Pixelated spatial light modulators (SLMs) can be em-ployed to encode complex modulation by means ofappropriate computer-generated holograms (CGHs).Because of the poor resolution of a pixelated SLM,each pixel of the complex function to be encoded mustemploy the minimum possible number of SLM pixels.Previously proposed CGH codes, appropriate for low-resolution SLMs, are based on either phase-only1–4 orreal-only modulators.5 These proposals employ eithersingle-pixel1,2,4 or double-pixel holographic cells.3,5,6
Here we present an extension to a previously pub-lished Letter.4 The proposal in the cited Letter was aphase-only CGH with on-axis reconstruction, employ-ing single-pixel holographic cells. This CGH is struc-tured to generate an on-axis signal field withoptimum reconstruction efficiency, maximum signalbandwidth, and high signal-to-noise ratio (SNR). Thelarge bandwidth and the high SNR are provided by a
binary phase periodic carrier that affects only theerror term in the CGH transmittance. For the appro-priate performance of this CGH, the SLM modulationfunction required a phase dynamic range of 2� radand a constant modulus. These conditions can be ful-filled by a zero-twist nematic liquid-crystal SLM ifthe thickness of the liquid-crystal layer is adjusted forthe required operating wavelength.
Here we propose to modify and to optimize theCGH code in Ref. 4 to implement it with nonidealphase SLMs for which the phase modulation is cou-pled with a nonconstant amplitude modulation.This type of modulation (known as phase-mostlymodulation) is provided by commercial and rela-tively inexpensive devices, e.g., the twisted-nematicliquid-crystal display (TNLCD). The CGH phasemodulation in Ref. 4 was obtained analytically. Incontrast, the CGH modulation in the modified codedeveloped here was obtained by selection, subject tocertain constraints, from the phase-mostly modula-tion values provided by the SLM. The encoding of anarbitrary on-axis complex function employing theconstrained complex modulation of a TNLCD waspreviously proposed and experimentally verified.6However, the resulting CGH described in Ref. 6 em-ployed two pixels of the SLM to encode each complexpixel of the desired modulation. For our current pro-posal, each pixel of the implemented complex func-tion is encoded by a single pixel of the SLM.
The experimental setup employed to obtain thephase-mostly modulation in a commercial TNLCD isdiscussed in Section 2. The derivation of the proposedCGH code is presented in Section 3. Some results in
V. Arrizón ([email protected]) is with Optica y Electrónica,Instituto Nacional de Astrofisica, Apartado Postal 51 y 216,Puebla, PUE 72000, México. L. A. González ([email protected]) is with the Departmento de Investigación en Fisica,Universidad de Sonora, Apartado Postal 5-88, Hermosillo SON83190, México. R. Ponce ([email protected]) and A. Serrano-Heredia ([email protected]) are with the Instituto Tecnológico yde Estudios Superiores de Monterrey, Av. Eugenio Garza Sada2501, Monterrey, NL 64849, México.
Received 31 August 2004; revised manuscript received 15 No-vember 2004; accepted 19 November 2004.
this derivation are developed in the Appendix. Ex-perimental support for our proposal is presented inSection 4 by the encoding of first-order Bessel beams.In Section 5 we present final remarks and conclu-sions.
2. Phase-Mostly Configuration of a Twisted-NematicLiquid-Crystal Display
Several arrangements have been employed to operatea TNLCD as a phase-mostly modulator. The simplerarrangement, referred to as the basic setup, employsa couple of linear polarizers appropriately oriented.Unfortunately, for this simple setup the phase mod-ulation is coupled with an amplitude modulation withsignificant variance. This amplitude variation ishighly reduced by the addition of two retardationplates to the basic setup.7,8 The first of these plates[Fig. 1(a)] together with the input polarizer �P1� isintended to generate an elliptic polarization state,with appropriate ellipticity, whose large semiaxis isbasically aligned with the optical axis at the inputplane of the liquid-crystal device. The second retar-dation plate, together with the output polarizer,forms a polarization detector that must be arrangedto transmit a complex field with minimum amplitudevariance (and maximum phase range). This arrange-ment is referred to as the optimal setup.
