Application of the fractional Fourier transform to the ... - OSA Publishing

Application of the fractional Fourier transform to the design of LCOS based optical interconnects and fiber switches Brian Robertson,1,* Zichen Zhang,1 Haining Yang,1 Maura M. Redmond,1 Neil Collings,1 Jinsong Liu,1,† Ruisheng Lin,1 Anna M. Jeziorska-Chapman,1 John R. Moore,1 William A. Crossland,1 and D. P. Chu1,2 1

Photonics and Sensors Group, Department of Engineering, University of Cambridge, 9 JJ Thomson Avenue, CB3 0FA, Cambridge, UK 2

e-mail:[email protected]

*Corresponding author: [email protected] Received 13 December 2011; revised 3 February 2012; accepted 6 February 2012; posted 6 February 2012 (Doc. ID 159779); published 20 April 2012

It is shown that reflective liquid crystal on silicon (LCOS) spatial light modulator (SLM) based interconnects or fiber switches that use defocus to reduce crosstalk can be evaluated and optimized using a fractional Fourier transform if certain optical symmetry conditions are met. Theoretically the maximum allowable linear hologram phase error compared to a Fourier switch is increased by a factor of six before the target crosstalk for telecom applications of −40 dB is exceeded. A Gerchberg–Saxton algorithm incorporating a fractional Fourier transform modified for use with a reflective LCOS SLM is used to optimize multi-casting holograms in a prototype telecom switch. Experiments are in close agreement to predicted performance. © 2012 Optical Society of America OCIS codes: 070.6120, 070.2575, 090.0090, 060.6718.

1. Introduction

Reconfigurable optical beam-steering switches based on liquid crystal on silicon (LCOS) spatial light modulators (SLMs) have been developed to the level where they are now capable of deployment in datacom and telecom networks [1,2]. In addition, optical interconnects utilizing LCOS technology have the potential to enhance computing and data storage systems due to their ability to adaptively align and redistribute optical data [3]. All these applications require low insertion loss and low crosstalk performance. In particular, for telecom networks, switches require < − 40 dB crosstalk suppression. To date, LCOS switches and interconnects have been typically based on 2f optical systems and the use of 1559-128X/12/122212-11$15.00/0 © 2012 Optical Society of America 2212

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blazed gratings, where ideally all the light should be diffracted into the 1 order at a position defined by the grating equation. An example of such a switch is shown in Fig. 1(a). However, LCOS SLMs can only display a quantized approximation to the blazed grating profile due to the finite size of the pixels (spatial quantization) and the limited number of available phase levels (phase quantization) [4,5]. As a result, light is also diffracted into higher and symmetric orders (m≠ 1), which may unintentionally couple into other output positions. This effect is exacerbated by phase display errors due to device nonuniformity [6], temporal variations [7], field fringing effects [8], and temperature changes. In an associated publication we demonstrated that by purposefully introducing defocus into the optical system, and displaying an off-axis holographic lens on the SLM to compensate for this wavefront error, it is possible to reduce worst-case crosstalk by

Fig. 1. (Color online) Experimental outline of wavefront encoding as applied to a 2f optical routing geometry. (a) Conventional 2f system where orders are focused along plane Q1 . (b) Defocused system where diffraction orders are focused along the plane Q2 .

>10 dB compared to an equivalent system that uses only conventional blazed gratings [9]; a technique we refer to as defocus based wavefront encoding [Fig. 1(b)]. The advantage of this method is that it has the potential of allowing the < − 40 dB crosstalk level to be reached with a lower quality LCOS SLM. Although we can predict the positions of the various replay diffraction orders in a defocused based wavefront encoded switch using geometric optics, it is still necessary to use a diffraction integral to calculate the replay field and corresponding signal and crosstalk power coupled into the output fibers to take into account the effects of spatial and phase quantization of the off-axis lens. This allows us to determine if the crosstalk meets the < − 40 dB target, and to optimize the holograms as required. In a standard 2f based switch we make use of the Fourier transform to calculate the replay field at the output plane. However, this approach is not suitable to the wavefront encoded switch as it calculates the field at the back focal plane of the focusing lens, not the fiber or interconnect replay plane. As we shall show, by ensuring simple optical system symmetry conditions are met, the replay field can be calculated using a fractional Fourier transform [10–12]. This has the advantage of calculation speed and the ability to fit seamlessly into known iterative hologram optimization algorithms such as the Gerchberg–Saxton routine [13] as demonstrated by previous works

[14–16]. In addition, as we shall demonstrate experimentally, this approach can not only be used to design a hologram to steer a single beam, but also to optimize a hologram that multi-casts (defined as routing an input signal to more than one output fiber). We shall first outline the fractional Fourier transform, and describe the conditions necessary for this algorithm to be employed in the analysis of an optical beam-steering switch. We present a comparison of the relative sensitivities of a blazed grating and wavefront encoded pattern based on defocus to uniform phase error based on a switch developed during the ROSES project [17]. Finally, this transform is used in conjunction with a Gerchberg–Saxton hologram optimization algorithm to design a multi-casting hologram that is experimentally tested in a prototype holographic telecom switch. 2. Design of a Wavefront Encoded Switch Based on Defocus

