Competition of Faraday rotation and birefringence in ... - OSA Publishing

Competition of Faraday rotation and birefringence in femtosecond laser direct written waveguides in magneto-optical glass Qiang Liu, S. Gross, P. Dekker, M. J. Withford, and M. J. Steel∗ Centre for Ultrahigh bandwidth Devices for Optical Systems (CUDOS) and MQ Photonics Research Centre, Department of Physics and Astronomy, Macquarie University, NSW 2109, Australia ∗ [email protected]

Abstract: We consider the process of Faraday rotation in femtosecond laser direct-write waveguides. The birefringence commonly associated with such waveguides may be expected to impact the observable Faraday rotation. Here, we theoretically calculate and experimentally verify the competition between Faraday rotation and birefringence in two waveguides created by laser writing in a commercial magneto-optic glass. The magnetic field applied to induce Faraday rotation is nonuniform, and as a result, we find that the two effects can be clearly separated and used to accurately determine even weak birefringence. The birefringence in the waveguides was determined to be on the scale of Δn = 10−6 to 10−5 . The reduction in Faraday rotation caused by birefringence of order Δn = 10−6 was moderate and we obtained approximately 9◦ rotation in an 11 mm waveguide. In contrast, for birefringence of order 10−5 , a significant reduction in the polarization azimuth change was found and only 6◦ rotation was observed. © 2014 Optical Society of America OCIS codes: (130.2755) Glass waveguides; (230.2240) Faraday effect; (230.3120) Integrated optics devices; (230.3240) Isolators; (230.3810) Magneto-optic systems.

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#222329 - $15.00 USD Received 4 Sep 2014; revised 25 Oct 2014; accepted 28 Oct 2014; published 4 Nov 2014 (C) 2014 OSA 17 November 2014 | Vol. 22, No. 23 | DOI:10.1364/OE.22.028037 | OPTICS EXPRESS 28037

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1.

Introduction

Isolators are indispensable devices in optical communication systems, since they prevent unwanted back reflections, which may cause signal degradation, instability or even damage to optical systems. A fundamental requirement for an isolator is to include an element of nonreciprocity [1]. In integrated optics, several elegant approaches to non-reciprocity based on direct temporal modulation have recently been proposed including silicon modulators [2, 3] and stimulated Brillouin scattering [4, 5]. Nevertheless, the most popular approach is still to use magneto-optic effects, either interferometers incorporating a non-reciprocal phase shift [6, 7], or commonly, Faraday rotation [8–13]. Among the Faraday effect schemes, the most common materials used are magneto-optical garnets and their doped variants. Although integrated techniques for optical isolators have shown great development in the past two decades, no optical isolator yet has been constructed for the integrated glass photonics platform. This might be possible by hybrid integration with alternative platforms such as magnetic garnet isolators or fiber-based devices which would have the advantage of exploiting the relatively mature technology and low propagation losses of these platforms. However, the insertion losses caused by mode- and index-mismatch between devices constructed in glasses and crystals, as well as the bulkiness of hybrid fiber solutions are significant issues that need to be considered. In particular, an increasingly important fabrication approach for glass-based integrated optics is the femtosecond laser direct-write (FLDW) technique. Various devices have been demonstrated for this platform [14, 15], but not yet optical isolators. The drawbacks of hybrid approaches to incorporating optical isolation are especially apparent for this platform: the modal and index mismatches between garnet and FLDW waveguides are quite severe and significant insertion losses would be expected; fiber-coupled devices could show much better insertion loss but a key advantage of the FLDW platform is compact integration, and this is defeated by such a hybrid scheme. More importantly, the family of FLDW devices now includes lasers, gratings, couplers and polarization manipulation elements [14, 15], and multiple elements can be incorporated in the same chip to produce fairly complex circuits. It will ultimately be necessary to include isolation within the FLDW circuit and this calls for the design of isolators themselves made with FLDW. Previously, Shih et al. [16] created single-mode waveguides at 1550 nm in a magneto-optical glass using a femtosecond oscillator. A 6◦ phase shift caused by the Faraday effect was observed in a 50 mm long sample under a 0.2 T magnetic field. A slight decrease in the glass Verdet constant was also reported. However, there has been no assessment of the variation in Faraday rotation caused by distinct waveguide properties such as birefringence and polarization-dependent loss. Some degree of birefringence is common in FLDW waveguides and has multiple causes including asymmetric waveguide profiles [17–22], stress due to local density changes around the waveguide [23–26], or nanograting formation [27–29], effects which are not yet completely understood. Polarization-dependent loss (PDL) is also likely to be associated with these phenomena, but again a detailed understanding of expected levels of PDL and how to reduce them is still to emerge. Whatever its origin, birefringence disrupts Faraday rotation by making the polarization eigenstates elliptical rather than circular. The competition between Faraday rotation and birefringence has been studied for a number of systems including single-crystal orthoferrites many years ago [30], as well as integrated garnet waveguides [13, 31–33] and microstructured fibers [34] more recently. However, there are few reports on this issue for integrated glass-based devices, let alone for FLDW waveguides. The competition between these effects is important in integrated glass-based devices since the Faraday effect in glasses is relatively weak. Though #222329 - $15.00 USD Received 4 Sep 2014; revised 25 Oct 2014; accepted 28 Oct 2014; published 4 Nov 2014 (C) 2014 OSA 17 November 2014 | Vol. 22, No. 23 | DOI:10.1364/OE.22.028037 | OPTICS EXPRESS 28039

