Tan et al.
`
`Vol. 18, No. 1 / January 2001 / J. Opt. Soc. Am. A
`
`205
`
`Dynamic holography for optical interconnections.
`II. Routing holograms with predictable
`location and intensity of each diffraction order
`
`Kim L. Tan*
`Department of Engineering, Cambridge University, Trumpington Street, Cambridge CB2 1PZ, UK
`
`Thomas Swan & Co. Ltd., c/o Department of Engineering, Cambridge University, Trumpington Street,
`Cambridge CB2 1PZ, UK
`
`Stephen T. Warr
`
`Ilias G. Manolis, Timothy D. Wilkinson, Maura M. Redmond, William A. Crossland,
`and Robert J. Mears
`
`Department of Engineering, Cambridge University, Trumpington Street, Cambridge CB2 1PZ, UK
`
`Thomas Swan & Co., Ltd. c/o Department of Engineering, Cambridge University, Trumpington Street,
`Cambridge CB2 1PZ, UK
`
`Brian Robertson
`
`Received October 19, 1999; revised manuscript received June 20, 2000; accepted July 25, 2000
`An analysis of dynamic phase-only holograms, described by fractional notation and recorded onto a pixelated
`spatial light modulator (SLM) in a reconfigurable optical beam-steering switch, is presented. The phase
`quantization and arrangement of the phase states and the SLM pixelation and dead-space effects are decou-
`pled, expressed analytically, and simulated numerically. The phase analysis with a skip–rotate rule reveals
`the location and intensity of each diffraction order at the digital replay stage. The optical reconstruction of
`the holograms recorded onto SLM’s with rectangular pixel apertures entails sinc-squared scaling, which fur-
`ther reduces the intensity of each diffraction order. With these two factors taken into account, the highest
`values of the nonuniform first-order diffraction efficiencies are expected to be 33%, 66%, and 77% for two-,
`four-, and and eight-level one-dimensional holograms with a 90% linear pixel fill factor. The variation of the
`first-order diffraction efficiency and the relative replay intensities were verified to within 1 dB by performing
`the optical reconstruction of binary phase-only holograms recorded onto a ferroelectric liquid crystal on a sili-
`con SLM. © 2001 Optical Society of America
`OCIS codes: 050.1950, 090.1760, 090.2890, 100.5090, 120.5060, 230.6120.
`
`INTRODUCTION
`1.
`In Part I,1 the effects of hologram illumination on the re-
`play beam profile, the on-beam-axis coupling efficiency,
`and the cross-talk isolation of a 4f holographic switch
`were presented. For dynamic holographic routing with a
`reconfigurable spatial light modulator (SLM), the conven-
`tional grating description in terms of its physical pitch
`length and analysis as a step-phase function2 are both
`cumbersome and inadequate:
`cumbersome because the
`physical parameters of a holographic replay are not nec-
`essarily required or known when one is analyzing a holo-
`graphic switch and inadequate because gratings (see Sub-
`section 2.A) form only a subset of all possible routing
`patterns with a pixelated SLM with inherent physical
`limitations. Another way to replay arbitrary hologram
`functions encoded onto SLM’s with pixel defects is to per-
`form a discrete Fourier transform (DFT; e.g., a fast-
`Fourier-transform algorithm3) on the zero-padded, over-
`sampled digital representation of the optical holograms.
`
`Clearly, the computing resources required for the DFT
`and for evaluating the efficiency of the main replay order
`and the potential cross talk of each noise order can be pro-
`hibitive for large hologram array sizes.
`Hence, here we treat the phase and the spatial effects
`that determine the intensity and location of each far-field,
`Fraunhofer diffraction order or peak (henceforth called
`replay order) separately in analytic forms. For the
`analyses we utilize the fractional hologram representa-
`tion described in Subsection 2.A.
`In the context of optical
`reconstruction of thin holograms, Subsection 2.B illus-
`trates the typically convoluted phase and spatial effects.
`The phase effect is concerned with quantization with a
`limited number of phase levels and the dynamic hologram
`patterns (i.e., the distribution of phase elements) within
`each hologram unit (or base hologram) at the digital
`synthesis–reconstruction stage. An equation that de-
`scribes the intensity of all the replay orders in an aliased
`digital replay (i.e., which comprises Dirac delta functions)
`
`0740-3232/2001/010205-11$15.00
`
`© 2001 Optical Society of America
`
`FNC 1009
`
`

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`206
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`J. Opt. Soc. Am. A / Vol. 18, No. 1 / January 2001
`
`Tan et al.
