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Efficient broadband frequency generation in composite crystals

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Journal of Optics J. Opt. 16 (2014) 062001 (6pp)

doi:10.1088/2040-8978/16/6/062001

Fast Track Communication

Efficient broadband frequency generation in composite crystals Genko T Genov1,2, Andon A Rangelov1,2 and Nikolay V Vitanov1 1

Department of Physics, Sofia University, James Bourchier 5 Blvd., 1164 Sofia, Bulgaria

E-mail: [email protected]fia.bg Received 6 March 2014, revised 31 March 2014 Accepted for publication 10 April 2014 Published 30 May 2014 Abstract

By using ideas from the technique of composite pulses in quantum physics we propose a highly efficient and broadband technique for sum and difference frequency generation with composite nonlinear crystals. The proposed technique works both with continuous-wave and pulsed lasers, as well as in the linear and nonlinear regimes of depleted and undepleted pumps, respectively. The feasibility of the technique is supported by numerical simulations for a composite Potassium Titanium Oxide Phosphate crystal. S Online supplementary data available from stacks.iop.org/JOPT/16/062001/mmedia Keywords: broadband frequency generation, composite pulses, quasi-phase-matched crystal PACS numbers: 42.65.Ky, 42.79.Nv, 82.56.Jn, 42.70.Qs (Some figures may appear in colour only in the online journal) 1. Introduction

has also been proposed [18]. Adiabatic frequency conversion has been reviewed recently [19]. Adiabatic approaches have the advantages of being robust and broadband. However, the adiabatic condition requires very high input intensity and/or very long nonlinear crystals [18, 19]. Usually, this is hard, or even impossible, to achieve due to the damage threshold of the crystal and hence the maximum conversion efficiency is far less then 100%. In the present paper, we propose to use the idea of another very efficient, flexible and robust technique from quantum physics—composite pulses—for SFG/DFG. Composite pulses are widely used in nuclear magnetic resonance [20, 21], quantum optics and quantum information [22–25]. An analogue of them—composite wave plates—is used successfully in polarization optics [26–30]. Here we demonstrate that highly efficient and broadband frequency generation can be achieved by using composite nonlinear crystals constructed in a similar fashion as composite pulses and composite wave plates. This technique is a viable alternative to the adiabatic approaches [19] because it requires much lower input intensity and shorter nonlinear crystals while delivering high efficiency and robustness to parameter variations.

Three-wave mixing is a widely used technique for frequency conversion in nonlinear optics with numerous applications across physics [1–3]. Phase matching of the waves involved in three-wave mixing is usually assumed as a crucial condition for the efficiency of this technique. Recent work has shown, however, that strict phase matching is not always necessary, e.g. in adiabatic second harmonic generation [4, 5], in linearly chirped quasi-phase-matched (QPM) gratings [6–8], and in cascaded processes [9, 10]. To this end, Suchowski et al [11–14] have recently used an aperiodically polled QPM crystal to achieve both high efficiency and large bandwidth in sum frequency generation (SFG) and difference frequency generation (DFG). They have applied the concept of rapid adiabatic passage in a two-state quantum system [15–17], which is mathematically identical to SFG/DFG in the undepleted pump approximation, in which the input pump laser field is little changed after the process. An extension of this technique to the depleted pump regime 2

G T Genov and A A Rangelov have contributed equally to this work.

2040-8978/14/062001+06$33.00

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© 2014 IOP Publishing Ltd Printed in the UK

J. Opt. 16 (2014) 062001

Another important advantage of composite crystals is that they preserve the main features from the linear case (the undepleted pump approximation) when applied in the nonlinear regime (depleted pump).

state atom [15–17]. The two amplitudes A2 and A3 correspond to the probability amplitudes of the ground and excited states, Ω to the Rabi frequency, and Δk to the laser-atom detuning [15, 16]. Now we choose the local modulation period Λ ( z ) as

2. Background

2π . Δ + δ ( z)

