spatial bandwidth analysis of fast backward fresnel …...diffraction within the paraxial...

$: Spatial bandwidth analysis of fast backward Fresnel …...diffraction within the paraxial approximation is jus-tified for referenceless holography. Thus, diffraction calculations related$
Spatial bandwidth analysis of fast backward Fresneldiffraction for precise computer-generated

hologram design

Jinyang Liang1,2 and Michael F. Becker2,*1Department of Biomedical Engineering, Washington University in St. Louis, St. Louis, Missouri 63130, USA

2Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas 78712, USA

*Corresponding author: [email protected]

Received 2 April 2014; revised 25 June 2014; accepted 27 June 2014;posted 27 June 2014 (Doc. ID 209345); published 31 July 2014

Designing near-field computer-generated holograms (CGHs) for a spatial light modulator (SLM) requiresbackward diffraction calculations. However, direct implementation of the discrete computationalmodel of the Fresnel diffraction integral often produces inaccurate reconstruction. Finite sizes of theSLM and the target image, as well as aliasing, are major sources of error. Here we present a new designprescription for precise near-field CGHs based on comprehensive analysis of the spatial bandwidth.We demonstrate that, by controlling two free variables related to the target image, the designed hologramis free from aliasing and can have minimum error. To achieve this, we analyze the geometry of thetarget image, hologram, and Fourier transform plane of the target image to derive conditions forminimizing reconstruction error due to truncation of spatial frequencies lying outside of the hologram.The design prescription is verified by examples showing reconstruction error versus controlled param-eters. Finally, it is applied to precise three-dimensional image reconstruction. © 2014 Optical Society ofAmericaOCIS codes: (090.1760) Computer holography; (050.1940) Diffraction.http://dx.doi.org/10.1364/AO.53.000G84

1. Introduction and Background

Holography has been used widely in many areas [1],including optical trapping and tweezing [2], laserbeam shaping [3], and biomedical imaging [4]. Toproduce a high-quality holographic image at a cer-tain distance, spatial amplitude and phase of thelight must be precisely controlled. Modern spatiallight modulators (SLMs) are becoming the standarddevice to create arbitrary complex wavefronts bymodulating the wave magnitude and/or phase. Meth-ods exist that utilize two modulators to achieve fullycomplex modulation [5–7], and some schemes canachieve complex modulation utilizing a single SLMwith perhaps some trade-offs such as aperture

sharing [8–10]. Having complex wave modulationreadily available, the design of a desired near-fieldcomputer-generated hologram (CGH) becomeslargely a matter of calculating the inverse diffractionfrom the target image back to the SLM. Backwardpropagation calculations can be performed using ex-act point-by-point algorithms [11,12] or various im-plementations of the Rayleigh–Sommerfeld (R-S)diffraction formula [13–18]. In addition, given thepixel size of current SLMs (typically 7–20 μm),diffraction within the paraxial approximation is jus-tified for referenceless holography. Thus, diffractioncalculations related to SLMs also can be performedwith the Fresnel diffraction formula using a singleFourier transform (FT) [19–22]. Taking advantageof the discrete Fourier transformation (DFT), a singleFT implementation of the Fresnel diffraction inte-gral (FDI) possesses a much faster computation

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speed, as compared with direct implementations ofthe R-S formula, and can have accuracy comparableto a R-S computation within the Fresnel region [23].

However, direct implementation of these computa-tional methods can yield inaccurate hologramreconstruction results. The first problem relates tosizing the two regions: SLM and target image. Fora finite-sized target image, its spatial bandwidthdetermines the size of the hologram generated frombackward propagation. Quite often, this area islarger than the finite size of the SLM, resulting intruncation of spatial frequency content. Those spa-tial frequency components that miss the SLM cannotbe reproduced by the CGH encoded onto the SLMand are therefore lost in the reconstruction process.Consequently, the reconstruction cannot reproducethe original target image. Second, when the back-ward FDI is digitized so that the DFT can be usedfor numerical computations, the possibility of errorsresulting from aliasing also arises. Previous work byLiu [21] discussed aliasing when using the forwardFDI with a fixed bandwidth source image. However,this work cannot be readily applied to backwardpropagation to design a precise CGH for two mainreasons. First, boundary conditions for the SLM (pri-marily its finite dimension; see Section 2 for a com-plete specification of the boundary conditions) werenot considered in a backward diffraction calculation.Second, the target image bandwidth is now a freevariable, and it will affect error and the settingof other parameters. For backward diffractioncalculation, either of these reasons could causesignificant error in hologram design and in the recon-structed image.