Figure 1(a) shows the standard 2f beam-steering switch geometry, where the input/output fiber array is positioned at the front focal plane of a Fourier transform lens (plane F 1 ), and a reflective phase-only LCOS SLM is placed at the back focal plane. The input beam enters the switch via the central fiber, is collimated by the Fourier transform lens, and is then incident on the LCOS SLM. If a blazed grating of 20 April 2012 / Vol. 51, No. 12 / APPLIED OPTICS

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period T is displayed on the SLM, the light is diffracted into a series of plane waves, each propagating at an angle of θm. These angles are defined by the grating equation, sin θm mλ∕T, where λ is the wavelength and m is the order of the diffracted beam. Due to the symmetry of the system, these orders are focused at an output plane, Q1 , which is coincident with the input plane, F 1 . Figure 1(b) shows the geometry of a wavefront encoded switch. The input/output fiber array is positioned a distance z2 f s from the collimating lens, whilst the SLM is positioned a distance z1 behind the lens. To focus the m 1 order at the fiber plane (F 1 ), it is necessary to add focusing power to the blazed grating. As described in reference [9], the resulting hologram displayed on the SLM can take the form of an off-axis lens. One of the consequences of this is that higher and symmetric diffraction orders are further defocused along plane Q1 , and are no longer coincident with plane F 1 [18]. As a result, optimum coupling is only obtained for order m 1. Note that, with respect to reference [9], we have swapped the indices of z1 and z2 to make the following analysis more compatible with previous papers in the field of fractional Fourier transforms. 3. Application of the Fractional Fourier Transform to a Wavefront Encoded Switch Based on Defocus

In the following section we shall describe how a fractional Fourier transform can be applied to the switch geometry of Fig. 1(b). To do this we shall first refer to the system P of Fig. 2, which consists of an input hologram, P H , a single lens of focal length f , and replay plane R . The hologram is illuminated by a plane wave to generate an input field, EH x; y, positioned a distance z1 in front of the lens. The output plane is located a distance z2 behind the lens, where a field, EI u; v is generated. The fractional Fourier transform, an integral that has been previously used in optics, signal processing, and quantum mechanics, is an ideal choice for this application. With relation to Fig. 2, a fractional Fourier transform of order a is given by [10] E1 u0 ; v0

Z

∞

−∞

Z

∞

−∞

K A x0 ; y0 ; u0 ; v0 EH x0 ; y0 dx0 dy0 ; (1)

Fig. 2. Relationship between Lohmann type I fractional Fouriertransform optical system and terms used in analysis. The input and output planes are a distance f s from a lens of focal length f .

and Aϕ

p 1 − i cot ϕ.

4

The term Aϕ is simply a system constant, and can be omitted in the rest of this analysis. When a 1 we have the standard Fourier transform, which directly relates an input field to the spatial frequency components making up that field. Note that implicit in Eq. (1) is a scaling factor, ξ, which relates the transverse scale of the two planes via the scaling x0 x∕ξ, y0 y∕ξ, u0 u∕ξ, and v0 v∕ξ. In the case of a Fourier transform, ξ λf 1∕2 . There are several references that describe how the fractional Fourier transform can be used to calculate the optical field at a plane displaced from the Fourier plane [10–12]. The derivation we shall use follows from the work of Bernado [19]. Any optical system consisting of an input plane, an output plane, and a set of optics in between can be represented by a characteristic ABCD matrix [20]. According to reference [21], diffraction through a rotationally symmetric lens system can relate the output field to the input field in terms of the ABCD matrix of the system as follows: k Ax2 y2 EH x; y exp −i E1 u; v 2 B −∞ −∞ 2 2 (5) Du v − 2xu yv dxdy. Z

∞

Z

∞

where K A x0 ; y0 ; u0 ; v0 Aϕ expiπcotϕu02 − 2 cscϕu0 x0 cotϕx02 × Aϕ expiπcotϕv02 − 2 cscϕv0 y0 cotϕy02 ;

ϕ 2214

aπ ; 2


(2)

(3)

If certain symmetry conditions of the ABCD matrix are met, we can rearrange Eq. (5) to the same form as Eqs. (1)–(4), and thus fractional Fourier transform theory may be used. There are two standard configurations, the Lohmann type I and II geometries [11]. It is the first (lens positioned halfway between the input and output planes as represented in Fig. 2) that we are interested in at the moment as it comes closest to representing our configuration for our wavefront encoded system. The ABCD matrix for this optical system, M, is