For implementation of the phase-mostly modula-
tion we employed a commercial SLM set (LC2000from HoloEye Photonics AG) formed by an array of600 � 800 pixels. We used this SLM in a simplifiedversion of the optimal setup that employs two po-larizers and a single retardation plate9 [see Fig.1(b)]. To obtain a phase modulation range close to2�, we employed a light beam with a wavelength of� � 457 nm, provided by an argon laser. The phase-mostly modulation, experimentally obtained, isshown in Fig. 2. For the discussion of the holographiccode in Section 3, it is useful to represent the SLMmodulation in the polar plot of Fig. 3, displaying thenormalized amplitude modulation (radial coordinate)versus the phase delay (azimuthal coordinate). EachSLM pixel provides a discrete set of complex trans-mittance values that are denoted as
Mg � �Mg�exp(i�g), (1)
where g (referred to as the gray level) is an integerindex with values of 0, 1, . . . 255. We obtained theplot in Fig. 3 by interpolating such discrete modula-
Fig. 1. Phase-mostly configuration based on a TNLCD employing(a) two retardation plates and (b) a single retardation plate.
tion values. The minimum amplitude |Mg| (corre-sponding to the phase �g � 286°) is approximately93.2% of the maximum amplitude (found at �g � 0°).The encoding of an arbitrary on-axis complex modu-lation, based on this constrained complex modulation(or phase-mostly SLM modulation), is analyzed inSection 3.
Let us assume that the TNLCD employed to encodethe complex modulation has a pixelated structurewith rectangular pixels of dimensions a and b, hori-zontal pixel pitch p, and vertical pixel pitch q. Inaddition, we assume that modulation Mnm
� |Mnm|exp�i�nm� in the �n, m�th pixel of the displayis given by the complex values in a phase-mostlymodulation plot, similar to the one discussed in Sec-tion 2 (Fig. 3). The transmittance of the CGH that canbe displayed in this SLM is
h(x, y) � �n, m
Mnmw(x � np, y � mq), (2)
where w�x, y� � rect�x�a�rect�y�b�. We assume thatthe CGH is intended to encode the spatially quan-tized complex function
c(x, y) � �n, m
cnmw(x � np, y � mq), (3)
where the complex modulation in the �n, m�thpixel, cnm � |cnm|exp�inm�, is subject to the condi-tion |cnm| 1. Each pixel in the complex functionc�x, y� is encoded by a single pixel in the CGH h�x, y�,i.e., the holographic cells have the minimum possiblespatial complexity. Our purpose is to design a CGHwith on-axis signal reconstruction, i.e., we assumethat the spectrum of c�x, y�, denoted C�u, v�, is cen-tered (exactly or approximately) at the zero frequency�u, v� � �0, 0�. Thus, the CGH transmittance must berelated to the encoded complex modulation c�x, y� bythe expression
h(x, y) � A0c(x, y) � e(x, y), (4)
where A0 is a positive constant and e�x, y� representsthe CGH modulation error. For an arbitrary CGHtransmittance h�x, y�, it is always possible to find anerror function e�x, y� and a constant A0 that fulfill Eq.(4). However, for a given complex function c�x, y�,error e�x, y� and constant A0 must be specified to op-timize the holographic performance (i.e., to maximizethe efficiency, SNR, and bandwidth). According to Eq.(4), the CGH Fourier domain field is given byH�u, v� � A0C�u, v� � E�u, v�, where H�u, v� andE�u, v� denote, respectively, the Fourier transformsof h�x, y� and e�x, y�. To obtain a high SNR, the errorspectrum E�u, v� must be negligible within the larg-est possible band, centered at zero frequency. To ful-
fill this requirement, we consider an error functiongiven by
e(x, y) � b(x, y)g(x, y), (5)
where it is assumed that the factor b�x, y� is a carriergrating whose first nonzero diffraction orders arefound as far as possible from the zero frequency, andg�x, y� is a function with the minimum possible band-width whose Fourier spectrum is centered on axis.We point out that, if such requirements for functionsb�x, y� and g�x, y� are fulfilled, the unavoidable con-sequence is that the error contribution in the CGHFourier spectrum tends to appear far from the zerospectrum frequency. From Eqs. (4) and (5) note thatboth functions b�x, y� and g�x, y� must have the pix-elated structure of h�x, y�, i.e.,
b(x, y) � �n, m
bnmw(x � np, y � mq), (6)
g(x, y) � �n, m
gnmw(x � np, y � mq), (7)
with the discrete modulations bnm and gnm. Consider-ing the SLM structure, we found that the optimalchoice of b�x, y� is the binary grating with discretemodulation
bnm � ( � 1)n�m. (8)
For this grating, the noise field E�u, v� is givenmainly by four off-axis replicas of the function G�u, v�(as illustrated in Fig. 4), centered at the spatial fre-quency coordinates �1�2p, 1�2q�, ��1�2p, 1�2q�,�1�2p, �1�2q�, and ��1�2p, �1�2q�. Thus, the prob-lem is reduced to finding the constant A0 that maxi-mizes the reconstruction efficiency and the errorfactor g�x, y� whose spectrum G�u, v� is centered onaxis and exhibits the minimum possible bandwidth.