`
`It is useful to analyze the underlying
`tensity pattern.
`spatial
`frequency properties of a general efficiency-
`optimized hologram as the base case. Noise-optimized
`holograms, such as those generated by Gerchberg–
`Saxton,6 direct binary search,7 simulated annealing,8 or
`error diffusion9 algorithms, retain the fundamental char-
`acteristics of their efficiency-optimized counterparts.
`
`Fig. 1. Free-space 1 3 N optical switch with a coherent 4f
`setup. The size of each replay replication, DR, is fl/d, a conse-
`quence of pixel periodicity.
`
`is presented in Subsection 3.B. The locations of these re-
`play orders are to be found by use of a skip–rotate rule
`such that the intensity expression is applicable. These
`intensities are exactly as given by a DFT of the hologram.
`Recording the optical holograms on a dynamically re-
`configurable SLM entails a further drop in replay inten-
`sity as a result of pixelation and dead space as well as
`phase errors (henceforth called the spatial effect).
`Throughout this analysis, the paraxial optical reconstruc-
`tion region (an area close to the optical axis at the replay
`plane where scalar diffraction theory is valid) is divided
`into subsections of size fl/d centered at the optical axis,
`where f is the focal length of the transform lens, l is the
`wavelength of illumination, and d is the pixel pitch, as
`shown by the free-space holographic replay system in Fig.
`1. These subsections of equal sizes are a result of the
`pixel periodicity of the pixelated SLM and are called re-
`play replications. Within all replay replications, the dis-
`tribution of replay orders is identical, except that their in-
`tensities are scaled by a sinc-squared roll-off that arises
`from the pixel transmittance function. For the simplest
`case of a rectangular pixel aperture, the intensity roll-off
`expression is given in Subsection 4.A. The effects of SLM
`pixel imperfection were analyzed for grating replay by use
`of fixed surface-relief diffractive optical elements (DOE’s)
`etched onto fused silica4 and optical correlation.5
`In Section 5 experiments using dynamic holograms re-
`corded onto SLM’s for fiber-to-fiber interconnects are de-
`scribed. The results for the relative replay intensities of
`a particular hologram and the first-order replay efficien-
`cies of several holograms are shown.
`
`2. PROGRAMMABLE DIFFRACTIVE
`OPTICAL ELEMENTS
`Routing holograms are dynamically reconfigurable phase-
`only holograms that are used in holographic free-space
`beam steering applications. A single spot replay from ei-
`ther noise- or efficiency-optimized routing holograms is
`required for one-to-one switch interconnections. An
`efficiency-optimized computer-generated hologram, ob-
`tained by methods such as the inverse DFT, produces
`only one solution of the phase distribution over a 1-D or
`two-dimensional (2-D) hologram plane. All phase-shifted
`or spatially shifted variants of this distribution are ei-
`gensets of the original hologram, replaying the same in-
`
`uhˆ ~xd , yd!u2 5 (
`
`j,z d@xd 2 ~^nj 1 sN&!,
`hm;n
`
`A. Fractional Representation of Routing Holograms
`With the DFT analysis and a replay order notation simi-
`lar to that of Dammann,2 the numerical intensity replay
`of an m quantized phase-level, N 3 N grid-size hologram
`is a series of Dirac delta functions:
`!‘
`!‘
`(
`(
`s!2‘
`t!2‘
`n
`yd 2 ~^nz 1 tN&!#,
`where n 5 gm 1 1, g being an integer, is the order of re-
`play; (xd , yd) is the discrete coordinate at the simulated
`replay plane; (j, z) is the coordinate of the first replay or-
`j,z is the simulated intensity of the nth replay
`der; and hm;n
`order, given that the hologram encoding is limited to m
`phase levels. This expression follows directly from the
`property of the DFT whereby an N-point function is trans-
`formed into an N-point replay. Depending on the holo-
`gram pattern, one or more numerical replay coordinates
`will have nonzero intensity. These coordinates are given
`by modulo-N functions, ^nj 1 sN& and ^nz 1 tN&, such
`that
`
`(1)
`
`2N/2 < ~^nj 1 sN&, ^nz 1 tN&! , ~N/2!,
`
`(2)
`
`where n and (s, t) can take any values as long as the ele-
`ments of the set (^nj 1 sN&,^nz 1 tN&) are distinct.