" M

1 − zf1 − 1f

# z1 z2 − z1fz2 . 1 − zf2

6

If we equate the exponential terms of Eqs. (5) and (1), it can be shown that for a fractional Fourier transform to hold, matrix terms A D, and as a result, z1 z2 . Moreover, it can be shown that we can express ϕ as z cos ϕ 1 − 1 . f

(7)

Similarly, we can solve for ξ to yield a scaling factor term of ξ2 f λ sin ϕ. The matrix of Eq. (6) can now be cast in the form M

cos ϕ − 1f

f sin2 ϕ . cos ϕ

(8)

This illustrates the ABCD matrix symmetry conditions required for the application of a fractional Fourier transform of order a 2ϕ∕π with a lens of focal length f . If the ABCD matrix for an arbitrary optical system consisting of a number of components can be cast in the form of Eq. (8), with f being an effective focal length, the output field will be related to the input field by an exact fractional Fourier transform. 4. Application of the Fractional Fourier Transform to a LCOS SLM: The Equivalent Model Representation

One of the advantages of using a fractional Fourier transform is that it can be expressed in terms of fast Fourier transforms, thereby allowing for rapid calculation and optimization of the replay field as described by Ozaktas et al. [22]. The design of diffractive elements by this fast fractional Fourier transform approach was reported by Zhang et al. [14], and Zalevsky et al. [15]. Their analyses showed that certain sampling criteria must be met to ensure an accurate representation of the replay field. In the following we will make use of an equivalent optical system approach developed by Testorf [16] that circumvents this restriction, and allows calculation of the replay field for any fractional order. In his analysis, the Lohmann type I system of Fig. 2 was replaced by an equivalent three lens system as shown in Fig. 3, which has an ABCD matrix of " f # − f F1 fF M fF 1 . 9 − f F − ff F1 f2 1

This matrix has the same form as matrix of Eq. (8) if we set the values for f F and f 1 as f F f sin2 ϕ;

(10a)

fF ; cos ϕ

(10b)

f1 −

Fig. 3. (Color online) Three lens equivalent model. The single lens arrangement of Fig. 2 can be replaced by the above equivalent optical system.

where f is the focal length of the lens of Fig. 2, and ϕ is given by Eq. (3). We can therefore calculate the replay field, EI u; v, using only four steps. Firstly the plane wavefront, which we denote as Ein x; y to take into account any amplitude profile, is incident on the SLM (shown in transmission in Fig. 3). The pixelated LCOS SLM is assumed to display a phase-only hologram represented by αx; y, where 0≤αx; y < 2π. The resulting transmitted wavefront, EH x; y, is the product of these two terms. In step 2, the first lens of Fig. 3, f 1 , imparts a quadratic phase curvature on EH x; y. In step 3, the central lens, f f performs a Fourier-transform on the wavefront exiting lens f 1 . Finally, lens f 2 imparts a quadratic phase curvature on the wavefront such that the output field, EI u; v, is given by −ikx2 y2 EI u; v FT Ein x; y expiαx; y exp 2f 1 2 2 −iku v . (11) × exp 2f 2 In the actual calculation we use an FFT for step 3, with a spatial sampling corresponding to the N × N pixels of the SLM plane. Thus the fields at all planes are uniformly spatially sampled on an N × N grid, with the sampled u coordinate being given by un

λf f ; NΔ

(12)

where Δ is the pixel size, and n is an integer varying from −N∕2 to N∕2. The same scaling factor relates v to y. 5. Correction for a Noncollimated Incident Beam

The previous two sections discuss the application of the fractional Fourier transform to SLMs that have a collimated beam incident on them. However, in the switch of Fig. 1(b) the incident beam is nonplanar. Let us consider Fig. 4(a), which shows the switch of Fig. 1(b) unfolded into a transmission system to more clearly relate how the optical system parameters affect the formulation of the fractional Fourier transform. The input fiber is positioned at plane PIN , whilst the output fibers are positioned at planes PR [both equivalent to F 2 in Fig. 1(b)]. Lenses L1 and L2 20 April 2012 / Vol. 51, No. 12 / APPLIED OPTICS

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Fig. 4. (Color online) Unfolded optical system representing switch of Fig. 1(b). (a) Full switch geometry. (b) Only first lens shown to illustrate the calculation of ρH .

are identical, and have a focal length of f . Let us consider the case where z2 f Ps, with s being positive. The wavefront incident on H is therefore convergent, and the beam focused a distance P do from the lens as shown. The distance from the H to the focal plane is given by dH do − z1. As a result, P the radius of curvature of the incident beam at H , ρH −dH , is given by ρ H z1 − f −

f2 ; s

(13)

where ρH is negative if the beamPincident on the hologram is focused to the right of H , and positiveP appears to come from a virtual focus to the left of H , Eq. (13) is derived by applying the thin lens formula to Fig. 4(b). According to the analysis Bernado and Soares [23], for a fractional Fourier transform to be valid when the hologram plane is illuminated by a nonplanar wavefront, z2 , must be related to z1 and ρH by z2