Fig. 4. Distribution of the main noise bands in the CGH Fourierdomain.
According to Eqs. (2)–(7), g�x, y� is specified by itsdiscrete modulation gnm, which is related to the CGHmodulation by the formula
Mnm � A0cnm � (�1)n�mgnm. (9)
To obtain an appropriate analytical solution for A0and gnm we assume, for the moment, that n � m haseven parity, in which case Eq. (9) reduces to
Mnm � A0cnm � gnm. (10)
Substitution of index n in Eq. (9) by q � n � 1 leadsto the relation Mqm � A0�cnm � �c� � gnm � �g, where�c and �g denote the increments of cnm and gnm thatare due to the increment ��n � 1� in index n. Sinceboth functions c�x, y� and g�x, y� have on-axis spec-trum bands, they must show negligible variationswhen the increment in x is of the order of the pixelpitch. As a consequence, we obtained the conditions|�c|�� 1 and |�g|�� 1, which lead to the relationMqm � A0cnm � gnm. According to this relation and Eq.(10) both complex vectors, A0cnm � gnm and A0cnm
� gnm, must belong or must be close to the SLMmodulation curve. Based on this result, we propose toestablish the discrete function gnm such that bothcomplex vectors,
Mnm1 � A0cnm � gnm, (11)
Mnm2 � A0cnm � gnm, (12)
belong to the SLM modulation curve or to interpo-lated values of it (as is explained below). This pro-posal is validated noting that Eqs. (11) and (12) arealso obtained if the above analysis is performed as-suming that n � m has odd parity.
We now discuss the determination of constant A0.From Eqs. (11) and (12) we obtain
A0cnm � (Mnm1 � Mnm
2)�2. (13)
Note that the center of curvature for each point in theSLM modulation plot is an interior point of this plot(see Fig. 3). Therefore, considering that the vectors
Mnm1 and Mnm
2 belong to the modulation plot, thepoint A0cnm obeying Eq. (13) cannot be an exteriorpoint. In general, to fulfill this condition the maxi-mum value of A0 must be given by the minimumvalue of the amplitude |Mg| in the SLM modulation[Eq. (1)]. From Eq. (4) we note that this maximumpossible value of A0 provides the maximum CGH ef-ficiency. For the SLM modulation in Fig. 3, it wasfound that A0 � min��Mg�� � 0.932.
According to Eq. (13), Mnm1 and Mnm
2 must be cho-sen as the couple of vectors in the SLM modulationplot whose middle point is A0cnm. The couple of pointsMnm
1 and Mnm2 always exist because A0cnm is an inte-
rior point of the SLM modulation curve. However itmust be noted that both vectors are chosen from thefinite and discrete set of modulation points defined byEq. (1). Hence, for most of the encoded modulationvectors A0cnm, any couple of points Mnm
1 and Mnm2
(from the available SLM modulation points) has amiddle point cnm
t � �Mnm1 � Mnm
2��2, which is notexactly equal to A0cnm. Thus, in practice we choseMnm
1 and Mnm2 such that � |cnm
t � A0cnm| takes itsabsolute minimum value. Because of the relativelylarge number (256) of points in the SLM plot, error εis close to zero for all the complex vectors cnm. FromEqs. (11) and (12) note that the vectors Mnm
1 and Mnm2
correspond to the two possible modulation valuesthat [according to Eq. (9)] can be assigned to the�n, m�th pixel in the CGH. The precise selection be-tween these two vectors is specified below.