Our objective in this paper is to analyze andminimize these limitations and error sources, so asto synthesize a desired wavefront for a referencelessFresnel hologram using backward propagation froma target image. This design process needs to becomputationally fast and efficient. The designedhologram can then be encoded onto an appropriateSLM, and the reconstructed holographic imageshould have minimum error. We solve the problemof sizing the SLM and target image by requiringthe target image to be accurate in irradiance but withits phase as a free variable. A quadratic phase isadded to the target image, so that, when a hologramis designed, its diffracted light fills the target imageregion with minimum error. We present a compre-hensive spatial bandwidth analysis of hologramdesign using backward FDI, which allows us to quan-tify the resulting hologram reconstruction errorsversus both target image bandwidth and quantityof added quadratic phase. The result is a design pre-scription for precise hologram generation with mini-mum reconstruction error. Example simulations areshown that demonstrate practical hologram designwithin specific limits.

This paper is organized as follows. In Section 2, wetake a detailed look at the boundary conditions of thebackward diffraction problem. First, the relationship

between the target image bandwidth, sizes of theSLM and target image, and their separation distanceis examined. Second, we derive the conditions foradding quadratic phase to the target image as a freevariable in order to properly scale between the twoplanes and to reduce error in the reconstructedimage. Third, the geometry of the light path fromthe target image, past the SLM, and onto the FTplane of the target image allows the path of particu-lar spatial frequency components to be traced. Weperform this geometrical analysis, which identifiesthe regions of the target image that experience trun-cation of some spatial frequency components, andderive conditions that minimize error in the recon-structed images. Last, the conditions to avoid alias-ing are derived. We will show that the conditions forminimum error due to truncation of spatialfrequency components at the SLM plane are morestringent than the conditions to avoid aliasing.Section 3 is devoted to numerical simulations wherethe backward diffraction method to design the holo-gram is verified and the errors are quantified. First,the backward diffraction routine derived in Section 2is validated against an exact point-by-point diffrac-tion computation. The major content of this sectionexamines hologram designs for different cases ofoptical layout. The root-mean-square (RMS) errorcalculated for each reconstructed image and thepoint of abruptly increasing error are compared withthe theoretical predictions of Section 2. A discussionthat summarizes our findings concludes the paper inSection 4.

2. Principles of Hologram Generation by BackwardPropagation

We first consider the problem in the continuous,scalar-wave domain. As illustrated in Fig. 1, plane-wave light, of wavelength λ, illuminates a pixelatedSLM in the z � 0 plane. The SLM is Lo × Lo in sizewith a square pixel pitch of Δo and a pixel numberof Mo ×Mo. The closest possible hologramreconstruction plane is equal in size to the SLM.At this minimum distance, every image point canbe reconstructed by a properly sampled Fresnelphase encoded lens at the SLM hologram plane

Fig. 1. General forward diffraction problem of generating adesired target wavefront (amplitude and phase) by propagationfrom an SLM to the target plane.

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(i.e., R-S scalar diffraction). In other words, there isno vignetting of the reconstructed image. This mini-mum distance is defined as twice the critical length,zc, defined by

zc �MoΔ2

o

λ: (1)

Previous authors have introduced the critical lengthand used 2zc to define the closest distance to employthe FDI method [14,21]. Subsequently, Davis andCottrell also identified zc as the minimum Nyquistsampled focal length of a phase-encoded lenscentered on axis [24].

However, the problem illustrated in Fig. 1 is some-what ill posed. The difficulty arises from the readilyapparent boundary conditions at the target imageand SLM planes; i.e., each is bounded in lateral ex-tent to a region of interest that will later be sampledfor digital computations. Specifically, for forward dif-fraction calculations from the bounded SLM plane,the extent of the diffraction solution, which stayswithin the right half plane from −π∕2 to �π∕2 indiffraction angle, extends outside the bounded targetimage, as shown in Fig. 1. When the SLM is pixe-lated, the referenceless hologram reconstructedimage is located at the zeroth diffraction order.Due to a relatively large pixel size, the majority ofthe zeroth-order diffracted light stays in a relativelysmall area. Strictly speaking, however, because of thesharp pixel edge, the diffracted light contains highspatial frequencies. This high-spatial-frequencylight, although weak, diffracts at large angles outsidethe bounded target image. This is most easilyvisualized using the R-S formulation, where the dif-fracted field is given by a convolution of the boundedwave, leaving the SLMwith the unbounded sphericalphase kernel. For the forward diffraction calculation,the area outside the region of interest is ignored as a“don’t care” region. Furthermore, contributions tothe convolution integral far from the center of thekernel are insignificant, and a practical implementa-tion with minimum energy outside the region ofinterest can be formulated [23]. For a paraxial dif-fraction problem that employs the Fresnel approxi-mation, bounded support at the SLM and targetimage planes creates a similar problem. But again,energy outside the region of interest can be mini-mized by correctly sizing the observation regionand/or source function bandwidth to fit the problem.