ρH f z . ρH z1 − f 1

(14)

Thus we can determine the optimum value of z1 such that the system of Fig. 4 performs a fractional Fourier transform by combining Eq. (13) with Eq. (14) and solving the resultant quadratic equation to give z1 f 2s. If we meet this condition, the system of Fig. 4 (non planar beam incident on the SLM) converts to case shown in Fig. 3 (planar beam incident on the SLM). However, we now have z1 z2 f 2s, with a is given by equation (7), and a modified lens focal length, f p , of fp

ρH f . ρH z1 − f

(15)

This scaled focal length and new value of φ takes into account the nature of the nonplanar beam incident on the hologram plane, and with these new 2216


parameters we can use the equivalent model representation described in section 3 to calculate the replay field of a quantized SLM in a wavefront encoded switch. With reference to Fig. 4(a), the focal length of the holographic lens, f H , required to focus an incident wavefront with radius of curvature −ρH at the replay plane must be such that the wavefront of curvature of the 1 order exiting the SLM is also ρH . This ensures that the light diffracted from the SLM is optimally focused into the output fibers. Thus, from geometric optics: f H −1∕2ρH.

(16)

Note that the above analysis is valid for a transmissive SLM. In the case the reflective SLM of Fig. 1(b), the required focal length is the negative of Eq. (16). 6. Calculation of the Relative Sensitivities of a Blazed Grating and Wavefront Encoded Switch to SLM Phase Error

As mentioned, in practice LCOS SLMs generate phase patterns that deviate from the ideal due to device spatial nonuniformity [6], temporal instabilities [7], pixel edge effects [8], and temperature changes. Thus any technique that makes the switch more tolerant to LCOS phase errors will be advantageous in telecom and interconnect applications. In the following section we compare the relative sensitivities of the replay fields of the switches of Figs. 1(a) and 1(b) to a uniform phase error. Here we define the hologram phase as 1 − γ × θH x; y, where θH x; y is the ideal phase and γ is the phase error. The actual phase error at each pixel will of course be a more complex function of temperature, time, and phase pattern, but this linear approximation will give us an indication of how robust both approaches are. Of particular importance is the value of γ that results in a crosstalk greater than the −40 dB telecom target. As a specific example, let us consider the system discussed in reference [9], which is based on a prototype fiber optic switch developed as part of the ROSES project [17]. It should be noted that the

-10

-1

-20 -0.5

-30 -40

0 -50 -60

0.5

-70 1

-80

(17)

where f H is the focal length of the holographic offaxis lens. In [9] we used a value of f H of 1 m to give a defocus of s 312.5 μm. Keeping the same defocus, let us now convert this design example to a fractional Fourier transform system with z1 f 2s. As a result, z2 25.3125 mm and z1 25.625 mm. It therefore follows that the order of the fractional Fourier-transform a 1.0159, ρH −1.9994 m, f p 25.3125 mm, and f H −0.9997 m. Note that the value of a we use is simply set by the fact that, for this particular example, s 0.3125 mm. In principal, any defocus s, and thus any order of fractional Fourier transform is permitted. From Gaussian propagation theory it can be shown that the 1∕e2 beam radius at the SLM plane is 2.37 mm. Placing a requirement that the off-axis lens covers at least three times the beam radius to capture the incident light we must therefore cover >475 × 475 pixels. In the following analysis we use 500 × 500 pixels for the hologram simulation. Finally the off-axis lens will be assumed to be quantized into 64 discrete phase levels spaced equally between 0 and 2π, typical of our experimental tests. The replay field when the 1 order is deflected through 140 μm was calculated using the modified equivalent optical system model described in sections 3–5, with a uniform phase error of γ 0.05 applied to show up the defocused higher orders. A log10 intensity simulation of the replay field is shown in Fig. 5, and the corresponding log10 intensity plot along the central y-axis in Fig. 6. From these simulations it can be seen that only the m 1 order is focused; the other orders are defocused, with the defocus increasing the further an order is from the mth 1 beam. Moreover, interference between the orders occurs at the replay plane leading to ripples in the intensity profile at the fiber plane, as is observed experimentally. Of particular interest is a comparison of the relative sensitivities to phase error of a Fourier transform switch (blazed gratings) and a wavefront encoded switch based on defocus (off-axis lenses). We shall therefore also evaluate an equivalent

-1

-0.5

0

0.5

-90

1

X-position

Fig. 5. Simulated replay field at fiber plane with 1 order deflected through 140 μm. SLM displaying off-axis lens with a γ 0.05 phase error to show up higher orders. Figure plotted as 10 log10 (Intensity).