After vectors Mnm1 and Mnm
2 have been establishedwith the above procedure, gnm can be obtained by therelation
gnm � (Mnm1 � Mnm
2)�2, (14)
which is derived from Eqs. (11) and (12). However,the precise specification of gnm depends on the posi-tions given to Mnm
1 and Mnm2 in the modulation plot.
Indeed, there are two possibilities, as illustrated inFig. 5. In Fig. 5(a), Mnm
1 performs a clockwise rotation(smaller than 180°) of the radial line that containscnm. In Fig. 5(b), Mnm
1 performs a counterclockwiserotation of this radial line. For further analysis, itis convenient to introduce the new symbols
Fig. 5. Two possible definitions for vectors Mnm1 and Mnm
2: (a) Mnm1 appears at the right position (and is redefined as Mnm
R) and (b) Mnm1
appears at the left position (and is redefined as MnmL). (c) Position of vectors Mnm
1, Mnm2� but in the specific positions depicted in
Fig. 5(c). For the first option in Fig. 5(a), Mnm1 and
Mnm2 are, respectively, identified with Mnm
R andMnm
L. For the option in Fig. 5(b), Mnm1 is identified
with MnmL (and Mnm
2 with MnmR). In general, consid-
ering these two possible definitions of points Mnm1
and Mnm2 as a degree of freedom for the final deter-
mination of the CGH modulation, Eqs. (13) and (14)can be rewritten as
A0cnm � (MnmR � Mnm
L)�2, (15)
gnm � dnm(MnmR � Mnm
L)�2, (16)
where dnm is a discrete binary function (with possiblevalues of �1 and �1). Comparing Eqs. (14) and (16)we note that the value of dnm � 1 corresponds to thechoice of Mnm
1 � MnmR, and dnm � �1 corresponds to
Mnm1 � Mnm
L. From Eqs. (9) and (16), the CGH dis-crete modulation is expressed as Mnm � A0cnm ���1�n�mdnm�Mnm
R � MnmL��2. Then, combining this
relation with Eq. (15) we determined that the CGHmodulation is given by
Mnm ��MnmR if (�1)n�mdnm � 1
MnmL if (�1)n�mdnm � �1
. (17)
In Eq. (17), MnmR and Mnm
L represent the couple ofpoints in the SLM modulation plot [see Fig. 5(c)]whose middle point gives the best approximation toA0cnm. For a given encoded modulation c�x, y�, dnm is abinary function that must be determined to obtainthe values of gnm [using Eq. (16)] that minimize thebandwidth of g�x, y�.
A. Complete Computer-Generated HologramDetermination for Soft Encoded Functions
The CGH pixel modulation adopts a more simpleform by assuming that dnm � 1 for every CGH pixel.In this case, Eq. (15) is unchanged and Eqs. (16) and(17) can be rewritten as
gnm � (MnmR � Mnm
L)�2, (18)
Mnm ��MnmR (n � m) even
MnmL (n � m) odd . (19)
It has been noted [below Eq. (5)] that gnm should bedetermined to minimize the bandwidth of functiong�x, y�. In the Appendix we show that the simplespecification of the CGH modulation in Eq. (19) isappropriate for a discrete modulation cnm
� |cnm|exp�inm�, where both the modulus |cnm| andthe phase nm are soft (or quasi-continuous) func-tions. For our purpose, phase nm is considered a softfunction if the phase difference |nm � ql|, modulus2�, for indices �n, m� and �q, l� corresponding to ad-jacent pixels, is small compared with �. In a similar
way, |cnm| is a soft function if the absolute value ofthe difference |cnm| � |cql| (for adjacent pixels) issmall compared with 1.