Fresnel diffraction may be expressed using the FTintegral as

U�x; y� �ejkz

jλzej

k2z�x2�y2�

ZZ �∞

−∞

�U�ξ; η�ej k2z�ξ2�η2�

�e−j

2πλz�xξ�yη�dξdη;

(2)

where k � 2π∕λ. The source amplitude distribution isgiven by U�ξ; η�, and the diffracted field at a distancez is given by U�x; y�. This forward Fresnel transform

can be expressed compactly using Fourier operatornotation as

U�x; y� � ejkz

jλzej

πλz�x2�y2�F

�U�ξ; η�ej πλz�ξ2�η2�

�: (3)

In Eq. (3), Ffg denotes the forward FT, and the spatialfrequency variables are given in terms of targetplane coordinates �x; y� as f x � x∕λz; f y � y∕λz. Wemake the following definitions for the quadraticphase or chirp functions: the term, ejπ∕λz�ξ2�η2�, withinthe FT is denoted as ϕ1�ξ; η�, and the premultiplierterm, ejπ∕λz�x2�y2�, is denoted as ϕ2�x; y�.

When using Eq. (3) for forward diffraction, thesource function U�ξ; η� has bounded support and sodoes the product of U�ξ; η� and ϕ1�ξ; η�. This productfunction cannot have a bounded spectrum (i.e.,bounded FT) and thus cannot be rigorously ex-pressed in a bounded region at the target imageplane. In practice, limiting the bandwidth of thesource function narrows the maximum diffractionangle, so that the source and diffraction regions ofinterest can be bounded and produce minimal error.The error arises from the fact that the source cannotbe band limited without making its spatial extent in-finite. To solve this problem in practice, the usual ap-proach is to apply an infinite impulse response filterto the source function that leaves support for thespectrum unbounded but still confines the majorspectral energy to a usefully limited spectral region.Truncation at the edge of the diffraction region ofinterest and disregarding the lost energy can nowresult in a tolerably insignificant error.

To design the hologram that will be coded onto theSLM to reconstruct the target image, the problem isreversed, and backward diffraction is used to deter-mine the desired light field at the SLM plane. Forbackward propagation, Eq. (3) is inverted [25] as

U�ξ;η��−jλz�ejkzej πλz�ξ2�η2�F−1

�U�x;y�ej πλz�x2�y2�

��z<0�;

(4)

where F−1fg denotes the inverse FT, and the spatialfrequency variables are given in terms of the SLMplane coordinates �ξ; η� as f ξ � ξ∕λz; f η � η∕λz.

As the target function and SLM are necessarilyfinite in size, the backward diffraction solution willextend beyond the SLM edges due to the samebounded support issue as faced by forward diffrac-tion. Thus, in practical design problems, the band-width of the target function must be subjected to abandwidth-limiting process. In order to have a trac-table target function, the hard edges of the imagealso must be low-pass filtered and zero padded. Animportant difference for hologram design by back-ward diffraction, as opposed to forward diffraction,is that some important information may extendbeyond the SLM boundary, and this information isnecessary for rigorously accurate target function


reconstruction when the SLM is illuminated. Han-dling and bounding this hologram reconstructionerror is an important result to be presented inthis paper.

A. Sizing the Hologram

Based on the discussion above, the first aspect ofhologram design, which can be dealt with in thecontinuous domain, is the sizing of the SLM andthe target image with respect to each other and theirseparation. For backward diffraction, a phase-flattarget image illuminated by a plane wave will notdiffract back to a smaller source area. A quadraticphase, equivalent to a positive lens in the Fresnelapproximation, is required in the target image. Thisrequirement is illustrated in Fig. 2, and it is equiv-alent to illuminating a phase-flat target image with aconverging spherical wave. An FT plane will exist atthe center of curvature of this converging wave. Inaddition, if this phase function is in the form of a pos-itive lens, it is opposite in sign to ϕ2�x; y�. By adjust-ing the quadratic phase in U�x; y�, the phase of theproduct fU�x; y�ϕ2�x; y�g can be reduced to zero orto a small residual quadratic phase of either a posi-tive or negative sign.

Figure 2 also illustrates the geometrical opticalpath of the highest spatial frequency in the target im-age. The SLMs shown in Fig. 2 are sufficiently wide,as to not truncate any of this high spatial frequencylight. When the SLM is encoded based on this com-puted backward-propagated light field, all the lightat this spatial frequency can be diffracted by theSLM to the reconstructed image plane. The heavysolid lines in Fig. 2 illustrate the minimum size ofthe SLM to avoid high-frequency truncation. Twogeneral cases are illustrated: the SLM to the rightof the FT plane (requiring quadratic phase radius ofcurvature larger than the separation between theSLM and target planes), and the SLM to the leftof the FT plane (requiring a radius of curvaturesmaller than that separation).

