolution possible for the shaping, using every pixel. The beam FWHM increases from 2.5 mm to a final FWHM of 8 mm to cover the entire screen. 39

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Figure 20: DMD dispersion compensation setup. The diffracted light is sent to an autocorrelator to measure the temporal length ∆τ of the pulses. To calculate the parameters of the setup we start from the diffraction equation (3.1). The approach is to compare the angular dispersion introduced by the grating with spacing dG with the angular dispersion of the DMD with spacing dD . Differentiating (3.1) for the grating and the DMD, we can calculate the ratio of angular dispersion introduced by the grating and the DMD mG dD cospθiD q dθiG {dλ “ dθmD {dλ mD dG cospθm Gq

(3.3)

where m is the mth order of diffraction for the grating (mG) and for the DMD (mD), the θsi are incident angles and θsm are diffracted angles of the mth order . To remove angular dispersion, the ratio has to be equal to the angular magnification of the second lens L2 so that it is compensated through the lens magnification and the dispersion from the grating MA “ ´

sL2 ´D sG´L2

(3.4)

where sL2 ´D and sG´L2 are the distances between L2 and the DMD and between the grating and L2 . The mirror M is introduced to change the sign of the angular dispersion. The last equation we need is the Guassian lens formula 1{fL2 “ 1{sG´L2 ` 40

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1{sL2 ´D to correctly image the beam on the DMD. We relate equations (3.3) and (3.4) in a system with the distances: s

2 ´D ´ sLG´L “ 2

mG dD cospθiD q mD dG cospθm Gq

(3.5)

1{fL2 “ 1{sG´L2 ` 1{sL2 ´D The parameters for the setup are: fL1 “ 0.100 m the focal length of the first lens, fL2 “ 0.175 m the focal length of the second lens, dG “ 1200 lines{mm the spacing of the grating, θmG “ 23 degrees the blaze angle of the grating, θiD “ 48 degrees the incidence angle on the DMD. Solving the system we obtain: sG´L2 “ 0.282 m and sL2 ´D “ 0.687 m. The first lens is situated on the focus of the second lens to image the laser on the DMD. If the dispersion compensation setup works properly, the spectral characteristics of the pulses must remain the same. One can do so by measuring the pulse duration before and after the DMD. It should not change. To measure the pulse duration we use an autocorrelator. Before the DMD, the pulse time duration at FWHM is ∆τ “ 222 f s. After the compensation setup and the DMD, the pulse duration was ∆τ “ 290 f s. The expected pulse time duration without the compensation setup is ∆τ « 320 f s [31]. We have mitigated the angular dispersion from the DMD but not completely. This may be caused by the use of non-achromatic lenses with ultrashort pulses. Each frequency component of the pulse has a different focusing length and is not correctly imaged on the DMD screen, causing the temporal chirp and stretching.

3.3.3

Spatial intensity shaping

Once the device is characterized, it can be used for our shaping purposes. Amplitude shaping is very straight forward with a DMD. We build binary masks for the screen of the DMD. This binary masks are black and white pictures (like the ones in figure (22)) corresponding to the two tilting states of the mirrors in the DMD.We will use the reflected light from black state mirrors as we know they are the most efficient ones. To characterize the shaped beams, we image the reflected light from the DMD on a camera with a setup as in figure (21). Playing with the distances from the DMD to L1 and from L2 to the camera, we are able to collimate the shaped beam and image it on the camera.

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Figure 21: DMD spatial amplitude shaping imaging setup. The focal lengths are fL1 “ fL2 “ 150 mm so that the image is ideally at 150 mm from L2 . To create the masks we directly draw a black and white picture with the desired shape for the intensity distribution on the black pixels. This will be a completely binary image with no gray-scale effects. If we want to simulate gray-scale pictures we can apply the error diffusion algorithm to the mask . This algorithm recreates the gray-scale effect in a binary picture [32] by reducing the difference between pixels in the gray-scale picture and the binary picture. A example of each can can be see in figure (22). Because of the size ratio of the screen, the masks seem compressed on the horizontal direction, but they are correctly proportional on the device. Error diffusion algorithm We want a normalized gray scale picture (where 0 is plain white and 1 plain black) to be completely binary. Every pixel is indexed with pm, nq with m row and n column. Then, the gray scale picture is defined as opm, nq. The algorithm scans opm, nq and evaluates its value. If opm, nq ą 0.5. it sets the value of a new pixel map bpm, nq to 1, otherwise to 0. After that, the error of this change is calculated as epm, nq “ bpm, nqópm, nq. This error is transferred to the immediate neighborhood of the pixel with weighting factor cpm, nq. The inmediate neighborhood are the pixels in positions: pm, n ` 1q, pm ` 1, n ` 1q, pm ` 1, nq and pm ` 1, n ´ 1q. The weighing