Λ ( z) =

Three wave mixing is described in the slowly-varying amplitude approximation [1–3, 19] by the symmetrized coupled-wave equations i∂zB1 + iv1−1∂tB1 + ∂ωv1−1∂t, tB1 = Ω͠ B2*B3e−iϕ ,

(1a)

i∂zB2 + iv2−1∂tB2 + ∂ωv2−1∂t, tB2 = Ω͠ B1*B3e−iϕ ,

(1b)

i∂zB3 + iv3−1∂tB3 + ∂ωv3−1∂t, tB3 = Ω͠ B1B2 eiϕ ,

(1c)

Then the effective phase mismatch becomes Δk ( z) = Δk 0 − Δ − δ ( z).

particular 0 3

i∂zB2 = Ω͠ B1*B3e

(2b)

,

i∂zB3 = Ω͠ B1B2e . iϕ

(λ

0 1

, λ 20 , λ30

)

with

0 2

(7)

zf

p = sin2 ( S 2), with S = ∫ Ω dz . Hence, complete energy z i

transfer between the signal and the idler occurs for S = π or odd multiples of π [1–3, 19]. The frequency generation efficiency is sensitive to both variations in S and the phase mismatch. This sensitivity is greatly reduced by replacing the bulk crystal by a composite crystal—a stack of crystals with different local modulation periods.

Usually equations (2) are simplified by assuming that the incoming pump wave B1 ( z ) is much stronger than the signal and the idler, and therefore, its amplitude is kept constant during the evolution (undepleted pump approximation). Then we make the substitutions A2 ( z ) = B2 ( z ) eiϕ ( z) 2 and

3. Composite crystals We consider a QPM crystal with N different segments. In each segment j we control the length of the modulation period Lj and δ ( z ) = δj . In the undepleted-pump regime the problem is linear and the total evolution matrix is a product of N evolution matrices (acting from right to left),

A3 ( z ) = B3 ( z ) e−iϕ ( z) 2 and obtain (3)

T with A ( z ) = ⎡⎣ A2 ( z ), A3 ( z ) ⎤⎦ , Ω ( z ) = 2Ω͠ ( z ) B1 ( z ) and

⎡ −Δk ( z) Ω⋆ ( z)⎤ ⎥. H ( z) = 12 ⎢ ⎣ Ω ( z) Δk ( z) ⎦

wavelengths

If initially A3 ( zi ) = 0 , then p = b 2 = 1 − a 2 is the efficiency of frequency mixing. In the phase matching case ( Δk = 0), which corresponds to resonant excitation in quantum physics, the frequency mixing efficiency is

(2c)

i∂zA ( z) = H ( z) A ( z),

0 1

of

⎡ a b⎤ U = ⎢ ⋆ ⋆⎥. ⎣ −b a ⎦

( j = 1, 2, 3), and equations (1) read (2a)

set

1 λ = 1 λ ± 1 λ for SFG/DFG; then we shall have perfect phase matching for this set of wavelengths. We show below that by using δ ( z ) as a free control parameter, we can achieve very efficient frequency conversion in a certain range around this set of wavelengths, without the necessity of phase matching over this range. The evolution of the two waves can be described by the propagator U , which connects the field amplitudes at the initial and final coordinates, zi and z f , A ( z f ) = UA ( zi ). Since U is unitary it is parametrized with two complex Cayley–Klein parameters a and b ( a 2 + b 2 = 1) as

is the time. Here vj are the group velocities, ∂ωvj−1 are the group velocity dispersions (GVD), ωj are the frequencies, kj are the associated wave numbers, and Bj are the respective amplitudes, where j = 1, 2, 3 refer to the pump, the signal and the idler. Finally, c is the speed of light in vacuum and χ ( 2) is the second-order susceptibility of the crystal. First, we consider the continuous wave regime; then the time derivatives of the electric fields vanish, ∂tBj = ∂t, tBj = 0

−iϕ

(6)

We choose Δ to be equal to the phase mismatch Δk 0 for a

where ϕ ( z ) = ∫ Δk ( z ) dz is the total accumulated phase and Δk ( z ) = Δk 0 − ΔkΛ ( z ) is the effective phase mismatch [19]. The latter is the difference between the dispersion phase mismatch Δk 0 = k1 + k 2 − k 3 and an artificial phase mismatch ΔkΛ ( z ) = 2π Λ ( z ), where Λ ( z ) is the local modulation period for the QPM gratings and may vary over the propagation distance z [19]. Furthermore, 2) 5 ( ͠ Ω = − χ ω1ω2ω3 c k1k 2k 3 is the coupling coefficient and t

i∂zB1 = Ω͠ B2*B3e−iϕ ,

(5)

U ( N ) = U ( L N , δN )…U ( L 2 , δ2 ) U ( L1, δ1).