To fit the backward diffraction field onto the finiteSLM area, we introduced a quadratic phase in thetarget image. The term in Eq. (4) to be inverseFourier transformed is given by

U�x;y�ϕ2�x;y��UR�x;y�exp��j

π

λ�zf zc�azc��x2�y2�

�

×exp�−j

π

λzf zc�x2�y2�

�; (5)

where a dimensionless propagation distance zf is de-fined as zf � jzj∕zc. The target image amplitude isnow the explicitly real function UR�x; y�. The param-eter a controls the curvature of the quadratic phaseadded in the target image. The sign of a can be either� or −, and, for a � 0, the quadratic phase added tothe target image exactly cancels the phase of ϕ2�x; y�.In this case, backward Fresnel propagation reducesto the inverse FT of the real target image.

Although placing the SLM at the FT plane is at-tractive because it has the minimum required sizefor the SLM (Fig. 2), it is difficult to implement inpractice. The strong DC component [26] of the FT re-quires a very large dynamic range for the light at thisplane and for the amplitude modulation capability ofthe SLM. For practical reasons, the SLM is bestplaced a short distance away from the FT plane toalleviate this dynamic range problem. Thus, theparameter a should be small, to avoid high-frequencytruncation, but not be exactly zero.

The truncation effect in Fresnel holograms wasdiscussed previously by Stern and Javidi [27]. Fromthe spatial-frequency perspective, the spectrum ofthe target image was low-pass filtered by thehard-edged aperture function, resulting in error inthe reconstructed image. Without phase added tothe object, the aperture effect was simple and spaceinvariant. In contrast, we introduced an additionalquadratic phase to the object that interacted withϕ2�x; y� to alter the overall phase profile. For thiscase, the effect of truncation on the reconstructed ir-radiance image is spatially dependent, since UR�x; y�with the added phase being low-pass filtered andconverted to irradiance. Phase was added to the tar-get image in order to scale the diffracted field to theSLM aperture. It also allows us to minimize the trun-cation problem and therefore ensure more preciseimage reconstruction.

We now proceed to analyze the requirements toavoid high-frequency truncation using geometricalanalysis. The lateral size of the target image andthe hologram are denoted by Lw and Lo. The boundson a to avoid high-frequency truncation can be de-rived using similar triangles shown in Fig. 3 for pos-itive a and negative a. For both cases, similartriangles have the common solid (green) and dashed(red) sides. The heights of the similar triangles are hand Lw∕2� Lo∕2 for the smaller and larger triangles,respectively. At the FT plane, the highest spatialfrequency in the target image diffracts to height x0

Fig. 2. Effect of a quadratic phase in the target image expressedas an equivalent lens. Backward propagation results in an FTplane located at the phase center of curvature. Two of the manypossible SLMplanes are shown that accommodate the highest spa-tial frequency light. These possible SLM sizes are bounded by theheavy solid lines in the figure that connect to the highest spatialfrequency (�f h) locations in the FT plane (red dots).


from the z axis. The value of amust allow the dashed(red) critical ray for the highest spatial frequency toreside within the SLM area.

From geometry, the condition on a to avoid highspatial frequency truncation is given by

jazcj ≤zf zc

Lw∕2Lo∕2−zf zcθmax

∓1; (6)

where the minus sign in ∓ should be used for a pos-itive value of a and vice versa. The meaning of thisinequality will be explored further when the problemis discretized.

B. Discretization

To implement our method by digital computation,the problem was digitized. The geometry of the dis-cretized diffraction problem is shown in Fig. 4. Thecomputation window isM pixels wide at both planes.The width of the active windows for the hologramand target image areMo andMw pixels, respectively.The computation window, an M ×M matrix, is intro-duced along with the zero padding factors μo �M∕Mo for the SLM plane and μw � M∕Mw for the tar-get image plane. Both padding factors are greaterthan or equal to 1. The sample interval at the targetimage in relation to the SLM pixel size is determinedfrom the FT spatial frequency definition as

Δ � λzMΔo

� Δozfμo

: (7)

The normalized bandwidth of the target image, b,determines the maximum diffraction angle from the

target, which is θmax � bθNyq, where θNyq � λ∕2Δ de-notes the paraxial approximation to the diffractionangle at the Nyquist sampling frequency.