Fourier transform based switch [Fig. 1(a)] where z1 f and z2 f , and the blazed grating phase pattern was designed using a modulo π algorithm [3]. This algorithm can be used to define a noninteger pixel pitch, thereby allowing us to accurately steer the signal beam to the required port positions [24]. For both systems we shall set the 1 order to be deflected to a position 35 μm from the optical axis as this is theoretically the worst crosstalk configuration. Figures 7(a) and 7(b) show the corresponding y-axis intensity distributions of the replay fields when (a) ideal phase patterns are displayed, and (b) when a uniform phase error of γ 0.05 is applied. As can be seen, when we display both an ideal blazed grating and wavefront encoded pattern the replay fields 0 -10 -20

10xlog10(Intensity)

f2 . s− 2f H

0

Y-position

ROSES switch used nonuniform spatial positioning of the output ports to control crosstalk. However, in this analysis we assume a uniform port spacing of 35 μm as we are interested in determining the effectiveness of wavefront encoding in a practical situation. We shall consider eight output ports arranged symmetrically about the input port with an input radius of 5.2 μm, a lens of focal length f 25 mm, and a pixel size 15 μm. The target positions are thus δ 35, 70, 105, and 140 μm from the optical axis. In reference [9] it is shown that if z1 f (a nonfractional Fourier transform system), the defocus, s, is given by

-30 -40 -50 -60 -70 -80 -90 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Position at fiber plane (mm)

Fig. 6. (Color online) Simulated replay field along y-axis at fiber plane with SLM deflecting 1 order through 140 μm. SLM displaying off-axis lens with a γ 0.05 phase error to show up higher orders. Figure plotted as 10 log10 (Intensity). 20 April 2012 / Vol. 51, No. 12 / APPLIED OPTICS

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are almost identical, with the exception of crosstalk peaks in the blazed grating replay field in the region of −100 to −150 μm. However, for both cases, the maximum crosstalk at the target locations is of the order of −43 dB. This level of crosstalk is primarily due to clipping of the incident Gaussian beam at the edges of both holograms. The truncation of the incident beams generate diffraction rings centered about the main peak, observable as an intensity oscillation in the replay fields of Fig. 7(a). Thus, in order to make a realistic estimation of worst-case crosstalk, the maximum crosstalk at a specific replay field position, u; 0, will be evaluated over the range of u 5 μm; 0. Introducing a γ 0.05 phase error causes the crosstalk to increase dramatically in the case of the quantized blazed grating based switch. Crosstalk peaks occur at regular positions corresponding to the focused m ≠ 1 diffraction orders, with a worst case value −25.3 dB below the 1 signal beam peak intensity. In the case of the quantized wavefront encoded hologram, the maximum crosstalk intensity falls well below that of the equivalent blazed grating. (a)

0

Blazed grating Wavefront encoding

-10

10xlog10(Intensity)

-20 -30 -40 -50 -60 -70 -80 -90 -0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2


(b)

0

Even when we apply a γ 0.05 phase error, the maximum crosstalk value is still −40.5 dB below the 1 order peak intensity at the port locations. To fully compare the two switch designs, we shall first investigate the maximum crosstalk at the output port positions as a function of number of quantized phase levels, N, in order to determine the minimum value of N that ensures our −40 dB target is met for this example switch. The results of our calculation are shown in Fig. 8. As can be seen, the maximum crosstalk for the blazed grating starts from 0 dB for a binary grating (power in −1 crosstalk order equals power in 1 signal beam), and then drops to around −43 dB with some fluctuations due to the particular form of quantization introduced by the modulo 2π algorithm. In the case of the wavefront encoded hologram, the worst-case crosstalk falls smoothly from −25.9 dB at N 2 to −42.4 dB at N 64. The values for N at which the worst-case crosstalk goes above our −40 dB target is N 8 for the wavefront encoded switch, and N 16 for the Fourier transform based switch. To compare the relative sensitivities of the two designs to uniform phase error, the variation in worst case intensity crosstalk in the replay field with γ was calculated and plotted in Fig. 9. Both switch designs have a worst-case crosstalk of around −43 dB when γ 0. As γ increases, the worst-case crosstalk increases for both switches. However, the wavefront encoded based switch is far more tolerant to linear phase error as the increase is far less. In the case of the blazed grating based switch, an error of γ 0.011 (1.1% error) results in −43 dB worst case crosstalk, whereas for the wavefront encoded switch, the equivalent phase error is 0.067 (6.7% error), approximately a six fold reduction in sensitivity. As a final test, the theoretical power coupled into a fiber positioned at the target port locations (maximum coupled power over a range of 5 μm with respect to each port location) was numerically


-10

0

-10

-30

10xlog10(Crosstalk)

10xlog10(Intensity)

-20

-40 -50 -60 -70 -80

-15 -20 -25 -30 -35 -40

-90 -0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2


Fig. 7. (Color online) Comparison of theoretical blazed grating and wavefront encoded replay fields along the beam displacement axis for a deflection of 35 μm with (a) ideal reconstruction, and (b) a uniform γ 0.05 phase error applied to the phase patterns. Dotted line—blazed grating. Solid line—wavefront encoded system. 2218


-5


-45 -50

10

20

30

40

50

60

Number of phase levels (N)

Fig. 8. (Color online) Theoretical maximum crosstalk for equivalent blazed grating and wavefront encoded switch as a function of number of phase levels, N.