`If
`( s, t) are now assigned as the normalized spatial fre-
`quency coordinates (henceforth called fractional coordi-
`nates) of the first replay order, (i.e., s 5 j/N, t5 z/N),
`criterion (2) becomes
`
`(3)
`
`
`2 12 < ~^ns 1 s&, ^nt1 t&! , 12 ,
`
`where ^ns 1 s& and ^nt1 t& are modulo-1 functions.
`The normalized coordinate of the first replay order is then
`rewritten as the simplest fraction between two rational
`numbers, i.e., s 5 x/x0 and t5 y/y0 , where x and y are
`integers and x0 and y0 are positive integers. All four in-
`tegers can be odd or even, as is m; and the denominators
`x0 and y0 may or may not be multiples of m. The grating
`is a special case for which the first-order replay fraction is
`(1/x0, 1/y0) for two dimensions [either (1/x0, 0) or (0, 1/y0)
`for 1-D routing], x0 and y0 are multiples of m, and Dam-
`mann’s efficiency expression [i.e., sin2(np/m)/(np/m)2; Ref.
`2] is applicable only for the 1-D cases without dead space.
`The ( s, t) base hologram requires only four integers for a
`complete representation of its form by x0 3 y0 sample
`points, its replay orders, and the intensities that charac-
`terize it.
`If the x0 3 y0 size base hologram is repeated on
`an SLM (with a high-pixel-count N 3 N array), generally
`in any noninteger multiples the number of addressable
`points is huge (@N2).10 Provided that there is adequate
`apodization (such as Gaussian) in the illumination, the
`optical replay of this hologram yields the first diffraction
`
`

`

`Tan et al.
`
`Vol. 18, No. 1 / January 2001 / J. Opt. Soc. Am. A
`
`207
`
`order at ( sf l/d, tf l/d) spatial frequency coordinates
`from the optical axis. Hence the fractional hologram rep-
`resentation eliminates the need to know the physical pa-
`rameters of the holographic replay system without com-
`promising any aspect of the hologram properties.
`
`B. Thin Holographic Elements
`A general overview of thin holographic elements for steer-
`ing an incoming beam to a single main order in the
`paraxial domain (i.e., when the incident angle and the dif-
`fraction angle are both small and the transverse feature
`of the diffractive element is much larger than the illumi-
`nation wavelength) is shown in Fig. 2. Both the micro-
`prism and the step-phase triangular gratings in Figs. 2(a)
`and 2(b) have the same physical pitch lengths (8d, where
`d is the feature size of the quantized phase steps). They
`steer the illumination to the desired fractional coordinate
`(1/8, 2/8, and 3/8 are shown) as a function of the phase
`depth of the unit element (2p, 4p, and 6p are shown).
`The difference is that in the latter case there are higher-
`order replications at multiple integers along the frac-
`tional coordinate (lighter arrows) as a result of phase
`quantization and spatial pixelation of the continuous
`phase-retardation profile in the former. Blazing the tri-
`angular phase profile for a higher order produces an arbi-
`trary steering function by exciting the required replay or-
`der (i.e., the main order diffracted to any of the x0
`locations for using x0 pixels per period). However, this
`
`presents severe difficulties in making surface-normal op-
`tical components with such considerable depths and num-
`bers of discrete phase steps, and the long optical paths ul-
`timately cause the grating function to depend on the
`polarization along and orthogonal to the grating line.
`One useful configuration of the step-phase grating blazed
`for a higher order is arrayed-waveguide gratings, for
`which the optical paths are laid out on a planar substrate
`and hence the path-length difference can be made several
`tens or hundreds of the center wavelength to disperse a
`broadband light to several locations.11
`In making surface-normal optical components one can
`exploit the modulo-2p property of the phase-retardation
`profile to wrap any quantized phase level beyond 2pback
`to the $0, 2p% range. Figure 2(c) shows the phase profiles
`of s 5 1/8, 2/8, 3/8 as well as their corresponding replay
`images.
`In most fabrication and encoding techniques for
`recording fixed DOE’s and programmable holograms on
`SLM’s, only a limited number of quantized phase levels m
`are available (m ! x0 , e.g., m 5 2 for symmetric replays
`and more for asymmetric replays).
`It is possible that
`phase quantization is out of sync with spatial pixelation;
`i.e., stepping from one pixel to the next does not involve a
`single phase step increase or decrease, as shown in Figs.
`2(d) and 2(e). Consequently, there are multiple diffrac-
`tion orders within the central replications, and similar
`but scaled replay orders that appear at other replications
`are a result of sampling the optical field at each pixel.