For backward propagation, correct sampling of thefU�x; y�ϕ2�x; y�g product in Eq. (5) is determined byits bandwidth. This in turn depends on the band-widths of UR�x; y� and the product of the two expo-nentials in Eq. (5). The product of the twoexponentials can be reduced to a single quadraticexponential with an equivalent radius of curvature,zeq, given by

1zeq

� 1z

�a

zf � a

�� 1

zf �a�: (8)

The total available sampling bandwidth at the targetimage plane, 1∕2Δ, must be greater than or equal tothe sum of the bandwidths of the two functions in theproduct fU�x; y�ϕ2�x; y�g. This relation is expressedas

MwΔ2λz

f �a� � b2Δ

≤12Δ

; (9)

where the first term represents the bandwidth of thecombined product of the quadratic exponentials, andthe second is the bandwidth allocated to the targetimage function. Equation (9) is the expression ofthe bandwidth constraint to prevent aliasing of thefU�x; y�ϕ2�x; y�g product with the boundary condi-tions noted above. This may be solved for the condi-tions on a for a given target image bandwidth, b, as

jaj ≤ zfzf

μwμo�1−b�∓1; (10)

Fig. 3. Light paths for backward propagation between the targetimage and SLM planes when there exists an FT plane located at afinite distance from the target image. θmax is the maximum diffrac-tion angle from the target. Critical rays for high-frequencytruncation are shown. In (a) a positive value a is shown and in(b) a negative value.

Fig. 4. Relationship of the hologram (SLM) plane �ξ; η� at z � 0with pixel pitch Δo to the reconstructed image plane �x; y� at z �zf zc with pixel pitch Δ. The computation window is M pixels wideat both planes. The active hologram window isMo pixels wide, andthe active window of the target image is Mw pixels wide.


where the minus sign in ∓ should be used for apositive value of a and vice versa, both here andin Eq. (11).

The condition to avoid high-frequency truncation[Eq. (6)] can also be converted to the variables ofthe discrete problem. Substituting for Lo; Lw, andθmax in Eq. (6), one obtains

jaj ≤ zfzf

μw�1−bμo�∓1: (11)

For μo � 1, Eq. (11) is identical to Eq. (10). In the DFTdomain, the function calculated at the SLM plane bythe inverse DFT is one period in a plane of sampled,replicated units [28]. Clearly, this makes no sensewhen the objective is to study the effects of high spa-tial frequency components that would miss the SLMedge. Instead, these high-frequency componentswould be aliased back toward the center. For thisreason, the case of μo � 1 is not of interest. Forμo > 1, Eqs. (10) and (11) are no longer identical,and Eq. (11) becomes a more stringent conditionon a. This implies that proper sampling offU�x; y�ϕ2�x; y�g is guaranteed when the conditionfor no high-frequency truncation at the SLM edgesis met.

At the SLM plane, the full complex field is neededto program the SLM values. This requires that theproduct of the pre-exponential ϕ1�ξ; η� and the in-verse FT, F−1fU�x; y�ϕ2�x; y�g, in Eq. (4) also be prop-erly sampled. This in turn requires that the SLMplane sampling bandwidth be sufficient as given by

MoΔo

2λz� 1

2μwΔo≤

12Δo

; (12)

where the first term represents the bandwidth re-quired for ϕ1�ξ; η�, and the second term is the band-width of the inverse FT of the fU�x; y�ϕ2�x; y�gproduct. The second term is independent of a but de-pends on the spatial width of the target image region.Choosing the target plane padding factor to beμw � 2 apportions the available bandwidth equallybetween the quadratic phase, ϕ1�ξ; η�, andF−1fU�x; y�ϕ2�x; y�g. After noting that z � zf zc, andsubstituting for zc in Eq. (12), this simply states thatzf ≥ 2. This is always true since z ≥ 2zc is required toavoid vignetting of the reconstructed image. It is im-portant to notice that the right hand side indicatesthat the full bandwidth of the SLM is utilized. Tocorrectly sample the hologram without aliasing forthe subsequent forward propagation calculation,however, it is required that the sampled data havebandwidth of half the Nyquist frequency of sampling[21]. This may be achieved by resampling the matrixfor the forward propagation calculation by a factor oftwo. At this point, we choose μo � 2 for equal size andsymmetry of the regions of interest. As a result ofEq. (11), the bandwidth of the target image mustbe strictly less than half the Nyquist frequency of

the sampling because, if (1 − bμo) goes to zero, theallowed range of a also goes to zero.

In summary, the conditions for applying a singleDFT implementation of the inverse FDI have beenderived to obtain the correctly sampled magnitudeand phase of the diffracted field. Besides the require-ment of padding factors μo and μw, proper choice of afor a given target image bandwidth b is sufficient forusing the FDI to solve the backward propagationproblem. A quadratic phase must multiply the flat-phase target image, and its curvature must satisfyEq. (11) to guarantee that no high-frequency trunca-tion occurs at the SLM and for proper sampling atthe target plane. Also, z ≥ 2zc guarantees propersampling at the SLM plane. Finally, note that theright side of Eq. (12) requires the use of the wholeavailable bandwidth of the SLM, and that this sam-pling period must be considered for any subsequentforward FDI diffraction computations.