there are several different methods to measure birefringence [35], its level in a FLDW waveguide may be too weak to measure directly and accurately in a device of a few centimeters length, yet still strong enough to disrupt the Faraday rotation. Finally, for a monochromatic measurement, the waveguide birefringence Δn can be unambiguously determined from polarization evolution results only if 2π (1) Δn L < 2π . λ Otherwise, the number of orbits of the polarization state on the Poincaré sphere caused by the birefringence is unknown, and the value of birefringence can not be determined. In this paper, we aim to expand the possibility of optical isolator incorporation for the FLDW glass platform by exploring the interplay between Faraday rotation and birefringence in these waveguides. The waveguide fabrication, post-treatment and the resultant waveguide properties were described in a previous report [36]. Here, the Faraday effect in the fabricated waveguide is demonstrated. The competition between Faraday rotation and waveguide birefringence is theoretically calculated and experimentally verified. Profiting from the nonuniformity of the magnetic field used to induce the Faraday rotation in our setup, the two contributions to polarization rotation can be separated and the small birefringence is determined quite accurately, breaking the limit in Eq. (1). 2. 2.1.

Background theory Governing equations

To calculate the competition between Faraday rotation and birefringence, we start with the time-harmonic wave equation for the electric field E in the form ∇2 E + k2 ε¯ E = 0,

(2)

where k = 2π /λ is the wavenumber (λ is the free space wavelength), and ε¯ is the dielectric tensor. For a lossless, magnetic gyrotropic material, the dielectric tensor has the form ⎤ ⎡ εxx iα (z) 0 0 ⎦, (3) ε¯ = ⎣−iα (z) εyy εzz 0 0 where εxx , εyy , εzz are real, and α (z) is the gyration parameter that represents the Faraday rotation in the glasses. (We address the issue of waveguide losses later). The gyration parameter is proportional to the magnetic flux density of the applied field:

α (z) =

V B(z)nλ , π

(4)

where V is the Verdet constant of the material. Note that for correspondence with our experiment, we have allowed the magnetic field and thus α to be z-dependent. We can extend this picture to treat waveguides by considering a transverse variation of ε (x, y) with a region of raised refractive index. Since the index changes in FLDW waveguides are typically on the scale of 10−3 , which is much smaller than the background glass index n, the variation in ε = n2 is small. As the magneto-optical response is also weak, it is appropriate to first solve the wave equation Eq. (2) for the modes with no magnetic effect and then add the Faraday rotation as a perturbation. We choose the direction of propagation along z, and since we are focusing on glass systems, now take the waveguide dielectric function ε (x, y) to be scalar but with a non-circular profile, so


that it will induce form birefringence. (Stress-induced birefringence also plays a role in FLDW waveguides, but the additional notational burden does not alter our conclusions and we do not include it explicitly). We insert into the wave equation the usual ansatz E=

1 Af(x, y)ei(β z−ω t) + c.c. , 2

where c.c. is the complex conjugate, and find the eigenvalue equation 2 −β + ∇t2 + k2 ε (x, y) f = 0.