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factors are taken from [32]: cp0, 1q “ ´7{16 cp1, 1q “ ´1{16 cp1, 0q “ ´5{16 cp1, ´1q “ ´3{16

(3.6)

Figure 22: On the left, a simple binary mask of an ellipse that maps a circle on the SLM. On the right, a binary mask of a Dali’s painting after the error diffusion algorithm. The pixels after the error is transferred are o1 pm ` a, n ` bq “ opm ` a, n ` bq ` cpa, bq ˆ epm, nq.

(3.7)

This generates the new picture o1 pm, nq with a highly ordered structure and minimized error epm, nq. 3.3.4

Spatial phase shaping - Holography

The DMD is an intensity only SLM. However, there are ways to shape the phase from the spatial components, known as holograms. A hologram is the recording on a physical medium of the interference pattern between two waves [13]. The first wave is a reference wave Ur and the second one is the target wave Ut that has the phase ϕ properties that we want. The transmittance tpx, yq of the hologram is obtained from tpx, yq “ |Ur ` Ut |2 . Computer generated holograms are built with algorithms in a computer and then translated into a physical medium. Combined with a versatile device like the DMD, 43

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Figure 23: The inset shows the setup to observe the Fourier transform of the hologram, where L1 and L2 are lenses. The main illustration shows the result of the Fourier transform, where u and v are the Fourier components of the spatial coordinates px, yq. we achieve fast and completely tailored holograms for any spatial phase distribution. We will follow Burch’s method as described in [33]. If we take the light from the hologram tpx, tq through the setup in the inset in figure (23), we will observe the Fourier transform of the hologram in the Fourier space. The Fourier transform will have different components that we represent in figure (23) as rectangles with the space that they take in the Fourier space. The axes pu, vq are the Fourier transform of the original space px, yq. The rectangles at u “ ˘α are the Fourier transforms of the complex target function Ut px, yq “ Apx, yqeiϕpx,yq and its complex conjugate. Those rectangles contain the phase information of the target function. U and V are the spatial bandwidths of the target wave function along the two coordinates. The rectangle in the center represents only spatial frequencies from the square term A2 px, yq in the transfer function. In the Burch method, the transmittance function of the hologram is tpx, yq “ 0.5p1 ` Apx, yqcosr2παx ´ ϕpx, yqsq.

(3.8)

This allows the carrier frequency α to be as low as U {2 and the sampling intervals to discretize tpx, yq are dx “ 1{2U and dy “ 1{V . With this method, we obtain phase only holograms that look like figure (24). 44

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Figure 24: Three holograms are shown. From the left to the right: the first one has a flat phase (no phase), the second one has a linear phase and the third one a quadratic phase. The jumps in the holograms masks are because of the 2π wrapping of the phase.

The last step is to measure the spatial phase of the shaped beams. We use Fourier-transform interferometry for the measurements. Fourier-transform interferometry The phase ϕ in (3.8) introduced to the beam with the hologram is a global phase. To be able to measure it we need to have another relative phase to compare with. An interferometry measurement gives the relative phase information between two spatially interfering beams. The Fourier-transform method is a fast method to extract the phase information from the interference fringe-pattern, with higher accuracy and sensitivity than other methods as it erases the low-frequency noise from the interference measurement [34]. We use an interferometry setup as shown in figure (25). The spatially dependent interference on the beam splitter can be written as gpx, yq “ apx, yq ` cpx, yqexpp2πif0 xq ` c˚ px, yqexpp´2πif0 xq

(3.9)

cpx, yq “ 0.5bpx, yqexppiϕpx, yqq.

(3.10)

with

The phase information is in the cpx, yq term. We will extract it through the use of the Fourier transform. As the phase information is only in one direction, we perform

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the Fourier transform in the interference direction. The Fourier transform on the x coordinate of (3.9) is Gpf, yq “ Apf, yq ` Cpf ´ f0 , yq ` C ˚ pf ` f0 , yq.