3. Simulations

To design a hologram and validate its design or tostudy the effects of encoding algorithms at the SLMplane, at least two approaches are possible. First,we validate the SLM design by doing a point-by-pointbackward R-S computation from the target function.This is computationally slow, so only a few cases werestudied. It is also important to propagate thedesignedhologram back to the reconstruction plane andcompare it with the original target image. In the nextsection (Section 3.A), our backward design FDI algo-rithm with added quadratic phase is validatedagainst anexact point-by-point diffraction calculationfrom points in the target image back to the SLMplane. Following that, example holograms are de-signed and reconstructed using the FDI method,and the RMS error dependence on the various param-eters is determined.

The following parameters were used throughoutthe simulations: λ � 633 nm, Δo � 13.68 μm, andfor backward propagation, μo � μw � 2 and M �1024. For forward propagation, the matrix (M ×M)was resampled by a factor of two, as described inSection 2.B, becoming 2048 × 2048. The quality ofreconstructed images was quantified by using theRMS error, defined by

σRMS � 1

It

��ZZ�Io�x; y� − It�x; y��2dxdy

s; (13)

where Io�x; y� and It�x; y� are the intensity of thereconstructed image and the target image, respec-tively, with equalized irradiance. It is the averagedirradiance of the target image.

A. Validating the Backward Propagation Algorithm

The algorithm for fast backward propagation using asingle DFT, as described in Eq. (4), and the samplingscheme of the previous section was validated for ac-curacy against an exact R-S diffraction calculation.


The exact computation used a point-by-point sum-mation of the backward diffracted fields from eachpixel of the target image. Although computationallyslow, this technique is rigorously correct. Complexwave fields at the SLM plane computed by bothmethods were compared using a double Lena targetimage, shown in Fig. 5. A fourth-order, digital Butter-worth low-pass filter was applied to the target imageto confine its bandwidth [29]. The irradiance trans-mittance of the filter was 0.1 at the cut-off frequencygiven by b, and 95% of the energy was containedwithin the circle of spatial frequencies less than b.When normalized to the same total energy, the com-parison diffracted wave amplitudes for all pixelsdiffered by less than one part in 10−12, and the phasediffered by less than 10−10 radians. When the back-ward and forward FDI algorithms were used to-gether without any truncation of the fields at theSLM plane, the exact target image was recovered(<10−10% RMS error).

B. Hologram Design Examples

To verify and illustrate the assertions of the previoussections, we conducted a series of hologram designsimulations using backward and forward diffraction.A complete cycle of backward and forward diffractionwithout SLM encoding recovers the exact target im-age. Truncating the forward wave to occupy the SLMwindow will result in small or large errors. Becausethe inequality governing high-frequency truncationis more stringent than the one governing aliasing,the reconstructed image will be examined with vari-ous values of added quadratic phase, a, and targetimage bandwidth, b, to see the effects of violatingthe inequality of Eq. (11) on the reconstructed imagequality.

Equation (11) specifies that, to avoid high-frequency truncation, the phase chirp factor a mustbe limited to a region that depends on target imagebandwidth, b. We plot this region as a function of thenormalized propagation distance zf in Fig. 6. Threedifferent target image bandwidth parameters(b � 0.25, 0.35, and 0.45) were chosen for simulation;all were less than the absolute limit of 0.5 indicatedby Eq. (11). These curves illustrate that a larger

bandwidth tolerates a smaller range of a. Forexample, at the fractional distance zf � 8, the rangeof a is Δa0.25 � �−0.88; 1.15� for b � 0.25, Δa0.35 ��−0.55; 0.65� for b � 0.35, and Δa0.45 � �−0.19; 0.21�for b � 0.45, respectively.

The effect of high-frequency truncation on thequality of the reconstructed image was first exam-ined by using one-dimensional (1D) spatial cosine im-ages at zf � 8. The amplitude of these cosine imagesis defined by U�x; y� � cos�π�xb∕Δo��. Cosine imagesserve as good candidates to verify Eq. (11) becausethe majority of their energy is concentrated at a sin-gle spatial frequency determined by b. Therefore, theconformity between the reconstructed and target im-ages will be broken when a specific value of a causeshigh-frequency truncation. Figure 7 shows the RMSerror of reconstructed cosine images versus chirp fac-tor a. The high-frequency truncation effect is pro-nounced for all three cosine frequencies, as allthree curves for RMS error show a sharp increasewhen the corresponding spatial frequency moves off

Fig. 5. Double Lena target image (512 × 512) with a b � 0.35low-pass Butterworth filter applied.

Fig. 6. Range of allowed phase chirp factor a to avoid high-frequency truncation for three bandwidths, b � 0.25 (blue), 0.35(red) and 0.45 (black). The minimum and maximum boundariesof a are plotted as dashed and solid lines, respectively.