`
`(a) gratings composed of a microprism array,
`Fig. 2. Beam steering with periodic diffractive optical elements with a full period shown:
`(b) step-phase gratings blazed for the desired main replay order, (c)–(e) thin optical elements that have only limited numbers of phase
`levels m, and up to 2pphase depth. With each beam-steering technique the phase profile and the associated replay plane image within
`the two central replications are depicted for three routing fractions ( s 5 1/8, 2/8, 3/8).
`
`

`

`208
`
`J. Opt. Soc. Am. A / Vol. 18, No. 1 / January 2001
`
`Tan et al.
`
`The far-field amplitude of the (nx , ny)th diffraction or-
`der of a complex-amplitude hologram, H, is often assumed
`to be12
`
`x021
`(
`kx50
`
`y021
`(
`ky50
`
`x0 y0
`
`H~kx , ky!
`
`h~nx , ny! 5H 1
`3 expF 2pjS nxkx
`
`nyky
`y0
`x0
`3 $sinc~pnx /x0!sinc~pny /y0!%,
`
`1
`
`DGJ
`
`(4)
`
`Fig. 3. Overlap of higher replay orders in the numerical replay
`grid of binary gratings. FFT, fast Fourier transform.
`
`A. Replay Intensities of Grating Holograms
`For a general 1-D grating period x0 , the orders that ap-
`pear within the numerical replay grid as a result of phase
`quantization and the distribution of x0 phase elements
`with m phase states have an aggregated intensity (by
`summation of the intensities at overlapped positions):
`‘
`
`sinc2S np
`
`m
`
`D ,
`
`x0p
`m
`
`1 g
`
`s 5 (
`hm;n
`
`g52‘
`
`(5)
`
`(6)
`
`S
`
`D 2
`
`.
`
`s 5 sinc2S npD (
`
`hm;n
`
`where sinc(x) 5 sin(x)/x and the number of pixels in each
`unit element of the hologram is x0 3 y0 . The phase
`term, within the normalized double summation [the first
`set of braces in Eq. (4)], is periodic with respect to x0 and
`y0 and can be evaluated with a DFT algorithm. The
`double sinc term (in the second set of braces) accounts for
`the intensity scaling that is due to pixelation without
`dead space. The orthogonal phase and spatial effects de-
`termine the replay intensity and location of each holo-
`gram replay order. The replay can be further simplified
`for routing holograms that steer the light to a single main
`order.
`
`3. PHASE EFFECT IN DIGITAL
`REPRESENTATIONS OF ROUTING
`HOLOGRAMS
`In a numerical DFT grating replay in which only a single
`sampling point for each pixel state is used, an aliased nu-
`merical replay will always result, as is consistent with
`Nyquist’s sampling criterion. For example, any integer
`multiples of a binary {0, p} phase grating unit will give a
`replay of 1 unit intensity at the first numerical replay
`point, corresponding to a 21/2 fractional coordinate, as il-
`lustrated in Fig. 3(a). Because two sample points are
`used for each grating period, the highest-frequency com-
`ponent that can be replayed by the DFT is the fundamen-
`tal frequency.
`If the fractional coordinate of the first re-
`play order is now 21/8 with the use of a binary
`{0 0 0 0 pppp% phase grating unit, the four times over-
`sampling gives up to four times the fundamental fre-
`quency replayed in the same output plane, resulting in
`four peaks, which appear at 61 and 63 orders. These
`two grating functions and their replays as well as those of
`s 5 61/4 and 61/16 are shown in Fig. 3.
`The higher replay orders, exceeding 61/2 of a simu-
`lated hologram replay grid, should not be thought to be
`spilling over to the higher replay replications, as each
`phase element is sampled only by a single point in the nu-
`merical DFT and no normalized spatial frequencies of
`,21/2 or > 1/2 are possible in the numerical replay.
`In-
`stead, those higher replay orders that overflow the 61/2
`fractional coordinate at one end must be rolled back into
`the same numerical replay grid through the opposite end.
`It is possible that these contributions from infinite higher
`replay orders overlap exactly the main order or other
`higher noise orders that are present in the numerical re-
`play grid.
`
`where n denotes the replay order and g is an integer.
`s :
`The summation gives the intensity, hm;n
`‘
`n
`n 1 gx0
`By use of the residue method, the convergent series sums
`to
`
`
`
`m
`
`g52‘
`
`s 5
`hm;n
`
`.
`
`(7)
`
`sinc2~np/m!