(5)

(6)

We can solve this to find a pair of orthogonal polarization states with profiles fx (x, y) and fy (x, y), along with the propagation constants βx , βy , and corresponding effective indices n¯ x,y = βx,y /(2π /λ ). We now expand the whole field in terms of these basis modes:

(7) E = 12 Ax (z)fx eiβx z + Ay (z)fy eiβy z e−iω t + c.c. , and look for the evolution of the tuple of slowly varying envelopes

A (z) . A(z) = x Ay (z)

(8)

By inserting Eq. (7) back into Eq. (2), it is easy to show that this field vector obeys the equation

2 ∂ 2¯ (9) + k ε eff A(z) = 0, ∂ z2 where we have introduced the effective dielectric tensor

n¯ 2x iα (z) , ε¯eff = n¯ 2y −iα (z)

(10)

which contains both modal birefringence and gyrotropy. (Stress birefringence would shift the values n¯ x and n¯ y somewhat but make no formal difference. Equation (9) is our main governing relation. 2.2.

Combined action of Faraday rotation and birefringence

Since the dielectric tensor is Hermitian and positive definite, a square root tensor η¯ can be introduced ε¯eff = η¯ η¯ , (11) which is itself Hermitian. Note that ε¯eff and η¯ are functions of z. Then Eq. (9) has the form ∂ ∂ + ikη¯ − ikη¯ E = 0. (12) ∂z ∂z Either term in this equation can be set to zero to calculate the solutions for backward or forward propagation. The latter term gives the equation

∂ E = ikη¯ E. ∂z

(13)


We now introduce the Stokes vector S = (S1 , S2 , S3 ), where S1 = |Ex |2 − |Ey |2 S2 = Ex Ey∗ + Ex∗ Ey S3 = i(Ex Ey∗ − Ex∗ Ey ) and define Δη = (ηxx − ηyy )/2, and a vector

(14)

⎡

⎤ Δη r ⎦ Ω = 2k ⎣ηxy , i ηxy

(15)

r + iη i . Then Eq. (13) may be written in the simple vector form where ηxy = ηxy xy

d S = Ω × S, dz

(16)

ˆ at a rate of |Ω Ω|. which has the natural interpretation of rotation of S around the unit vector Ω Finally, we also set ε± = (εxx ± εyy )/2 and defining the ratio γ = ε− /α , the vector Ω becomes ⎡ ⎤ 2 2 k ε + + α γ + 1 − ε+ − α γ + 1 γ ⎣0⎦ , Ω= γ2 + 1 1 ˆ ) of rotation represented by Eq. (16) tilts on the indicating that the axis (i.e. the unit vector Ω Poincaré sphere depending on the ratio between waveguide birefringence and the Faraday rotation as illustrated in Fig. 1. We remark that since we are allowing the magnetic field and Faraday parameter α to vary with position, Eq. (16) can not generally be solved in closed form. Figure 1 shows the polarization state of light propagating in a waveguide with Faraday effect and birefringence, for linearly polarized launched light oriented along yˆ into the waveguide. Each path shows the evolution of the polarization with position z for a given γ . If the waveguide is symmetric and has no birefringence (γ = 0), the propagation maintains linear polarization in the waveguide but with a rotation due to the Faraday effect. Once birefringence is introduced, the modes and states of light become elliptically polarized and are confined to the upper or lower hemisphere. The Faraday rotation is also reduced (never reaching the far side). As the birefringence, and therefore γ , is increased, the linear rotation of the light becomes increasingly diminished and the Faraday rotation in the waveguide may not be detectable. These properties of Faraday rotation are familiar, and indeed plots very similar to Fig. 1 are to be found in works by Wolfe et al. from the 1980s on integrated garnet waveguides [31, 32]. Observe however, that Fig. 1 depicts the change of polarization state in a waveguide under a uniform magnetic field, which is inconsistent with our experimental situation. As we see in the following sections, due to the nonuniformity of the actual magnetic field in our experiment, the polarization evolution becomes considerably more complex, but with the result that the effects of Faraday rotation and birefringence can be separated, allowing us to extract an unambiguous value for Δn, free of the restriction observed in Eq. (1). 2.3.