(4)

(8)

Our objective is to construct a composite crystal, which completely converts the signal into the idler and is robust to variations in the dispersion phase mismatch Δk 0 and the coupling Ω around a selected value Ω0 of this coupling. Such a composite crystal will tolerate dispersion phase mismatch in

Ω ( z ) can be made real by attaching its phase to the amplitude B1 ( z ). By mapping coordinate onto time, z → t , equation (3) becomes the time-dependent Schrödinger equation for a two2

J. Opt. 16 (2014) 062001

Figure 1. Dependences of γ ( z ) and φ ( z ) on z for the U5 composite

sequence, i.e. N = 9, n = 5, with phases given in table 1, Ω0 = π ⎡⎣ nLo + ( n − 1) Le ⎤⎦ N , and Le = Lo 5.

Table 1. Phases φ2 m − 1 (m = 1, 2, … , n ) of universal composite crystals wherein the number of segments n is incorporated in the respective name Un.

CP

Phases ( φ1, φ3, …, φ2n − 1)

U3

( 0, 1, 0) π

U5

( 0, 5, 2, 5, 0) π

U7 U9

2 6

( 0, −1, 10, 5, 10, −1, 0) π 12 ( 0, −0.37, −0.65, −0.45, 0.30, −0.45, −0.65, −0.37, 0) π

Figure 2. Frequency generation efficiency p( N ) versus deviations in Ω

from its desired value Ω0 and the phase mismatch Δk 0 − Δ for different composite crystals with the phases given in table 1 for Le = Lo 5 .

U11

0, −0.43, 0.09, −0.62, 0.55, 0.11, 0.55, −0.62, 0.09, −0.43, 0) π U13 ( 0, −15, −6, −13, 8, 13, 2, 13, 8, −13, −6, −15, 0) π 24

a certain range of wavelengths, i.e., it will be broadband. Mathematically, our goal is to maximize the frequency generation efficiency P (N ) = U (21 ) N

2

for as wide as possible var-

iations of Ω and Δk 0 . We construct sequences that correspond to the recently introduced universal composite pulse sequences for two-state systems in quantum optics [31] by using segments of two different thicknesses, Lo and Le . A unique feature of these sequences is the robustness to unknown deviations in any experimental parameter. Here we use δj and Lj of each segment j as control parameters instead of the constant phase shift in the driving field in [31]. We choose a sequence with an odd number of segments N = 2n − 1 and thicknesses ⎧ ⎪ Lo Lj = ⎨ ⎪ ⎩ Le

where

Lo

and

( j = 1, 3, …, 2n − 1), ( j = 2, 4, …, 2n − 2), Le

⎡⎣ nL + ( n − 1) L ⎤⎦ = Nπ , o e

are i.e.

chosen maximum

Figure 3. Numerical simulation of SFG efficiency versus the pump

intensity and the signal wavelength. Left frames: undepleted pump, B1 2 = 100 B2 2 ; right frames: depleted pump, B1 2 = 4 B2 2 . Top frames: standard periodically poled structure designed to achieve perfect phase matching at 1550 nm; middle frames: composite crystal, U13 sequence with local modulation periods given in table 2; bottom frames: same as middle frames but with a random error with amplitude 5% in the thickness of each segment compared to the values in table 2.