`sinc2~np/x0!
`Equation (7) gives an accurate description of the intensity
`of each frequency component that is present in the nu-
`For the s 5 61/2, 61/4, 61/8,
`merical replay grid.
`61/16 binary phase gratings shown in Fig. 3, the first-
`order intensities are
`
`s51/2
`hm52;n561
`
`5
`
`s51/4
`hm52;n561
`
`5
`
`s51/8
`hm52;n561
`
`5
`
`sinc2~p/2!
`sinc2~p/2!
`
`sinc2~p/2!
`sinc2~p/4!
`
`sinc2~p/2!
`sinc2~p/8!
`
`5 100%,
`
`5 50%,
`
`5 42.68%,
`
`s51/16
`hm52;n561
`
`5
`
`sinc2~p/2!
`sinc2~p/16!
`
`5 41.05%,
`
`(8)
`
`

`

`Tan et al.
`
`Vol. 18, No. 1 / January 2001 / J. Opt. Soc. Am. A
`
`209
`
`respectively. The number of orders that appear within
`the N numerical replay points is x0 /m, with a minimum
`of 1 regardless of m values. As an example, optimization
`of a four-level phase hologram to route to a 21/2 frac-
`tional coordinate will always produce a binary grating.
`Although x0 is 2 and m is 4 in this case, it should really be
`considered m8 5 2, because that is the underlying holo-
`gram property for this routing. For a grating that con-
`sists of as many encoding elements as there are phase lev-
`els (e.g., s 5 1/4, m 5 4; s 5 1/5, m 5 5, etc.), the
`grating’s numerical replay will always contain a single or-
`der of 100%. This means that the phase effect produces
`100% efficiency [x0 5 m in Eq. (7)].
`In the actual grating encoding that uses rectangular- or
`square-aperture pixels and optical grating replay by the
`use of a transform lens, each harmonic frequency compo-
`nent will be revealed with appropriate intensity scaling
`by the transform of the pixel aperture function.
`In the
`ideal situation in which the grating is an infinite repeti-
`tion of an m-level phase ramp, the pixel aperture is rect-
`angular or square without dead space, and the grating il-
`lumination is an infinite-expanse plane wave, the reverse
`process of the summation in Eq. (5) takes place. The spa-
`tial term of a grating effectively cancels out the denomi-
`nator of Eq. (7) to yield Dammann’s expression.2 Decou-
`pling the phase effect
`from the overall efficiency
`expression is important because many arbitrary 2-D holo-
`gram patterns in addition to 1-D gratings are often re-
`quired, the parameters of the intensity scaling term are
`dependent on the recording device, and the spatial effect
`in general does not cancel out the denominator of Eq. (7)
`to yield Dammann’s expression of hologram replay effi-
`ciency.
`
`B. Replay Intensities of General Holograms
`At the digital stage, the numerical replay field of a gen-
`eral 1-D phase-only hologram of a s 5 x/x0 first-order re-
`play fraction contains the same fundamental properties
`as the 1/x0 phase-only hologram. The relative intensities
`of all the replay orders that are present are redistributed
`according to which harmonic of the basic 1/x0 hologram is
`excited. Extending the analysis to general 2-D holo-
`grams, we use the least-common multiple of m, x0 , and y0
`
`Fig. 4. Modulo-1 skip–rotate rule used to locate higher replay
`orders of a s 5 1/10 quaternary hologram replay fraction.
`
`[Eq. (7)] to enhance the intensity of each overlapped loca-
`tion within the numerical replay grid:
`
`s,t 5
`hm;n
`
`sinc2~np/m!
`sinc2@np/lcm~m, x0 , y0!#
`
`,
`
`(9)
`
`where lcm(m, x0 , y0) takes the least common multiple of
`m, x0 and y0 , either x0 or y0 is assumed 1 for general 1-D
`holograms, and m > 2. There are lcm(m, x0 y0)/m non-
`zero intensity orders within x0 3 y0 points in the numeri-
`cal replay grid.
`It is important that 2-D Fourier (con-
`tinually shifted) phase-only holograms rather
`than
`crossed holograms be replayed. The crossed phase-only
`holograms can be considered to be producing independent
`routings in the x and y directions, giving many additional
`orders in the numerical replay grid.
`
`C. Replay Locations of General Holograms
`One can predict the location of each replay order by fold-
`ing higher replay orders with fractional
`locations of
`,21/2 or >1/2 back to the numerical replay grid.