Practical considerations

It is helpful to identify the the birefringence corresponding to the curves in Fig. 1. Using ε− = εxx − εyy = n2xx − n2yy ≈ 2n · Δn, γ = ε− /α and Eq. (4) we find Δn =

V Bλ γ . 2π

(17)


Fig. 1. Polarization evolution in waveguides with Faraday rotation and birefringence on the Poincaré sphere for various values of γ . All curves have an initial polarization state of | ↑ ≡ |y ˆ ≡ (0, 1).

Substituting the Verdet constant of the glass TG20 (see section 3), V = −74 rad/T·m, and the strongest field intensity of the ring magnets used, B ≈ 0.2 T, the estimated waveguide birefringence corresponding to γ = 0.5 (orange curve), at which point the Faraday rotation has been significantly distorted, is Δn ≈ 1.5 × 10−6 . Finally, let us consider the effect of losses for a moment. A polarization-independent loss may be included by adding a common imaginary term in to the diagonal elements in Eq. (10), with the effect that the overall intensity decays with z, but the normalized Stokes vector s = S/|S| obeys the same dynamics of Eq. (16). Polarization-dependent loss (linear dichroism), represented by differing imaginary terms inx and iny , is more important since the differential attenuation of the linear polarization states can create the illusion of polarization rotation. We return to this issue after describing the experiment. 3. 3.1.

Experimental configuration Waveguide fabrication and properties

The properties of the magneto-optical glasses used in this work research and the characteristics of fabricated waveguides under different femtosecond laser writing conditions have been systematically studied in [36]. Thermal annealing was applied to the fabricated waveguides in order to reduce their propagation loss and to minimize any possible reduction on the glass Verdet constant. After waveguide fabrication, characterization and post-annealing, we selected two waveguides which were fabricated in a magneto-optical glass (known as TG20 [37]), using a regeneratively amplified Ti:Sapphire femtosecond laser system (λ =800 nm, 1 kHz repetition rate, 120 fs pulse width, circular-polarized) laser system for examination of Faraday effect and birefringence. These waveguides were created with 300 nJ pulse energy, written in two passes of the writing beam (referred to as WG-1) and 325 nJ pulse energy, four passes (referred to as WG-2) respectively. The waveguides had approximately Gaussian index profiles and Gaussian mode profiles with measured mode field diameters of 6.0μ m and 5.0μ m respectively. The peak index contrast was estimated at 1.3 × 10−2 , and the waveguides had relatively low losses


(1.8 dB/cm for WG-1 and 2.3 dB/cm for WG-2). 3.2.

Characterization of polarization evolution

The experimental setup for polarization measurement was built on a 6-axis positioning stage and is shown in Fig. 2. After pigtailing with a polarization-maintaining (PM) fiber held in a ferrule, the glass sample (length Lwg =10.65 mm) was placed in a set of ring magnets (OD 14×ID 7 × Lm 15 mm, with the length chosen based on calculations for obtaining the largest rotation). Linearly polarized laser light (λ = 635 nm) was launched into the waveguide via the PM fiber. The output light was collected and collimated by an aspheric lens, and then directed to a polarimeter, which could measure the light polarization state, including azimuth, ellipticity and Stokes parameters. An iris was placed between the waveguide and the lens as an alignment aid and the extra power disturbance caused by interference around the waveguide mode could be minimized by closing this iris. Once the system was aligned, all the components were fixed for experimental accuracy except for the magnets, which could be translated with respect to the beam path in order to observe the rotation behavior as the magnetic field configuration seen by the waveguide was varied.

Fig. 2. Schematic diagram of experimental setup for polarization examination. The orange arrow in the figure indicates the translation direction of the magnets. GL is a Glan-Laser polarizer and AL the aspheric lens.

The waveguide was first aligned in the middle of the ring magnets, a point we label zm = 0. Then the magnets were moved away from the glass sample in steps of 1 mm until reaching the furthest spot (labeled zm = 26 mm), which was limited by the stage travel range. The output polarization state from the waveguide was recorded at each step. For reference, we replaced the waveguide with an unmodified magneto-optical glass sample (Lglass =15.08 mm) and repeated the same measurement procedures. For these observations, the fiber-coupled laser source was replaced by a linear-polarized HeNe laser (λ =632.8 nm). The small difference in wavelength of the two sources means we can neglect any wavelength variation in Verdet constant or birefringence. The polarization of light propagated through the waveguide with a reversed magnetic field was also measured by flipping the magnets and repeating the procedures. Realignment of the system was required only after changing the sample waveguides or the magnetic field direction. The effect of the waveguide being aligned away from the axis of the cylindrical ring magnets was checked and found to be insignificant. The deformation of the stage or components caused by movement of the magnets and the corresponding effect on measurement results was also tested and found to be negligible. To allow quantitative analysis of the experimental results, we next theoretically calculate the B field induced by the magnets, and then determine the expected output polarization states according to the analysis described in Section 2.