(3.11)

Capital letters denote Fourier transforms of the lowercase functions and f is the spatial frequency in the x direction. Gpf, yq presents three lobes in the spatial

Figure 25: Interferometry setup. After the DMD, the shaped beam goes through an iris in the Fourier plane (FP) to let only pass the u “ α component with the phase information. The shaped beam and a reference coming directly from the laser are interfered on a beam splitter (BS) and the interference pattern is sent to a camera for its analysis.

Figure 26: Shape of the Gpf, yq function. The three different peaks of the function can be seen. One of the side peaks is selected, erasing the rest, to extract the phase information. 46

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frequency domain. The center peak is the DC components Apf, yq and the side lobes are Cpf, yq and its complex conjugate, as shown in figure (26). Both side peaks contain the phase information. If we select one of them and Fourier transform it back to spatial domain, we obtain cpx, yq “ 0.5bpx, yq ` iϕpx, yq,

(3.12)

where we finally can access the phase by simply extracting the angle of the imaginary function cpx, yq. 3.3.5

Finding the single mode - Genetic algorithm optimization

Now that we have a means to shape the spatial intensity distribution of the pump, we look to optimise its single mode propagation in the bigger waveguides. The goal is to find a 850 nm intensity distribution that only couples to the fundamental mode in the waveguide, not exciting any higher modes. As we don’t know the distribution that we seek, an optimization algorithm can be used. One promising method is a genetic algorithm. This algorithm draws inspiration from natural selection and genetics to search for optimization problems solving. They work very well for the optimization of problems with big parameter space [35]. We can say that our parameter space is every mirror on the DMD, as every combination of mirrors leads to a completely different intensity distribution. Genetic (or evolutionary) algorithms have been used in many different fields of science and physics in particular [36]. The reason is that genetic algorithms are able to deal with dynamic changing parameters within the problem and they work in big parameter spaces where other algorithms quickly stuck in local minima solutions or don’t converge at all [36]. Genetic algorithms have been use for spectral shaping [37, 38] and to optimize the design of integrated structures [39, 40, 41]. However, this is the first time it is used to tailor spatial modes in waveguide propagation. To implement the genetic algorithm we need the method to shape the spatial intensity, a fitness function to compare the waveguide modes with the desired coupled mode and a feedback loop from this fitness function to the shaping setup, to optimize the shaping. We characterized the spatial shaping in section 3.3.3. In this section, we will describe the fitness function and the feedback loop to complete the genetic algorithm.

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The fitness function In section (3.1), on figure (13), we saw several different modes propagated at 888 nm. The goal is to find the correct input field such that power is coupled into only one waveguide mode. As the spatial phase of each propagated mode is different, they interfere an we cannot know the actual intensity distribution of each mode from the pictures. We have to find a random intensity distribution to couple the fundamental mode. A fitness function is used to quantize how close the distribution is to the desired coupled mode. The fitness function is an important part of a genetic algorithm as it leads the evolution of the problem towards convergence. The characteristics depend on the problem and there are many options [36]. The main operation has to calculate a difference between a target and each attempt to obtain it from the experiment. In our problem, the target is the fundamental mode of the waveguide and the attempts are the coupled modes from different intensity distribution. To analyze the coupled modes we take pictures with a camera. The fitness function has to compare the pictures with the target. There are two main approaches: comparing directly the spatial modes or comparing the Fourier transform. To compare the picture with a picture of the target there are several ways. The easiest is to calculate the overlap integral between them to maximize it. Compare the number of maxima is another way, looking for only one maximum. In our case, the target is the fundamental mode for 850 nm. We can use a simulation of this mode as a picture of the target. A target picture and one coupled mode can be seen in figure (27) in a)

Figure 27: a) is a simulation of the fundamental mode used as a target picture. b) is a picture of a coupled mode. A fitness function calculates the overlap integral between a) and b) to quantize how far is b) from being like a). c) is the absolute value of the Fourier transform of b). The fitness value is simply the radius Rk in the Fourier space k. Minimizing Rk is another way of optimizing single modeness. 48