(9)

to

S = Ω0

frequency 3

J. Opt. 16 (2014) 062001

2

Figure 4. Intensity of the pump B1 (red line), the signal B2

2

(green line) and the idler B3 2 (blue line, multiplied by a factor of 3 for the sake

of visibility) as a function of the propagation distance z for 1064 nm pump with maximum input intensity of B1 2 = 10 GW cm −2 and 1550 nm signal with maximum input intensity of B2 2 = 10 GW cm −2 . Different frames represent different time instants. The input pump and signal have Gaussian temporal shapes with FWHM of 5 ps and 500 fs, respectively. The broadband nature of this conversion technique can be further seen in the animation of this figure in the supplementary material (see supplementary online material).

generation efficiency is achieved in the phase matching case. We explain below how Le and Lo are chosen. Next, we define γ ( z ) = δ ( z ) Le and we choose γj ( z ) = const in each segment j of the crystal. We also choose

Table 2. Local modulation periods Λ for even-numbered segments in the U13 composite sequence. Odd-numbered modulation periods are Λ = 15.26 μm . The thicknesses of odd- and even-numbered segments are Lo = 645 μm and Le = 129μm , i.e. Le = Lo 5.

γj = 0 for all odd j = 1, 3, … , 2n − 1. This choice ensures

Segments

2

4

6

8

10

12

15.84

14.93

15.53

14.51

15.08

15.68

14

16

18

20

22

24

14.86

15.45

16.09

15.00

15.60

14.72

z

that φ ( z ) ≡ ∫ δ ( z′) dz′ = φ2 m − 1 (m = 1, 2, … , n ) is con0

Λ ( μm )

stant during the odd-numbered segments (see figure 1). As a result, the composite propagator takes the form U ( N ) = Uo ( 0) Ue ( γ2n) Uo ( 0)…Uo ( 0) Ue ( γ2 ) Uo ( 0).

Segments Λ ( μm )

(10)

We choose γ2m and Le so that ⎡ −iγ 2 m 2 0 ⎤, Ue ( γ2 m ) = ⎢ e iγ 2 m 2 ⎥ ⎦ ⎣ 0 e

γ2 = 5π 6, γ4 = − π 2 , γ6 = π 2, γ8 = −5π 6. An example of γ ( z ) and φ ( z ) for the U5 sequence is given in figure 1. Because any physical implementation demands Le > 0 , the jumps of φ are not instantaneous and therefore, the fulfillment of equation (11) is only approximate. Figure 2 demonstrates the performance of the composite

(11)

and so that the phases φ2 m − 1 correspond to the phases of the universal composite pulses in [31]. The values of φ2 m − 1 are given in table 1, while the corresponding values of γ2 m are calculated by the formula γ2 m = φ2 m + 1 − φ2 m − 1.

crystals of table 1. The high-efficiency region ( p( N ) > 0.9) of a single segment is greatly expanded by composite crystals, even with just a few segments. Clearly, the efficiency increases as the number of segments in the composite crystal increases.

(12)

For example, for the U5 sequence of table 1 we have 4

J. Opt. 16 (2014) 062001

number of segments, for which the bandwidth of the highefficiency region increases by a factor of f, scales as f 2 . Therefore, in order to double the bandwidth we have to increase the number of segments by a factor of 4. In order to test the operation of the composite crystals for ultrashort pulses, we have solved the full nonlinear equation (1) for a short crystal (1 mm) and the U5 composite sequence with local modulation periods given in table 3 . The pump pulse at 1064 nm is assumed to be 5 ps long (full-width-athalf-maximum, FWHM), while the signal is 500 fs. The results in figure 4 show new effects: the walk off between the pulses, which is due to the different group velocities of the pump, the signal and the idler, and the spreading of the pulses due to GVD. These effects set an upper limit on the total length of the crystal, which in turn makes it hard for adiabatic techniques to be used; composite crystals have a great advantage in this case because they require much shorter crystals. Our simulations show very similar SFG results for a signal in the range of 1500 nm–1600 nm, which indicate the broadband nature of this technique (see supplementary online material at stacks.iop.org/JOPT/16/062001/mmedia).

Table 3. Local modulation periods Λ for even-numbered segments in the U5 composite sequence. Odd modulation periods are Λ = 15.26 μm . The thicknesses of odd- and even-numbered segments are Lo = Le = 111 μm .