4.

Magnetic field of the ring magnets

The ring magnets shown in Fig. 2 in this research were magnetized axially through their body, thus the direction of the field inside the magnets is parallel to the ring axis. The magnetic field for a ring magnet, whose internal radius is a (3.5 mm in our case) and outer radius is b (7 mm), can be treated as the difference of the fields from a solid cylindrical magnet of radius b and a smaller cylindrical magnet of radius a. A routine application of the Biot-Savart Law provides the following expression for the axial component of the magnetic field along the ring axis: ⎡ ⎤ Lm Lm Lm Lm − z + z − z + z μ0 M ⎣ ⎦, 2 + 2 − 2 − 2 Bring (z) = 2 Lm 2 Lm 2 Lm 2 2 2 2 b + (z − ) b + (z + ) a + (z − ) a2 + (z + Lm )2 2

2

2

2

(18) where z is the position along the cylindrical axis, Lm is the length of the ring magnet, μ0 is the permeability of free space and M is the magnetization of magnets. Based on the measured data for Faraday rotation in bulk glass, the inferred magnetization M was (8.7 ± 0.3) × 105 A/m. The implied axial B field of the ring magnets used in the experiment, which had an overall length of 15 mm, is plotted in Fig. 3 based on Eq. (18). The gray box gives a reference for the length and position of the magnets with respect to the magnetic field. The green lines in Fig. 3 illustrate the relative positions between the waveguide and magnetic field for each measurement step. The orange line indicates the configuration where the entire waveguide has passed the sign reversal point of the magnetic field, which is significant below.

Fig. 3. Magnetic field and the relative positions of the waveguide in the experiment. The orange line indicates the position when the entire waveguide passed the sign reversal point of the magnetic field. The gray box gives a reference for the length and position of the magnets. Note that in practice, the magnets were moved and the waveguide remained fixed.

5.

Simulation for the expected experimental results

We constructed code to find the expected output polarization states according to the experimental procedure and the theory established above. Given the experimental situation, the parameter z in Eq. (13) could not be treated as a variable since the length of the waveguide was fixed. However, the square root tensor η¯ (z) changes depending on the waveguide birefringence Δn, the relative position between the waveguide and the magnets zm , and the longitudinal variation in the magnetic field expressed by Eq. (18). Since the magnetic field is non-uniform, the solution of Eq. (13) has no analytic form and is integrated numerically. We label the output field


E(zm ; Δn, φ , δ ) which is associated with a given configuration of the magnetic field B(z; zm ) and launch polarization Jones vector E0 (φ , δ ) = [cos φ , sin φ eiδ ]. To understand the expected behavior, an initial set of calculations was performed for a range of plausible waveguide birefringence values: Δn = 0, 1 × 10−7 , 1 × 10−6 , 1 × 10−5 , 5 × 10−5 and 1 × 10−4 . The calculations give the output state evolution on the Poincaré sphere, which corresponds to the state that should be recorded by the polarimeter. It is important to emphasize that each point on the Poincaré sphere trajectories in Figs. 5 and 6 below represents the output state for a different magnet configuration, and the dotted paths do not refer to a single propagation through the waveguide. To clarify this, Fig. 4 illustrates the calculation process. Each solid curve in Fig. 4 shows the polarization evolution along the waveguide (i.e. z = 0 to Lwg ) for a given B(z, zm ) configuration. The crosses at the end of these curves denotes the output polarization. The initial launch state is linear vertical polarization, marked with the red star. The dotted curve joining the output states (shown as crosses) is thus the trajectory of output state evolution as the magnetic field is translated, which is the quantity which can actually be observed. For clarity of display, in Figs. 5 and 6, only these trajectories of output polarization state are shown.