Another interesting case is encoded complex func-tions of the type
cnm � rnm exp(inm0), (20)
where both rnm, which is a real factor, and phase nm0
are quasi-continuous functions. For the encoded com-plex modulation in Eq. (20), in the Appendix we provethat the appropriate binary function dnm is given by
dnm ��1 if rnm � 0�1 if rnm � 0. (21)
Thus, for the domain of the encoded complex functionin Eq. (20) with rnm � 0, the CGH modulation Mnm isequal to Mnm
R for the pixels with even n � m (andMnm
L otherwise). The choice is inverted if rnm � 0.This distribution of values of the CGH modulationMnm is illustrated in Fig. 6 for a simple encoded com-plex function of rank 6 � 4. The top half of Fig. 6(domain �1) corresponds to rnm � 0, and the bottomhalf (domain �2) to rnm � 0. This example suggeststhat for the CGH encoding complex function given inEq. (20), the distribution of values Mnm
R and MnmL
form a checkerboard array that is affected by an orderdislocation in the transition from domain �1 to do-main �2.
The type of function in Eq. (20) is frequently foundin optical processing. Specifically, several solutions ofthe paraxial wave equation (e.g., Bessel, Gaussian,Hermite, and Laguerre beams) have the form of Eq.(20). A particularly interesting case occurs when cnm
corresponds to sampled values of a continuous func-tion of the type f�r, �� � R�r�exp�iN��, where �r, �� arepolar coordinates, and R�r� is a real function of theradial coordinate. The CGH for the synthesis of first-order Bessel beams that belong to this type of func-tion is designed and evaluated in Section 4.
Fig. 6. Distribution of vectors MnmR and Mnm
L for the top region �1
(with rnm � 0) and the bottom region �2 (with rnm � 0) of a CGH.
B. Efficiency and Signal-to-Noise Ratio of the Hologram
In the proposed CGH both the phase and the ampli-tude of the desired signal are independently prespeci-fied. Therefore, the signal phase is not a degree offreedom, as is frequently assumed in the design ofCGHs. For the proposed code, the CGH transmit-tance h�x, y� is related to the encoded complex signalc�x, y� by the relation h�x, y� � A0 c�x, y� � e�x, y� [Eq.(4)], where e�x, y� is the CGH modulation error. Thus,the CGH efficiency is given as
� �
A02
s
�c(x, y)�2 dxdy
s
�h(x, y)�2 dxdy
, (22)
where integration is performed in signal domain s.For the discussed CGH, the possibility of generat-
ing the desired signal field c�x, y� with high fidelitydepends on how much the error Fourier spectrumE�u, v� is disjointed from the signal Fourier spectrumC�u, v� in a specified on-axis spectrum domain S.Now, let us assume that we will recover the desiredcomplex field c�x, y� by using a low-pass filter in thespectrum plane and then perform Fourier transfor-mation of the filtered spectrum. In general, fieldct�x, y� recovered after this operation differs from de-sired field c�x, y� (in the specified signal domain s).The signal fidelity (in domain s) can be measured bythe conventional definition for the SNR,10 given by
SNR �
s
|c(x, y)|2 dxdy
s
|c(x, y) � �0ct(x, y)|2 dxdy
, (23)
�0 �
s
�c(x, y)�2 dxdy
s
�ct(x, y)�2 dxdy
. (24)
Equations (22) and (23) are used in Section 4 to com-pute the efficiency and SNR of two CGHs encodinghigh-order Bessel functions.