Fig. 7. RMS error of reconstructed 1D spatial cosine imagesversus phase chirp factor a. The range allowed for a to avoidhigh-frequency truncation for each bandwidth is identified asΔa0.25 for b � 0.25, Δa0.35 for b � 0.35, and Δa0.45 for b � 0.45.


of the SLM region and becomes truncated. The sim-ulation results are in good agreement with Eq. (11).

To implement our proposed method with a gray-scale image, we conducted simulations using thedouble Lena target image at a distance of zf � 8.The target image was band limited by the Butter-worth low-pass filter. The RMS error of the recon-structed images versus a is plotted in Fig. 8.Different from the RMS error curves for cosine target

images, no abrupt increase in error was observed atthe high-frequency truncation boundary. At theboundary of the high-frequency truncation region,the reconstructed images for all three bandwidthshave around 0.1% RMS error. This is because themain spatial frequency content of the target imageis concentrated at low frequencies, and the smallresidual error arises from the spectral energy beyondthe cut-off frequency of the Butterworth low-passfilter. High-frequency truncation does not cause asignificant degradation of the reconstructed imagenear the onset value of a. The RMS error curvesincrease steeply when major spectral energy in thelow-frequency range (from the DC component [26]to b � 0.1) is truncated. This is illustrated in Fig. 8,where the RMS error increases to 10%–30% whenthe truncated frequency reaches b � 0.1. Finally,the three RMS error curves converge when trunca-tion of the DC component begins at the image edgesresulting in an even larger RMS error.

To examine the reconstruction quality for differentcases, a mosaic of representative reconstructedimages is displayed in Fig. 9(a). At the onset of thehigh-frequency truncation region, reconstructedimages resemble the target image. Truncationcauses minor ripples to appear at the outer edgesand blurs some fine spatial structures, resulting ina small RMS error. For example, Figs. 9(b)–9(d) showreconstructed images for a � 1 and bandwidths of

Fig. 8. RMS error of the reconstructed image versus phase chirpfactor a. The allowed range of a without high-frequency truncationfor each bandwidth is identified as Δa0.25 for b � 0.25, Δa0.35 forb � 0.35, andΔa0.45 for b � 0.45. The truncation region for the low-est 10% of spatial frequencies is marked in gray. The truncationregion of the DC component is marked in light green.

Fig. 9. (a) A mosaic of representative reconstructed images for various values of the phase chirp factor a (columns) and fractional band-width b (rows). Below this, in the bottom row, left to right are images (b)–(d) that show zoomed views of reconstructed images [green dashedsquares in mosaic (a)] for a � 1 and bandwidths of b � 0.25 in (b), b � 0.35 in (c), and b � 0.45 in (d), respectively. Red arrows indicate theripples caused by high-frequency truncation. In the bottom row, further to the right, images (e)–(g) are zoomed views of reconstructedimages [red dashed squares in mosaic (a)] for a � −2 and bandwidths of b � 0.25 in (e), b � 0.35 in (f), and b � 0.45 in (g), respectively.


b � 0.25, 0.35, and 0.45. For a � 1, the target imagewith a bandwidth of b � 0.25 remains within thetruncation-free region, while target images withthe other two bandwidths experience high-frequencytruncation during the reconstruction process. As aresult, small ripples are visible in Figs. 9(c) and 9(d)near the outer edges. The small, single ripple visiblein Fig. 9(b) is also present in the target image andresults from the Butterworth low-pass filter, nothigh-frequency truncation.

At larger values of a, low frequency and then DCcontent are lost. In Fig. 9(a), this truncation resultsin severe degradation at the edges of the recon-structed images. More specifically, Figs. 9(e)–9(g)show reconstructed images for a � −2 and band-widths of b � 0.25, 0.35, and 0.45. All three imagessuffered a severe aperture effect. The upper edgeand left edge are completely lost, and the entireimage is affected by significant ripples.

By examining the reconstructed images for differ-ent values of a, we notice that the truncation effect isspatially dependent. At the onset of high-frequencytruncation, the minor ripples are only observedat the outermost image boundaries as seen inFigs. 9(b)–9(d). Similarly, truncation of the DC com-ponent also starts at the outer edges and proceedstoward the center. In addition, the ripple fringes intruncation of the DC component have the largestperiod at the corner, and the period decreases as theyapproach the center as seen in Figs. 9(e)–9(g). Thisphenomenon is predicted by Fresnel diffractionand also by the geometric analysis, shown in Figs. 2and 3, which approximates Fresnel diffraction.