Fig. 4. Illustrative calculations of polarization evolution within a waveguide (color curves) and output states (crosses) for discrete steps in the magnetic field position. Left: zoom-in view. Right: overview of the Poincaré sphere. The magnetic field and Verdet constant are those for the experiments and the waveguide birefringence is set at Δn = 2.0 × 10−5 . The initial launch state of linear vertical polarization is marked with the red star.

5.1.

Analysis of theoretical results

Figure 5 displays a calculation of the expected output polarization as a function of the relative position between the waveguide and the magnetic field for different degrees of birefringence. The other calculation parameters, such as the magnetization and dimension of the magnets, as well as the length of the waveguide, are set to be the same as those in the actual experiment. The launch polarization state is set as vertically-oriented linear, marked as the red star in the figure. The start points, which correspond to the position zm = 0 in Fig. 3, are marked as squares. The distinct turning points visible in some of these tracks (marked with triangles and correspond to the orange line in Fig. 3) are due to the nonuniform axial profile of the magnetic field provided by the ring magnets. These turning points provide useful characteristics for determination of #222329 - $15.00 USD Received 4 Sep 2014; revised 25 Oct 2014; accepted 28 Oct 2014; published 4 Nov 2014 (C) 2014 OSA 17 November 2014 | Vol. 22, No. 23 | DOI:10.1364/OE.22.028037 | OPTICS EXPRESS 28046

waveguide birefringence and correspond to the positions at which the entire waveguide has passed the reversal point of the magnetic field.

Fig. 5. Polarization changes due to the Faraday effect and different levels of birefringence. The red star (S2 = 0, S3 = 0) indicates the launch polarization, which is vertically-oriented linear. Squares mark the start point for each curve and corresponding to zm = 0 in Fig. 3. Triangles correspond to the situation marked by the orange line in Fig. 3 when the entire waveguide passed the reversal point of the magnetic field.

As the simplest case, we firstly discuss the Faraday rotation in a waveguide with no birefringence (black track). Since linearly polarized light is launched into the waveguide, the variation path of the light output states follows the equator of the Poincaré sphere. These are of course elementary properties of Faraday rotation, but encountered in an unusual way since the evolution on the Poincaré sphere here is associated with shifts in the relative position of the waveguide and non-uniform magnetic field, as opposed to the familiar picture of an increasing length of propagation through a fixed uniform field. Note that the zero birefringence path turns back on itself (at the triangle), since the applied magnetic field changes sign at that point. Once birefringence is introduced, the curves start tilting and curling. We see from Fig. 5 that if the birefringence Δn is on the scale of 10−7 –10−6 (green and red), then the Faraday rotation is not seriously disrupted. Once the birefringence reaches a level of 10−5 , the deviation of the tracks becomes obvious: the rotation of the azimuth of the polarization reduces, and the change in ellipticity becomes apparent. As the birefringence further increases, the tracks coil around the point signifying the input vertically-oriented linear state. The complicated nature of the large birefringence tracks can be understood in terms of the simple circular orbits in Fig. 1. As the magnetic field in the experiment is non-uniform and changes sign several times, the rotation vector in Eq. (16) is constantly moving on the Poincaré sphere, and the polarization state precesses around this vector to create complicated motion both inside the waveguide for a single configuration, and also in the evolution of the output state with magnetic field position.


5.2.

Unambiguous determination of birefringence

We’ve now established that the distinctive shapes of the polarization variation trajectories result from the nonuniformity of the magnetic field, and are determined by both the waveguide birefringence and the initial launch state. Since the polarization explores the two-dimensional manifold of the Poincaré sphere rather than a line, the calculation results are free of the limitation of Eq. (1) and we can expect to identify the birefringence fairly accurately. Therefore, we next attempted to create fitting curves for the measured results based on the established calculations in order to determine the birefringence value of the waveguide under examination. 6. 6.1.