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and b). The coupled mode and the target can also be compared in the Fourier space. The Fourier components of the coupled mode have to equal the ones of the fundamental target mode. A single mode only has one component in the Fourier space, it is ideally a single point. Higher modes have more Fourier components that enlarge its size in the Fourier space as it can be seen in figure (27) b). In this case, the fitness value is the radius Rk of the mode in the Fourier space and we want to minimize it. The Fourier transform can be performed optically with the use of a lens or numerically through the Fourier transform of the coupled modes pictures. The selection of the final fitness function will be decided upon results in simulations and experimentation. Feedback loop - The genetic algorithm With the genetic algorithm, we will find a mask for the SLM to shape the pump spatial intensity. The resulting intensity distribution from this mask will be propagate only on the fundamental mode in the waveguide. In order to find it, we connect the genetical algorithm in a feedback loop between a camera looking at the coupled modes of the waveguide and the SLM. The algorithm will keep evolving until the fitness function gives a maximized result. The loop in which the algorithm works is described in figure (28).

Figure 28: The flow chart of the genetic algorithm. Starting from the top left and following the arrows. The working flow of the genetic algorithm is simple though powerful. The algorithm works with a population of individuals. Each individual in our system is a random intensity mask for the SLM. We begin with a population of randomly generated individuals. The population will evolve to the next generation by the operations 49

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of the algorithm. Each individual is sent to the SLM and the shaped light coupled into the waveguide. The output of the waveguide is recorded with a camera and analyzed by the fitness function. The individuals are then sorted by their fitness value from the closest to the target to the worst individuals. In the next stage, a percentage of the sorted individuals are selected in the retain operation. The objective is to keep the best individuals from the original population. We can choose to keep a bigger or smaller part of the original population. In this step, individuals with worst fitness values can be randomly included with a low probability called random select coefficient. Mutation operations can also be performed also with low probability on the retained individuals. Mutations are random changes on characteristics of the individuals (for example, turning off a pixel that was on). With the random select and the mutation operations we introduce diversity to the

Figure 29: Crossover operation between two pairs of parents in a simulation of the genetic algorithm. The target is in c). Parents are cut in half and each parent contributes with one half to the new individual as seen in a) and b). The new individual 2 has a better fitness with the target and will be maintained over the new individual 1. 50

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selected individuals, to avoid falling into local minima solutions. After the population has been reduced to a number of better individuals, these individuals ”breed” to generate new individuals up to the size of a new population. The breeding is performed through crossover operations. Crossover operations are different ways to combine the characteristics of individuals to form new. Again, the choice of the crossover operation relies on trial and error on the real experiment. There are different options: one-point , multipoint and uniform among others. [35]. Figure (29) contains examples of a crossover operation. With the new population the algorithm starts again in a feedback loop between the SLM and the camera fitting the outputs with the target. This continues until the fitness of the population with the target is within a threshold, triggering the end of the algorithm and returning the best individual generated. The setup used to run the genetic algorithm is shown in figure (30). Parameters as the size of the populations, the coefficient of randomly added individuals and the coefficient of mutated individuals affect the converging time of the problem and the quality of the solution reached.

Figure 30: The feedback loop setup to run the genetic algorithm. Masks generated by the genetic algorithm are send to the DMD. The shaped light is coupled into the waveguide and the output modes are outcoupled and imaged on a camera. The output modes are analysed with the fitness function in the computer.

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4

Results on the SLM coupling optimization

Results of the amplitude and spatial phase shaping setups are shown. Then, the amplitude shaping results are used to run and obtain the first results from the genetic algorithm. The results are all on the SLM approach. The design and fabrication of the integrated SFG device is out of the scope of this master’s thesis and will be addressed in future projects. Nonetheless, results from the SLM experiments will be useful for the characterization of the integrated device.

Figure 31: The intensity distribution of a painting by Salvador Dali imaged on the camera. The picture has been processed with error diffusion for the gray scale effect.

4.1

Spatial amplitude shaping

With the setup described in section (3.3.3), we are able to shape the incident laser with any intensity distribution and with high fidelity to the desired distribution. Some examples can be seen below in figure (31) and figure (32). The shaped intensity distributions correspond exactly with black pixels in the mask on the SLM. We can group the pixels in blocks to changer their state in blocks, changing the resolution as in figure (32) a) and b). In figure (31) we can see the intensity distribution shaped after a mask with fake error diffusion gray scale. The image is very detailed and the mask is able to shape the beam in the same way than the binary masks. This results are very good for the genetic algorithm. It is possible to have well defined and propagating shaped beams with any intensity distribution. This is the base on which the main experiment relies. 52

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4.2 Spatial phase shaping

Figure 32: Four examples of masks and the measured shaped intensity distribution. These are examples of what is used in the experiment and the genetic algorithm later. a) and b) both represent the same gaussian distribution, but b) with a higher resolution. c) and d) are two randomly generated masks, individuals of a population.