-15 Blazed grating Wavefront encoding

10xlog10(Crosstalk)

-20

-25

-30

-35

-40

-45

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Linear phase error

Fig. 9. (Color online) Theoretical dependence of maximum replay field crosstalk for equivalent blazed grating and wavefront encoded switches as a function of uniform phase error, γ.

evaluated using a mode overlap integral as γ was varied from 0 to 0.1. A single mode fiber with a mode field radius of 5.2 μm at 1550 nm was assumed in this calculation. The resulting maximum crosstalk as a function of γ is plotted in Fig. 10. As can be seen, when γ 0, the respective crosstalk values are approximately −60 dB for the wavefront encoded switch and −54 dB for the Fourier transform switch. These values are lower than the replay field crosstalk calculations [Fig. 7(a)] as we must couple light into the fundamental mode of the test fiber, which tends to average out the fluctuations in the replay field due to clipping of the Gaussian beam. The higher crosstalk level for the blazed grating pattern defined by the modulo 2π algorithm is due to the peaks in the replay field in the region −150 to −100 μm. However, we can clearly see that as γ increases, the coupled power for the wavefront encoded switch is always far less than the equivalent Fourier transform switch. The point at which crosstalk goes 0

Blazed grating Wavefront encoded

10xlog10(Crosstalk)

-10

-20

-30 -40

-50

-60

-70

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Linear phase error

Fig. 10. (Color online) Theoretical dependence of maximum fiber crosstalk for equivalent blazed grating and wavefront encoded switches as a function of uniform phase error, γ. Output fiber has a mode field radius of 5.2 μm at 1550 nm.

above our −40 dB target level is γ 0.0109 for the blazed grating and γ 0.0645 for the wavefront encoded pattern; again a six fold reduction in sensitivity. We can therefore conclude that the wavefront encoded system is more tolerant to linear phase error. This is because the crosstalk power is spread over a large area as opposed to being localized at points corresponding to positions of other output ports. More importantly, we can pre-determine a value for uniform phase error that ensures the crosstalk does not exceed −40 dB. In practice, phase error is more complex than a simple linear offset as it depends on phase value, field fringing effects, device nonuniformity, and temporal effects. However, to a first approximation it gives us an idea of how sensitive a particular system will be to LCOS hologram display errors. 7. Experimental Demonstration of Multi-Casting in a Telecom Switch Using a Fractional Fourier Transform Based Hologram Design Algorithm

One the advantages of using a hologram in a telecom switch or an interconnect is that we can direct an optical signal to more than one output port, which is referred to as multi-casting. As discussed by Testorf [16], it is possible to design a diffractive element for the fractional Fourier regime by replacing the Fourier transform in the standard Gerchberg–Saxton algorithm with a fractional Fourier transform of order a, and the inverse FFT with a fractional Fourier transform of order 2 − a. In the following section we use such an optimization algorithm to design a multicasting hologram for an experimental telecom switch similar to the one depicted in Fig. 1(b), which was modified to account for the reflective nature of the SLM. Figure 11 shows a more detailed layout of the 1 × 10 space-switch used in our tests. It consists of a 12-way MTP fiber ribbon, a lenslet array (250 μm pitch and 743 μm focal length), lens L1 (focal length of f 175 mm), a mirror M 1, and a nematic LCOS SLM. Note that the mirror was simply used to fold the optical path, allowing the system fit the available bench space. The LCOS SLM was a 720 × 1280 pixel array device, with pixels on a pitch of 15 μm (0.5 μm gap between pixels). The device was assembled in-house using a process developed by the group [25] with a non anti-reflection coated coverplate. To determine the phase delay characteristics of the SLM as a function of gray level (applied voltage) a binary-phase grating measurement technique [6] was employed in situ. This phase response data was incorporated into the design algorithm to set the quantized phase levels of the final hologram. In operation, light at 1554.13 nm was launched into the switch via one of the central fibers (port 6) and the corresponding lenslet converted the 5.2 μm input beam waist to a 50 μm beam waist at the input to the switch. This mode-conversion allowed us to better match the capabilities of the SLM in terms of pixel size and maximum beam deflection angle. An external polarization controller was used to align 20 April 2012 / Vol. 51, No. 12 / APPLIED OPTICS

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Fig. 11. (Color online) Layout of telecom test switch. System consists of an input/output fiber array, a lenslet array, lens L1 , mirror M 1, and a nematic LCOS SLM.