Analysis of experimental data Competition between Faraday rotation and birefringence

The birefringence of the waveguide Δn is unknown, so was set as a free parameter. Although the polarization and extinction ratio of the input PM fiber was directly measured by the polarimeter, there was still a small degree of uncertainty in the launch state E0 because the axis of the fiber does not in practice necessarily perfectly match the fast/slow axis of the inscribed waveguide. Therefore, we also introduced the launch state as a parameter with a small range of freedom around y-oriented linear polarization. Note that these parameters hold the same value for all 26 positions of the magnetic field. The measured results of polarization in the waveguides and bulk glass under forward (magnetic field direction is parallel to the light propagation direction when the glass sample was located at zm = 0) /reverse magnetic fields are plotted on the Poincaré sphere in Fig. 6, marked in dots for the forward field and crosses for the reverse. The polarization in the waveguides or bulk glass was right-rotated (clockwise when looking against the direction of transmission) as the intensity of the forward field increased (the relative position between the waveguide and the magnets changed from zm = 14 mm to zm = 0, see Fig. 3), consistent with the definition that a negative Verdet constant (VTG20 = −74 rad/T·m) corresponds to right-rotation when the direction of propagation is parallel to the applied magnetic field. Note that the bulk glass (Lglass = 15.08 mm) was longer than the waveguides (Lwg = 10.65 mm), thus the net rotation in the bulk glass is larger, appearing as a 0.1 extension along the S2 direction for each black line in Fig. 6. To generate curve fitting plots, we calculated the output state of propagation in the waveguide using the code described in the previous section with the waveguide birefringence and the initial launch state set as variables. By tuning the values of these two variables in order to fit the experimental data, we obtained the best-fit curves marked in green for WG-1, blue for WG-2 and black for the bulk sample. We determined the polarization of the launch states E0WG1 for the measurement of WG-1, and E0WG2 for that of WG-2. The input Jones vectors E0WG1 and E0WG2 were found as

cos(84.2◦ ) cos φ1 = , (19) E0WG1 = sin(84.2◦ ) ei0.66π sin φ1 eiδ1 and E0WG2 =

cos(84.3◦ ) cos φ2 = , sin(84.3◦ ) ei0.76π sin φ2 eiδ2

(20)

where φ1 and φ2 had an uncertainty of ±0.5◦ , and that for δ1 and δ2 was within 0.05π . These parameters, φ1 and δ1 , φ2 and δ2 , determined the position of the fitting curves on the Poincaré sphere and slightly affect the tilt of the curves. Realignment of the system after measurement of each set of data also accounts for the translation of the overall curves. #222329 - $15.00 USD Received 4 Sep 2014; revised 25 Oct 2014; accepted 28 Oct 2014; published 4 Nov 2014 (C) 2014 OSA 17 November 2014 | Vol. 22, No. 23 | DOI:10.1364/OE.22.028037 | OPTICS EXPRESS 28048

Fig. 6. Polarization variations in waveguides and unmodified glass on a Poincaré sphere. Experimental data have uncertainties of ±0.008 on each axis.

The waveguide birefringence is the main determinant for the tilt, curvature and the folding degree of the fitting curves in the calculations. Since glasses have homogeneous structures, the birefringence of unmodified glass is approximately zero and the corresponding fitting curves present as orbits that are parallel to the equator of the sphere in Fig. 6. Since the two waveguides were created from the same sample, if any weak material birefringence is present in the bulk glass, it should be common to both waveguides. In contrast, the laser writing process certainly introduces birefringence into the waveguides through changes in form and stress, and the induced birefringence can be expected to increase with laser pulse energy and the number of overwriting passes [18, 25, 28]. This increase of waveguide birefringence is clearly apparent in the strong enhancement in tilt and curvature, as well as the enlargements of the degree of folding around the turning points in Fig. 6. The extracted best-fit values of the waveguide birefringence are 4.8 × 10−6 for WG-1 (created with 300 nJ pulses and two overwriting passes) and 1.56 × 10−5 for WG-2 (created with 325 nJ pulses and four passes), with uncertainties of ±0.8 × 10−6 . As expected then, the birefringence of WG-2, which was written with higher energy and more overwiting, is larger than that of WG-1 and would cause a greater reduction of Faraday rotation in the waveguide. We also performed calculations to assess the impact of polarization-dependent loss (PDL). The level of PDL in our waveguides is unknown and not easily characterized in our current setup. Therefore given that the measured total loss was 1.8 dB/cm for WG-1 and 2.3 dB/cm for WG-2, we performed calculations assuming values for differential loss within 30% of the overall measured losses. In all cases, the largest shift in the predicted state was less than the uncertainties in Fig. 6, and we conclude that the PDL does not significantly affect our results. 6.2.