Segments Λ ( μm )

2

4

6

8

14.43

15.80

14.75

16.18

A very important feature of the composite crystals is that they work equally well in the depleted and undepleted pump regimes. Moreover, this technique can be used in the femtosecond domain, which is an important advantage over adiabatic techniques. We show below numerical simulations of these features by using the parameters of a real nonlinear crystal—Potassium Titanium Oxide Phosphate (KTP)— which demonstrate the feasibility of composite frequency conversion for SFG.

4. KTP simulations Potassium Titanium Oxide Phosphate (KTP) has a broad transparency range, high optical damage threshold, large nonlinear optical constants and it is suitable for QPM [32, 33]. We have examined the most commonly used e–ee interaction for KTP crystal (pump, signal and idler polarized along the extraordinary axis), and we have calculated our crystal to work with the typical pump wavelength 1064 nm of the Nd:YAG laser [34]. We have assumed a 10 mm long KTP crystal, with U13 composite sequence from table 1 and local poling periods given in table 2. Finally, we have used the Sellmeier coefficients from [33]. The contour plot in figure 3 compares the SFG efficiency for a standard periodically poled KTP crystal designed to achieve exact phase matching at 1550 nm (top frames) and a composite KTP crystal (bottom frames) versus the input pump intensity and the signal wavelength. The left frames are for the linear undepleted-pump regime and the right frames for the nonlinear depleted-pump regime. The figure shows the greatly enhanced efficiency and robustness of SFG by the composite crystal compared to the ordinary QPM KTP crystal, both in the linear and nonlinear regimes. The frequency bandwidth is increased from just a few nm to over 20 nm. Furthermore, in order to test the sensitivity of the composite crystals to the prescribed segment lengths, we have conducted the numerical simulations in figure 3 by artificially adding a random error in the segment lengths listed in table 2. As the comparison of the middle (without error) and bottom frames (with a random error that varies up to 5%) this error, which is far worse than what can be achieved experimentally, does not change the conversion efficiency dramatically. The bandwidth of about 20 nm observed in figure 3 can be increased by using longer composite sequences. We can estimate the required number of segments by using the analytical formula for the transition probability for the broadband composite pulses in [24] and [25]. We thereby find that the

5. Conclusions In this paper, we have used ideas from the technique of composite pulses in quantum physics to design a highly efficient technique for broadband frequency generation. In contrast to adiabatic approaches, this composite technique does not require high input intensity or very long crystal length and therefore, it can be a strong alternative for efficient and broadband SFG/DFG. This composite technique is particularly interesting for frequency conversion of ultrashort laser pulses because in this case very short nonlinear crystals are required in order to avoid the effects of GVD and group velocity mismatch. While the standard short QPM crystals can handle the problem of GVD, they operate at low conversion efficiencies and low bandwidth. The adiabatically chirped QPM crystals are a good solution for efficient frequency conversion of picosecond pulses [13], but not for femtosecond pulses. For the latter, a set of many different poling periods should be manufactured within a very small crystal length, which is a nearly impossible task. The composite crystals can be an efficient device for efficient frequency conversion in the femtosecond regime because they provide high conversion efficiency and bandwidth. It is interesting to compare the composite crystals introduced here with the broadband second harmonic generation technique demonstrated experimentally by Yu et al [35]. This has been achieved by matching of the group velocities of the second harmonic and the fundamental, while the phase velocities are QPM. However, the approach of Yu et al can be used only around a specific spectral region that is determined by the unique dispersion of the crystal (by proper choice of MgO-doping concentration in the lithium niobate). The composite crystals approach presented here does not have this limitation, i.e. it is not limited to a specific wavelength range 5

J. Opt. 16 (2014) 062001

and a nonlinear crystal; it also allows for higher-order compensation of deviations in the wavelength, pump intensity, etc. Therefore, the present technique can be considered more flexible and universal.

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Acknowledgments This work is supported by the Bulgarian NSF grant DMU-03/ 103 and the Alexander-von-Humboldt Foundation.

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