4.2

Spatial phase shaping

We now move onto the results from the spatial phase shaping experiments. We generate several holograms with different spatial phases. One is a flat mask with no phase that we will use to compare with the rest of the measured phases in the interferometry setup. The results can be seen in figure (33). We measured a flat hologram with no phase shaping and use it as a reference to measure a linear and a quadratic phase holograms. Spatial phase shaping is achieved. The different phases are related to the linear or quadratic nature of the hologram fringes. This represents another proof of the versatility of the DMD, although it is an amplitude only SLM, we are able to shape phase components through holography. We will not use this in the genetic algorithm at first but phase shaping will show useful in the future, to for example compensate the effects of slopes on the facets of waveguides or chirps from bulk optics components.

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4.3 Tailoring waveguide mode coupling

Figure 33: The intensities shown are from the blue line cut on the interference fringe pictures. The phase results are from the Fourier analysis of the intensity.

4.3

Tailoring waveguide mode coupling

Before running the genetic algorithm experiment, we have to make sure that the premise is correct. We want to check that different intensity distributions propagate different modes inside the waveguides. This can be done by generating some random masks for the DMD, feeding the shaped beams into the waveguide and looking at the output modes. The results are shown in figure (34). Every masks has produced a different output mode in the waveguide. We are actually overlapping different modes with the shaped input field. The energy is distributed differently between the modes. The genetic algorithm will work on this result.

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4.4 Genetic algorithm

Figure 34: Four different masks and the output mode produced from the waveguide.

4.4

Genetic algorithm

Due to its versatility, genetic algorithms need to be made for each particular problem. There are many options and parameters to control, and they affect the algorithm in different ways depending on every aspect of the experiment. Therefore, we will first perform some simulations to understand the behavior of the algorithm. After that, we will have more data to go to the experiment and successfully apply the method. 4.4.1

Computer simulations

Time efficiency is very important in a genetic algorithm, as it works with a big space of parameters. It is also critical when working with hardware in a feedback loop, as it introduces delay in each step. We first want to decide our parameters and see how each characteristic of the algorithm affects the computing time: the size of the population and the retain, random select and mutate coefficients. 55

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We divide the masks into 320 smaller 20ˆ16 pixels submatrices. Each sumbatrix consists entirely of 0s or 1s (black or white). This defines our space of parameters. An individual of the population is then defined as the 320 submatrices randomly choosen to be fully 0 or 1. Masks like the ones in figure (32) are generated. For the simulation we start with a population of random masks like c) in figure (32) and the target is a). The breeding is done as in the example in figure (29) and we use the difference between the masks and the target as the fitness function. The genetic algorithm will run with different characteristics and we measure the converging time and count how many generations are needed. Figure (35) shows the results when changing individual parameters, leaving the rest constant. We only change the parameter on each graph, leaving the rest constant. When not changed, this are the values used: 1000 individuals population, 0.3 retain coefficient, 0.1 random select coefficient and 0.1 mutation coefficient. Graphs on the left show the dependence of the generations until convergence on each parameter. Graphs on the right show the converging time over each parameter. Increasing the population decreases the number generations needed and leaves the computing time mostly unchanged around 2 and 1 minutes. The retain coefficient has two minimums both in generations and time, and after 0.7 the algorithm never converges. This is expected as when a big part of the original population is kept for breeding, the diversity will be poor and on the opposite, when we retain too many from the original population, we keep bad individuals too and the fitness is not improved. Random select coefficient and the mutation coefficient both show oscillations in the performance and therefore finding the correct value for each will depend on the experiment mostly. These results are very interesting. Now, the range of parameters for the implementation of the genetic algorithm are known. To have a good compromise between generations and time needed, the algorithm will likely run faster with a population between 1000 and 1500 individuals and a retain coefficient about 0.2 to 0.5. The information on this section is the foundation for the enhancement of SFG processes through the used of an SLM. Knowing how each parameter affects the algorithm is key to adapt it to the experiment. However, it is important to remark that the simulated times are not the expectation values for the experimental genetic algorithm. The time needed for the hardware in the feedback loop for the SLM to change the 56

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masks, the camera to capture the pictures and the computer to analyze the images, will increase the converging time. The advantage over manual alignment coupling, is that the process will be self-driven by the algorithm and provides an stable configuration for the whole setup.