Finally, we demonstrated three-dimensional (3D)holographic image reconstruction. As illustrated inFig. 10(a), two Lenas in the target image are sepa-rated in distance along the z axis. The upper-left Lenaimage is placed at zf � 4; the lower-right image isplaced at zf � 16. Both images have a normalizedbandwidth of b � 0.35. In the simulation, the chirpfactor a was set to be a � 0.6, so that the backwardFDI calculation for both images stayed within thetruncation-free region defined by Eq. (11). The twohologram fields generated by these images wereadded and encoded on the SLM. The holographicimages were reconstructed at the two preset propaga-tion distances of zf � 4 and zf � 16, as shown inFigs. 10(b) and 10(c). At each distance, the corre-sponding Lena image was successfully reconstructedwith minimum error while the other image wasblurred.

In summary, these simulations validate the fastbackward diffraction routine that we implemented.When the computed hologram field was not trun-cated at the SLM edge, the RMS error of the recon-structed image was less than 10−10%. The size ofthe field distribution at the SLM was controlled bythe phase chirp added to the target image using theparameter a. The chief error mechanisms wererelated to truncating information at the SLM edgesas a became larger. High frequencies were lost first at

the edges of the reconstructed image. As lowerfrequencies became truncated, larger errors ap-peared, mostly in the form of ripples and resolutionloss localized at the image edges. Severe errorsresulted when low spatial frequencies and the DCcomponent were truncated for large values of a. Thiserror also began at the image edges and progressedinward, blocking more of the reconstructed image.The limits on a derived in Section 2 were demon-strated in the simulated examples. The onset ofaliasing was not reached in any of the test cases,since it was a less severe limit, as shown by Eqs. (10)and (11). Most importantly, limiting the bandwidth ofthe target image was shown to be important in reduc-ing error in the reconstructed image. Target imagebandwidth is strictly limited to be less than halfthe Nyquist frequency of the image sampling, and,for practical cases, it should be less than this.

4. Conclusion

We have presented a design prescription and simu-lated the performance of fast backward Fresnel dif-fraction for precise CGH design. Our method used asingle DFT implementation of the FDI. Careful con-sideration was given to the spatial bandwidths of thetarget image and the hologram encoded onto theSLM and their boundary conditions. The target im-age bandwidth and added quadratic phase weretreated as free variables that could be adjusted tominimize reconstruction error.

Truncation of the backward diffracted field at theedges of the SLM was found to play a major role in

Fig. 10. Illustration of 3D holographic image reconstruction.(a) The target image with two Lena images displaced along thez axis at zf � 4 and zf � 16. (b) The reconstructed image at theplane zf � 4. (c) The reconstructed image at the plane zf � 16.Intensity images are shown with their dynamic range expandedto fill the entire gray scale, 0–255.


introducing error in the reconstructed image. Thiserror resulted from the spatially limited support(finite width) of the target image and the SLM.The conditions to avoid this error were derivedand shown to be more stringent than those used toavoid aliasing of the sampled functions. The quad-ratic phase added to the target image can reduce thiserror by sizing the backward diffracted light to betterfit the SLM area. The bandwidth of the target imagealso played a role in the spatial distribution of thebackward propagated light at the SLM. This band-width was strictly limited to less than half theNyquist sampling frequency of the target image.An inequality was derived that computed the onsetconditions for error due to truncation of spectral fre-quency components at the SLM edge. The truncationonset depended on the parameters governing theadded phase and the target image bandwidth as wellas the zero-padding factors used at the target andSLM planes. Zero padding was required at bothplanes. Finally, proper sampling in the backwardFDI algorithm required that the distance betweenthe target image and SLM must be greater thantwice the critical distance, z ≥ 2zc. We showed thatthe designed hologram pattern used the entire spa-tial bandwidth of the SLM. To correctly sample thisSLM pattern for forward diffraction computationwith the FDI, it must be resampled at the SLM bya factor of two to avoid aliasing when the recon-structed image was computed.

The accuracy of the fast FDI method for hologramdesign was verified by precise agreement betweenthe simulations of backward diffraction by the fastFDI method and by an exact point-by-point R-Scalculation. Simulations of typical optical hologramdesign arrangements illustrated reconstructedimage quality in the low error region and for mildand severe spatial frequency truncation error. RMSerror for the images was used to quantify the errorversus the control parameters of target image band-width and added phase chirp. For the threebandwidths that were investigated, the low recons-truction error region for the phase chirp parameterwas narrow for a target image normalized bandwidthof 45% of the Nyquist frequency, but the low error re-gion increased substantially as the image bandwidthwas decreased. Regions of RMS error ≤0.1% could befound for all target image normalized bandwidthsless than 45%. Careful choice of the bandwidthreduction method for the target image is required,so that this process does not introduce additional un-wanted image artifacts. Finally, a 3D target imagewas encoded using our prescription, and accurate3D reconstruction at the different in-focus imageplanes was demonstrated.

This work was supported by one grant jointlyfrom the Army Research Office and the DefenseAdvanced Research Projects Agency (DARPA) aspart of the Optical Lattice Emulator Initiative(OLE) program.

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