`Iden-
`tifying the location of each order (or equivalently the or-
`der number at each nonzero location) enables the intensi-
`ties to be calculated from Eq. (9). The modulo-1 skip–
`rotate rule is illustrated in Fig. 4 for a 1-D numerical
`replay with s 5 1/10 and m 5 4.
`It has been assumed
`that the phase-matching condition is satisfied (i.e., that
`there is a 2p(m 2 1)/m phase depth between the lowest
`and the highest phase levels.
`The key to obtaining the correct number of n orders
`that appear in the numerical replay grid is to draw an
`ideal saw-toothed blaze along the n axis passing through
`n 5 0 and the center of the 1-D numerical replay grid
`with a slope of x and a period of 1/s. The orders that do
`appear in the numerical replay grid have decreasing in-
`tensities in accordance with Dammann’s criterion, n
`5 gm 1 1,
`where
`g 5 0, 21, 11, 22, 12, 23, 13... .
`Each integer g is taken successively until a high order be-
`gins to overlap one of the existing replay orders (i.e., only
`the lowest lcm(m, x0)/m orders at 6 frequencies need to
`be considered). The overlap of a higher replay order with
`an existing replay order occurs only after every
`lcm(m, x0)-order separation, or 20 separations for the ho-
`logram shown. We take the example of n 5 27 order; its
`replay location (i.e., n 3 s or 27/10) is folded by the ideal
`saw-toothed blaze to 3/10 along the unit replay grid. The
`intensity of this aliased replay peak has contributions
`from n 5 27, 13, 227, 33,... . Thus the aggregate inten-
`sinc2(27p/4) 1 sinc2(13p/4) 1 sinc2(227p/4)
`sity
`is
`1 sinc2(33p/4) 1 ..., or 2.52%, as given by the infinite
`sum in Eq. (9) with y0 5 1.
`Extending the skip–rotate rule to the replay of 2-D
`Fourier holograms, we apply the modulo-1 function to
`both the x and the y directions, resulting in an (xi 1 yi)
`base vector and its multiples folding back to the $21/2,
`1/2% unit replay grid from top to bottom and right to left,
`and vice versa. The locations of the first ten replay or-
`ders for a (1/10, 3/8) base hologram are shown in Fig. 5.
`
`

`

`210
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`J. Opt. Soc. Am. A / Vol. 18, No. 1 / January 2001
`
`Tan et al.
`
`H~xh , yh!
`
`‘
`
`p52‘
`
`‘
`
`(
`
`q52‘
`
`5HF (
`d~xh 2 p, yh 2 q! 3 Hˆ ~ p, q!G
`D 1F (
`* arect S xh
`3 bexp~ jc!G *F 1 2 rect S xh
`
`yh
`r
`
`,
`
`r
`
`‘
`
`p52‘
`
`‘
`
`(
`
`q52‘
`
`yh
`r
`
`,
`
`r
`
`d~xh 2 p, yh 2 q!
`
`DGJ 3 Hi~xh , yh!,
`
`where ( p, q) denote the summation variables of the sam-
`pling pulse train and * denotes the convolution operator.
`The replay is the optical Fourier transform of Eq. (10),
`which yields
`
`(10)
`
`Am;nar2 sinc@rp~ p 1 ^s 1 ns&!#
`
`(n
`(s
`
`,t
`
`h~xr , yr!
`
`5 H (
`
`p,q
`
`3 sinc~rp@q 1 ^t 1 nt&!# 1 b~1 2 r2!exp~ jc!
`2 bexp~ jc!( 8
`( 8
`
`p
`
`q
`
`r2 sinc~rpp!sinc~rqp!J
`
`(11)
`
`* hi~xr , yr!,
`where uAm;nu2 is equivalent to hm;n
`s,t , and the primes on
`the second summation indicate that p and q can take any
`integer values but not zero simultaneously. Assuming
`that the hologram aperture is large and that the trans-
`form of the hologram illumination is a delta function, fur-
`ther simplification yields the replay intensity of nonzero
`(n (cid:222) 0) orders:
`
`uh~xr , yr!m;n(cid:222)0u2 5 (
`
`n
`
`‘
`
`(
`
`p52‘
`
`‘
`
`(
`
`q52‘
`
`s,t;p,q
`hm;r;n(cid:222)0
`
`,
`
`(12)
`
`
`
`s,t is as given bywhere hm;r;n(cid:222)0s,t;p,q 5 hm;n(cid:222)0s,t 3 hr;n(cid:222)0s,t;p,q , hm;n
`
`
`
`
`
`Eq. (9), and
`
`Fig. 6. 1-D dynamic hologram recording as depicted by the con-
`volution of each calculated value of an infinitely repeated base
`hologram with the clear aperture transmittance of a single SLM
`pixel. A constant dead-space transmittance is appended to the
`pixel (inset), and the resultant spatial distribution is then mul-
`tiplied by the finite hologram illumination. The reconstructed
`optical beam has its profile determined by this finite illumina-
`tion, whereas the distribution of the diffractive power is depen-
`dent on both the hologram and the pixel functions.