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Figure 35: Results of the timing simulation.

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5

Conclusions and outlook

We have successfully achieved the goals of this master’s thesis project, clearing the path for future experiments that will be based on the results of this work. We presented a promising design for a fully integrated SFG device that takes care of the multimode field with a tapered structure. The spatial modes of the signal and pump fields have been characterized identifying the expected spatial modes behavior. The SLM is completely characterized for spatial intensity and phase shaping with great results. We proved the tailoring of coupled modes in the waveguide through shaping of the input field. Finally, the genetic algorithm has been investigated in depth and adapted to our waveguide coupling problem, with a working setup and feedback loop. The parameters and characteristics of the algorithm are known and several options for the fitting of the individuals and target are open. It is ready to be implemented in experiments. The close future looks bright from this results. Concerning the integrated approach, upon fabrication of the device, the taper must be taken care of with the technology available as it will be one of the most demanding parts of the design. The polarization of the signal and pump fields have to be taken into account for the design of the directional coupler. Finally the polling will relay on well known results from other projects within the research group. As for the SLM approach, the genetic algorithm has to be demonstrated together with the intensity shaping to find single mode pump field coupling in the waveguide.The presented phase shaping method offers advanced solutions to compensate spatial phase chirping in the experiment. The angular dispersion compensation setup must be improved with achromatic lenses. The genetic algorithm will be then applied on the SFG process to enhance the non-linear process efficiency.

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6

References

[1] C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” Theoretical Computer Science, vol. 560, pp. 7–11, dec 2014. [2] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einsteinpodolsky-rosen channels,” Physical Review Letters, vol. 70, pp. 1895–1899, mar 1993. [3] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger Nature, vol. 390, pp. 575–579, dec 1997. [4] N. Gisin and R. Thew, “Quantum communication,” Nature Photonics, vol. 1, pp. 165–171, mar 2007. [5] J. L. O’Brien, A. Furusawa, and J. Vuˇcković, “Photonic quantum technologies,” Nature Photonics, vol. 3, pp. 687–695, dec 2009. [6] P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Physical Review Letters, vol. 75, pp. 4337–4341, dec 1995. [7] B. Brecht, D. V. Reddy, C. Silberhorn, and M. Raymer, “Photon temporal modes: A complete framework for quantum information science,” Physical Review X, vol. 5, oct 2015. [8] P. C. Humphreys, W. S. Kolthammer, J. Nunn, M. Barbieri, A. Datta, and I. A. Walmsley, “Continuous-variable quantum computing in optical time-frequency modes using quantum memories,” Physical Review Letters, vol. 113, sep 2014. [9] M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Quantum memory with millisecond coherence in circuit QED,” Physical Review B, vol. 94, jul 2016. ˜ 1 tz, K.-H. Luo, H. Suche, and C. Silberhorn, “Towards a quantum in[10] H. RA 4 terface between telecommunication and UV wavelengths: design and classical performance,” Applied Physics B, vol. 122, jan 2016.

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[11] A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Optics Express, vol. 19, p. 13770, jul 2011. [12] R. Paschotta, Encyclopedia of Laser Physics and Technology. Berlin: WilleyVCH, 2008. [13] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics. John Wiley & Sons, Inc., aug 1991. [14] W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Buechter, S. Reza, W. Grundkoetter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, “Integrated optical devices in lithium niobate,” Optics and Photonics News, vol. 19, p. 24, jan 2008. [15] H. Kogelnik, “An introduction to integrated optics,” IEEE Transactions on Microwave Theory and Techniques, vol. 23, pp. 2–16, jan 1975. [16] S. J. Orfanidis, Electromagnetic Waves and Antennas. Rutgers University, 2016. [17] A. Yariv, Quantum Electronics. Wiley, 3rd ed., January 1989. [18] S. E. Miller, “Integrated optics: An introduction,” Bell System Technical Journal, vol. 48, pp. 2059–2069, sep 1969. [19] K. Oda, T. Okumura, J. Kasai, S. Kako, S. Iwamoto, and Y. Arakawa, “Crystallinity improvements of ge waveguides fabricated by epitaxial lateral overgrowth,” Japanese Journal of Applied Physics, vol. 55, p. 04EH06, mar 2016. [20] K. Abe and T. Ishigure, “Fabrication for low loss graded-index polymer crossed optical waveguide using the soft-lithography method,” oct 2016. [21] J. D. Bierlein, A. Ferretti, L. H. Brixner, and W. Y. Hsu, “Fabrication and characterization of optical waveguides in KTiOPO4,” Applied Physics Letters, vol. 50, pp. 1216–1218, may 1987. [22] S. Saha, S. S. Yohanes, D. Jun, A. Danner, and M. Tsang, “Fabrication and characterization of optical devices on lithium niobate on insulator chips,” Procedia Engineering, vol. 140, pp. 183–186, 2016. [23] D. Marcuse, Theory of Dielectric Optical Waveguides. Quantum Electronics, Academic Press, INC., 1st ed., January 1974. 61