the incident polarization such that it was parallel to the director of the nematic SLM. The distance from the input plane to lens L1 was set at 200 mm, giving a value of s 25 mm, and the distance from lens L1 to the SLM was adjusted to 225 mm to ensure the z1 f 2s condition for a fractional Fourier transform was met. This results in a fractional Fourier transform order of a 1.1845, ρH −1.175 mm, f p 200 mm, and f H −0.5875 m. Under these conditions, the beam incident on the SLM has a Gaussian beam radius (1∕e2 ) of wSLM 1.66 mm , with a full beam diameter (3wSLM ) covering 330 × 330 pixels. The SLM was tilted such that the 1 order was coupled into port 7, and a series of tests were run to determine that the actual value of s using an on-axis holographic lenses of varying focal length. By fine-tuning the holographic lens focal length it was found that the optimum hologram required to 6 50 100

minimize insertion loss corresponded to a defocus of s 24.7 mm as opposed to 25 mm, resulting in a value for the fractional Fourier-transform order, a, of 1.1822. This slight difference can be explained by alignment errors, and the fact that the focal length tolerance of the lens was only quoted to 1%. A modified Gerchberg–Saxton algorithm incorporating the fractional Fourier transform and accounting for the irradiance distribution of the incident Gaussian beam was then used to design the multi-casting holograms. An example of such a hologram, optimized to equally split the input signal between ports 4 and 8, is depicted in Fig. 12, and the theoretical replay field is shown in Fig. 13. By using a suitable starting phase distribution, the algorithm converged to a stable solution after only five iteration steps. It should be noted that this approach can be also used to implement more complex interconnection patterns that require weighted multi-casting or full broadcasting. The intrinsic insertion loss of the switch due to Fresnel losses, SLM reflectivity, and SLM diffraction efficiency was measured to be −3.0 0.1 dB when

5 0

150 4

-10

250

3

300 2

350 400

1

10xlog10(Intensity)

Y-axis

200

-20

-30

-40

450 -50

500

50

100

150

200

250

300

350

400

450

500

0

X-axis

Fig. 12. Example of a multi-casting hologram designed to deflect light to two output ports and optimized using a modified Gerchberg–Saxton algorithm. The scale to the right shows phase retardation (0 to 2π). 2220


-60 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2


Fig. 13. (Color online) Normalized intensity distribution at the replay field of multi-casting hologram.

Table 1.

Port/Value Target (dBm) Optimized(dBm) Measured (dBm)

Multi-Casting Performance—Comparison of Target, Predicted, and Measured Power Levels.

a

1

2

3

4

5

6

7 (0th)

8

9

10

11

12

−40 −46.4 −39.8

−40 −35.9 −22.8

−40 −40.6 −31.8

−3 −3.93 −4.25

−40 −40.5 −22.8

IN IN IN

NA −35.5 −15.8

−3 −3.94 −4.15

−40 −43.3 −30.8

−40 −35.4 −22.8

−40 −48.0 −41.8

−40 −13.5 −15.1

a

Fiber 6 was the input fiber (IN), and we set no target for the 0th order power during optimization; we set the corresponding target power as nonapplicable (NA).

off-axis lenses deflected light to both ports 4 and 8. The theoretical insertion loss for our optimized multi-casting pattern to ports 4 and 8 was calculated to be −6.93 dB and −6.94 dB respectively when we take into account the intrinsic losses. Experimentally this loss was measured as −7.25 0.1 dB and 7.15 0.1 dB, which is in very close agreement to our predicted values. The initial target, theoretically predicted, and measured crosstalk matrices are presented in Table 1. These values are calculated with respect to a simple off-axis lens deflecting all the light to port 4 (loss of 0 dB). Note that light was launched into the switch via fiber 6 (denoted IN). As port 7 is not used we set the target level as nonapplicable (NA). Ideally the power split is −3 dB; however, due to quantization of the phase pattern, the insertion loss increases by approximately 1 dB to ports 4 and 8, which is also reflected in the experimental data. In addition, both the theoretical and experimental multicasting wavefront encoded holograms generate strong crosstalk at port 12 (−13.5 and −15.1 dB respectively). This high level of crosstalk when multi-casting is equivalent to the levels observed in a standard Fourier-transform based switch [Fig 1(a)], showing that there is not the same intrinsic improvement when routing to more than one port as there is when routing to just one output port. However, this crosstalk can, in principal, be reduced through, for example, the use of an additional simulated annealing optimization step. In future we plan to include the actual SLM pixel edge response during the hologram optimization [26], and use more advanced optimization techniques [27]. These should be effective techniques as it was noted that the multicasting performance of the switch of Fig. 11 is far better than an equivalent Fourier transform based switch utilizing the same device, particularly with regards to the uniformity of the power split. This improvement is attributed to the fact that the incident and diffracted beams in a Fourier-transform switch are planar, and undergo multiple reflections that interfere at the replay plane and degrade the hologram performance. In the case of a wavefront encoded switch, the multiple reflected beams are focused at different planes and this interference effect is mitigated. 7. Conclusions