`
`Fig. 5. First ten orders of a (1/10, 3/8) quaternary base holo-
`gram. Diamonds, locations of the orders present, with the num-
`ber of contour lines proportional to the intensity values; circles,
`absent orders. These diamonds do not represent the beam pro-
`file of the replay peaks.
`
`4. SPATIAL EFFECT IN OPTICAL
`RECONSTRUCTION OF ROUTING
`HOLOGRAMS
`Dynamically reconfigurable routing holograms are re-
`corded onto pixelated SLM’s. Each SLM pixel contains a
`clear aperture and some dead space. The sampling of the
`optical field by the pixel aperture, its ensuing multiple re-
`play replications, and further loss due to the nonunity fill-
`factor are appended to the expression that describes re-
`play intensities. The nonunity fill factor merely causes
`additional power loss without altering the reconfigurable
`spectrum of the diffractive power. The pixel pitch has
`been normalized to 1, and its geometry is taken to be a
`square, to free up the dependence on the physical param-
`eters as in Section 3. The ratio of the clear aperture to
`the pixel pitch is r in both the x and the y directions, as
`shown in Fig. 6; aand bare the amplitudes of complex
`transmittance in the pixel and in the dead space, respec-
`tively.
`
`A. Replay of Pixelated Holograms
`The hologram written onto a SLM is the convolution of
`the pixel transmittance function (in the pixel clear aper-
`ture and in the dead space) and the calculated hologram
`or constant dead-space modulation at each pixel. The
`N-point 1-D SLM contains a number of hologram repeats
`(i.e., repetition of a base hologram), as shown in Fig. 6.
`This repetition can be assumed to be infinite, since the il-
`lumination profile within the hologram aperture,
`Hi(xh , yh), has been taken into account.1 Mathemati-
`cally, the 2-D pixelated hologram replay is represented as
`follows:
`
`

`

`Tan et al.
`
`Vol. 18, No. 1 / January 2001 / J. Opt. Soc. Am. A
`
`211
`
`s,t;p,q 5 (
`hr;n(cid:222)0
`
`s,t
`
`a2r4 sinc2@rp~ p 1 ^s 1 ns&!#
`
`3 sinc2@rp~q 1 ^t 1 nt&!#.
`
`(13)
`
`A simple interpretation of Eqs. (12) and (13) is that the
`power of each replay order that is due to the phase effect
`alone will be distributed among the replicas centered at
`multiples of the normalized spatial frequency. Within
`each ( p, q) replay replication, the locations of the replay
`orders are identical and are similar to those that are due
`to the phase effect alone, but their intensities are scaled
`differently by the sinc-squared spatial effect term.
`The scaling by Eq. (13) is inaccurate for very high
`spatial-frequency orders, as Fraunhofer scalar wave
`propagation is invalid. Nevertheless, for typical dynamic
`holograms recorded onto several micrometer-size pixels,
`knowing the intensity and the location of the main replay
`order of a digital hologram is sufficient for predicting the
`optical hologram replay intensities within the central and
`several lower replay replications.
`In a system implemen-
`tation, only the replay orders that fall within the central
`replay replication are of any use. The replay intensities
`within the central replay replication can be described by
`
`G
`
`play locations close to the central zero order, e.g., within
`60.15 fractional coordinate. When the fill factor is re-
`duced because of the presence of dead space, the central
`region where the spatial effect has nearly uniform scaling
`becomes larger, at the expense of reduced replay inten-
`sity. For SLM without dead space (r 5 1), the repli-
`cated replay orders in 6(0.5 ! 1.0) normalized replay co-
`ordinate are much lower than to those in the central
`replay replication. When the fill factor becomes non-
`unity, the replay intensities in these regions are boosted.