Faraday rotation in the waveguides

To verify the conclusion above—that the reduction in Faraday rotation is more considerable in the waveguide exhibiting larger birefringence—we now only consider the Faraday rotation


in the waveguides ignoring the birefringence. Under this assumption, the expected rotation is given by

θ =V

Lglass

B(z) dz.

(21)

0

We assigned a reference value of 0◦ to the rotation in the waveguide for the zm = 26 mm setting. Then the measured rotation angles in the two waveguides, at the different positions (zm = 1, 2, 3 mm...) and for forward and reversed magnetic fields, were adjusted based on this reference and are shown in Fig. 7. The rotation in bulk glass is not shown. As expected, the largest rotation angle in the waveguide was obtained at zm = 0 with a value of 9.0◦ ± 0.4◦ for WG-1 and 6.3◦ ± 0.4◦ for WG-2. (Here, the error estimates reflect fluctuations in the polarimeter’s reading of the azimuthal orientation during measurement of the collimated output beam.) The nonlinear dependence on z results from the variation of the magnetic field in the z direction. Based on Eqs. (18) and (21), we theoretically integrated the B field along the waveguide and calculated the expected rotation of the transmitted light. The calculated reference curves are plotted in black in Fig. 7. Referring back to Fig. 3, the amplitude of the magnetic field is close to but still not equal to zero when zm = 26 mm, suggesting that a small Faraday rotation still takes place in the waveguide. Therefore, each reference curve has been given a linear shift of θ to remove the differences between the two curves at zm = 26 mm, where the rotation reference was defined as zero. Obviously, the experimental data of WG-1 follow the reference curves within error bars at most of the data points. A small deviation of rotation angles within 0.9◦ is only noticed at points closed to zm = 0. We attribute these small rotation decreases to the waveguide birefringence, and also to reduction of the Verdet constant after laser exposure and thermal annealing. In contrast, the rotation reduction is more noticeable in WG-2, and the largest difference between the reference and the measured results is 4.1◦ . Based on the analysis established above and the previous study in [36], the decreases in Verdet constant for WG-1 and WG-2 should be similar, we deduce that the larger reduction of rotation is mainly caused by stronger birefringence in WG-2.

Fig. 7. Experimental results of Faraday rotation in the waveguides.


7.

Conclusion

The competition between the Faraday effect and the birefringence in FLDW waveguides in a 10.65 mm-long magneto-optical glass was theoretically calculated and experimentally verified. Based on the analysis, we determined the birefringence of the waveguide created using 300 nJ and two overwriting passes was Δn ≈ 4.8 × 10−6 , and the reduction of Faraday rotation in the waveguide was insignificant. Though the changes of fabrication parameters were minor, the birefringence of waveguide fabricated with 325 nJ and four overwriting passes was Δn ≈ 1.56 × 10−5 and induced some decrease in the Faraday rotation. The largest rotation was 9.0◦ in the former waveguide, but 6.3◦ in the latter. The two obvious challenges towards enhancing the rotation are reducing losses and increasing waveguide length. Laser-written chips of 7 cm or more in length are common, so the 5 cm waveguides that would have yielded 45◦ rotation here should certainly be possible. In our case, however, the losses in such a waveguide would have been prohibitive. We are optimistic that a more extensive survey of available glasses, writing conditions and annealing protocols will reduce the losses to an acceptable level, as well as allow minimization of the induced birefringence. The ability to sensitively determine the birefringence by this method is interesting, but perhaps limited in application given that it requires a significant Faraday response in the waveguide under study. However, we note that certain glasses noted for other properties may be susceptible to this method. For instance, many chalcogenide glasses, although primarily valued for the high nonlinearity and mid-infrared transparency, possess a considerable Verdet constant in the visible [38], and this method could serve as a tool to characterize weak birefringence in laser written waveguides, even if the Faraday response is not itself of interest. Acknowledgment This research was conducted by the Australian Research Council Centre of Excellence for Ultrahigh bandwidth Devices for Optical Systems (project number CE110001018). This work was performed in part at the OptoFab node of the Australian National Fabrication Facility (ANFF); a company established under the National Collaborative Research Infrastructure Strategy to provide nano and microfabrication facilities for Australian researchers.