Calculation of the replay field in a wavefront encoded optical interconnect or fiber switch based on LCOS technology can be achieved using a fractional fast Fourier transform if certain optical system symmetry

conditions are met. Telecom switches have stringent requirements: high diffraction efficiency and worstcase crosstalk down by −40 dB compared to the signal beam. Although the holograms for a wavefront encoded switch based on defocus can be analytically designed by treating them as off-axis lenses (or more accurately, quantized mirrors), modifying the geometry of the switch to allow the use of a fractional Fourier transform simplifies the calculation of the replay field. We can determine the trade-off between number of available quantized phase levels, focal length and spatial and phase quantization of the holographic lenses, include physical effects such as electrode fringing fields and liquid crystal dynamics in the hologram optimization algorithm. The equivalent optical system method for implementing a fractional Fourier transform as developed by Testorf is well suited to switches based on pixelated holograms, such as a LCOS SLM, as it can be expressed in terms of a standard FFT, is fast, and blends seamlessly into a Gerchberg–Saxton iterative optimization algorithm. For this technique to be applicable to our switch design, the effect of a nonplanar incident beam was incorporated. Our analysis shows that for a given defocus, s, a fractional Fourier transform can be applied if the distance from the fiber plane to the main lens (focal length f ) is f s, and the distance from the main lens to the SLM plane is f 2s. Simulation of the replay fields of a wavefront encoded and Fourier-transform based switch showed that the wavefront encoded based switch requires only N 8 as opposed to N 16 phase levels to meet the target −40 dB crosstalk level. In addition, a theoretical study of the effect that LCOS SLM linear phase-errors have on the performance of both switches show that the wavefront encoding technique is more robust. For our specific example, a wavefront encoded switch is six times less sensitive to linear phase error than an equivalent Fourier transform based switch. Future work will be extended to include accurate modeling of temporal effects and pixel edge effects. A multi-casting experimental test was performed on a ten-port wavefront encoded fiber switch. It operated at 1554.13 nm, and the holograms were designed using a modified Gerchberg–Saxton algorithm employing a fractional Fourier transform modified for use with a reflective LCOS SLM. Experimental data is in good agreement with the theoretical replay field. In particular, it was noted that the uniformity of the signal beams was far better than achieved in 20 April 2012 / Vol. 51, No. 12 / APPLIED OPTICS

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equivalent Fourier-transform switches, which was attributed to the fact that multiple interference effects were mitigated as the beams were nonplanar. Future work will focus on the optimization of the hologram designs for multi-casting performance. This work was supported by the Cambridge Integrated Knowledge Centre (CIKC), by the UK Engineering and Physical Sciences Research Council (EPSRC) platform grant (Liquid crystal photonics) and follow on fund (ACCESS project), and by the Research Council UK (RCUK) Global Uncertainties program. †Current address: Wuxi OptonTech Ltd, 16 Changjiang Road, Wuxi New District, Jiangsu 214028, China References 1. T. A. Strasser and J. L. Wagener, “Wavelength-selective switches for ROADM applications,” IEEE J. Sel. Top. Quantum Electron. 16, 1150 (2010). 2. S. Frisken, “Advances in liquid crystal on silicon wavelength selective switching,” in Proceedings of OFC/NFOEC 2007, pp. 1–3 (2007). 3. D. Gil Leyva, B. Robertson, C. Henderson, T. Wilkinson, D. O’Brien, and G. Faulkner, “Crosstalk analysis of a free-space optical interconnect based on a spatial light modulator,” Appl. Opt. 45, 63–75 (2006). 4. K. L. Tan, S. T. Warr, I. G. Manolis, T. D. Wilkinson, M. M. Redmond, A. A. Crossland, R. J. Mears, and B. Robertson, “Dynamic holography for optical interconnections II. Routing holograms with predictable location and intensity of each diffraction order,” J. Opt. Soc. Am. A 18, 205–215 (2001). 5. E. Hällstig, L. Sjövist, and M. Lingdron, “Intensity variations using a quantized spatial light modulator for nonmechanical beam steering,” Opt. Eng. 42, 613 (2003). 6. Z. Zhang, H. Yang, B. Robertson, M. Redmond, M. Pivnenko, N. Collings, W. A. Crossland, and D. P. Chu, “Static phase compensation for a phase-only liquid crystal on silicon (LCOS) device,” Appl. Opt. (to be published). 7. A. Lizana, A. Márquez, I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Wavelength dependence of polarimetric and phase-shift characterization of a liquid crystal on silicon display,” JEOS RP 3, 08012 (2008). 8. L. Xu, J. Zhang, and L. Y. Wu, “Influence of phase delay profile on diffraction efficiency of liquid crystal optical phased array,” Opt. Laser Technol. 41, 509–516 (2009). 9. B. Robertson, Z. Zhang, M. M. Redmond, N. Collings, J. S. Liu, R. S. Lin, A. M. Jeriorska-Chapman, J. R. Moore, W. A. Crossland, and D. P. Chu, “The use of wavefront encoding in optical interconnects and fiber switches,” Appl. Opt. (to be published).

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