`
`C. Upper Bound of the First-Order Diffraction
`Efficiency
`The most important characteristic of a discrete hologram
`replay is the predictable replay intensity of each ( s, t)
`base hologram, with the account phase and spatial effects
`taken into account. The 2-D expression that predicts the
`first-order (n 5 1) replay intensity of an m-level ( s, t)
`base hologram is
`
`s,t;p50,q50 5
`hm;r;n51
`
`sinc2~p/m!
`sinc2@p/lcm~m, x0 , y0!#
`3 a2r4 sinc2~rsp!sinc2~rtp!.
`
`(15)
`
`The first-order diffraction efficiency of 1-D holograms is
`plotted in Fig. 8 against the main replay fraction s with
`these assumptions: a5 1, r 5 0.9, and m 5 2, 4, and 8.
`Note that within the window of the 60.15 normalized co-
`ordinate, the binary, quaternary, and octonary holograms
`have approximately 33% 66% and 77% (or 24.8, 21.8,
`and 21.1-dB) first-order diffraction efficiencies, respec-
`tively.
`If the first-order replay fractions are chosen close
`to the edges of the central replay replication (e.g., 61/2),
`there is an additional 3.1-dB loss owing to the spatial ef-
`fect with r 5 0.9. For all first order replay fractions, the
`replay intensity of the quaternary hologram is twice that
`of the binary hologram replay except for s 5 61/2 binary
`gratings. Similarly, the replay intensity of the octonary
`
`Fig. 7. Composite effects of spatial and phase quantization on
`the intensity of a phase-matched 3/8 quaternary replay fraction.
`Small open circles, the nonzero replay orders within the central
`replay replication along the intensity roll-off (dotted curves). 3,
`intensity of the central zero order. Dashed curve, intensities of
`the zero orders in other replications.
`(n, p), order and replica-
`tion number.
`
`sinc2F
`
`(s
`
`,t
`
`s,t;p50,q50 5 (
`hm;r;n(cid:222)0
`
`n
`
`sinc2~np/m!
`np
`lcm~m, x0 , y0!
`3 a2r4 sinc2@rp~^s 1 ns&!#
`
`3 sinc2@rp~^t 1 nt&!#.
`
`(14)
`
`Note the remarkable similarity to the diffraction of an
`N-slit diffraction grating13 when the hologram’s phase
`and spatial effects are treated separately.
`
`B. Numerical Simulations of 1-D Hologram Replay
`The composite sinc-squared scaling for various replay or-
`ders across all replay replications can be shown to give ac-
`curate intensity values by numerical simulations. The
`1-D version of Eq. (14) is applied to the replay of a phase-
`matched quaternary phase hologram with its first replay
`order at s 5 3/8. The pixel aperture transmittance is as-
`sumed to be unity; the deadspace is transparent and com-
`prises ;28% of a pixel. Each point of the calculated base
`hologram is represented by 32 samples with 9 samples as-
`sumed to be constant transmittance.
`There is a separate composite sinc-squared scaling for
`each replay order n, as shown in Fig. 7. The first-order
`intensity has been reduced to 34.53% from 85.36% after
`the phase effect (with exaggerated dead space to show
`higher replication orders with some clarity). However,
`the third-order intensity at 21/8 is reduced only slightly,
`from 14.64% to 7.37%.
`In addition, the first replay order
`that belongs to the 21 replay replication system is quite
`pronounced at 21.61%. This and all other replay orders
`at multiples of integers from the first replay order in the
`central replay replication originate from the phase-effect-
`only first replay order. The scaling that is due to the spa-
`tial effect, for replay orders located far away from the op-
`tical axis (zero order in the central replay replication), is
`quite severe. Therefore it is advantageous to choose re-
`
`

`

`212
`
`J. Opt. Soc. Am. A / Vol. 18, No. 1 / January 2001
`
`Tan et al.
`
`14) SLM used in these preliminary experiments was filled
`with CDRR8 FLC.15 The switching time and tilt angle of
`a device filled with this FLC mixture are 500 ms and
`;34°,16 respectively, at 25 °C. The SLM had ;220 (4.4-
`mm) continuous pixels with ;2/3 of a pixel length (4.5
`mm) working in a section of the array.
`
`A. Replay Plane Mapping
`For any realistic comparison with diffraction theory, de-
`termination of SLM dead space is of the utmost impor-
`tance. With the SLM blanked high and low at ;1 kHz so
`as to minimize FLC scattering, several replay orders that
`were due to dead space and pixel reflectance were mapped
`with the fiber-scan setup. With the above-mentioned l, f,
`and d, the separation between two successive replay or-
`ders is ;1.24 m