Jano Gil López

[24] J. E. Hoffman, S. Ravets, J. A. Grover, P. Solano, P. R. Kordell, J. D. WongCampos, L. A. Orozco, and S. L. Rolston, “Ultrahigh transmission optical nanofibers,” AIP Advances, vol. 4, p. 067124, jun 2014. [25] J. Shafii, J. D. Cobb, and R. J. Vernon, “Optimal design of overmoded waveguide tapers using the pontryagin minimum principle with application to te(0n) circular waveguide taper designs,” 20th International conference on Infrared and Millimiter Waves, December 1995. [26] J. Love, W. Henry, W. Stewart, R. Black, S. Lacroix, and F. Gonthier, “Tapered single-mode fibres and devices. part 1: Adiabaticity criteria,” IEE Proceedings J Optoelectronics, vol. 138, no. 5, p. 343, 1991. [27] M. Lawrence, “Lithium niobate integrated optics,” Reports on Progress in Physics, vol. 56, pp. 363–429, mar 1993. [28] M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,” Applied Physics Letters, vol. 62, pp. 435–436, feb 1993. [29] S. Kraprick, Cascaded Wavelength Conversion Processes in Lithium Niobate Waveguide Structures. PhD thesis, University of Paderborn, 2015. [30] B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper—engineered frequency conversion in nonlinear optical waveguides,” New Journal of Physics, vol. 13, p. 065029, jun 2011. [31] J. Cheng, C. Gu, D. Zhang, and S.-C. Chen, “High-speed femtosecond laser beam shaping based on binary holography using a digital micromirror device,” Optics Letters, vol. 40, p. 4875, oct 2015. [32] R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proceedings of the Society of Information Display, vol. 17, no. 2, pp. 75–77, 1976. [33] W.-H. Lee, III Computer-Generated Holograms: Techniques and Applications, pp. 119–232. Elsevier BV, 1978.

62

Jano Gil López

[34] M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringepattern analysis for computer-based topography and interferometry,” Journal of the Optical Society of America, vol. 72, p. 156, jan 1982. [35] K. Man, K. Tang, and S. Kwong, “Genetic algorithms: concepts and applications [in engineering design],” IEEE Transactions on Industrial Electronics, vol. 43, no. 5, pp. 519–534, 1996. [36] R. K. Ursem, Models for Evolutionary Algorithms and Their Applications in System Identification and Control Optimization. PhD thesis, University of Aarhus, 2003. ˜ 1 hle, [37] S. Chatterjee, W. Wohlleben, C. Lange, W. Stolz, M. Motzkus, and W. RA 4 “Optimizing the performance of a vertical-cavity surface-emitting laser,” Applied Physics Letters, vol. 89, p. 151122, oct 2006. [38] D. Zeidler, S. Frey, K.-L. Kompa, and M. Motzkus, “Evolutionary algorithms and their application to optimal control studies,” Physical Review A, vol. 64, jul 2001. [39] L. Sanchis, A. Hakansson, D. Lopez-Zanon, J. Bravo-Abad, and J. SanchezDehesa, “Integrated optical devices design by genetic algorithm,” Applied Physics Letters, 2004. [40] S. Preble, M. Lipson, and H. Lipson, “Two-dimensional photonic crystals designed by evolutionary algorithms,” Applied Physics Letters, vol. 86, p. 061111, feb 2005. [41] Q. Shi, T. G. Nguyen, A. Mitchell, and A. Song, “Waveguide optimization via evolutionary algorithms,” dec 2011.

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