As an application of the synthetic holographic codediscussed in Section 3, we implement CGHs to gen-erate first-order Bessel beams with finite support.First, we encode a first-order Bessel beam with com-plex amplitude
B(r, �) � AJ1(2��0r)exp(i�)circ(r�R), (25)
where �r, �� are polar coordinates, J1 is the first-orderBessel function of the first kind, �0 is the radial spa-tial frequency of the beam, R is the radius of thecircular beam support, and A is a normalization con-stant. Modulation cnm, required to establish the pix-elated complex function in Eq. (3), is obtained byuniform sampling of B�r, ��. The intensity and phasemodulation of the first two rings of the Bessel beamare shown in Figs. 7(a) and 7(d). The sampling reso-lution in this plot ��x � 33 �m� corresponds to thepixel pitch of the SLM employed to implement theCGH (the SLM set LC2000 from HoloEye PhotonicsAG). The radius of the beam support is R � 67�x� 2.21 mm. The intensity and phase modulation ofthe CGH encoding the sampled complex modulationof the Bessel beam are depicted in Figs. 7(b) and 7(e).This CGH transmittance is designed, by use of Eqs.(17) and (21) in Section 3, for the experimental phase-mostly SLM modulation shown in Fig. 3. We dis-played intensity instead of amplitude to improve thevisibility of the low variance function in Fig. 7(b).First we performed the computational reconstructionof the Bessel beam generated by the CGH. This pro-cess starts with the Fourier transformation of theCGH transmittance, which is displayed in Fig. 8. Thisspectrum image shows the signal contribution C�u, v�centered on axis and the more significant noise bandsat the corners of the image. A circular aperture isemployed to transmit only the signal contribution ofthe spectrum. The filtered spectrum is subject to an-other Fourier transformation to generate the synthe-sized Bessel beam whose intensity and phase are,respectively, shown in Figs. 7(c) and 7(f). The SNR ofthe reconstructed field, computed with Eq. (23), isapproximately 105, and the efficiency [computed withEq. (22)] is � � 25.1%.
We also synthesized an enhanced first-order Besselbeam whose complex amplitude is
Be(r, �) � Ar J1(2��0r)exp(i�)circ(r�R), (26)
where the factor r is intended to prevent the atten-uation of high-order signal rings. Considering thesame SLM parameters specified above, we imple-mented the first two rings of the function in Eq. (26).In this case, the intensity and phase of the sampledcomplex modulations are shown in Figs. 9(a) and 9(d).The intensity and phase transmittance of the CGHencoding the Bessel beam are, respectively, shown inFigs. 9(b) and 9(e). Finally, after low-pass spatialfiltering of the CGH, we obtained the synthesizedsignal whose intensity and phase are shown in Figs.9(c) and 9(f). In this case the SNR is 80 and efficiency� � 45.6%. The higher efficiency in this case is due tothe enhancement of the beam given by the factor r inEq. (26). For the two considered CGHs, the SNR andefficiency were computed in the circular support thatcorresponds to the two rings of the encoded complexfields (in Figs. 7 and 9). The relatively low computed
efficiencies are typical of CGHs for which prespecifiedphase and amplitude of the signal field need to beencoded. This is a drawback of the discussed holo-graphic codes in comparison with CGHs for whichonly the signal intensity is a concern and the phase isa free parameter for design.
For the experimental implementation of theseCGHs, the two-dimensional noise carrier modulationbnm � ��1�n�m was replaced by its one-dimensionalversion bnm � ��1�n. The reason is that the available
SLM performed an accurate display only of high-frequency binary modulation in the direction corre-sponding to the low-frequency SLM scanning signal.The CGH signal field was separated from the noise byuse of a 4�f spatial filtering setup whose transforminglenses had an f � 4.0 cm focal length. Images of theexperimentally generated first-order Bessel beamsare shown in Figs. 10 and 11. For each case, theintensity of the beams was recorded at z � 5 cm and90 cm from the second transforming lens of the 4�fsetup. The experimentally reconstructed fields showa reduced quality compared with the numerically re-constructed beams. This difference can be explainedin part in the deficient transference of high-frequencyvideo signals along the horizontal scanning axis ofthe SLM. Nevertheless, the experimental patterns inFigs. 10 and 11 make evident the high invarianceunder propagation of the synthesized beams.
5. Final Remarks
The purpose of the discussed CGH is to synthesize afully complex field in which both the amplitude andthe phase are independently preestablished. Thus,our CGH is different from those CGHs intended togenerate a signal with the phase as a freeparameter.11–15 The encoded fully complex modula-tion c�x, y� in the CGH plane is related to the CGHmodulation h�x, y� (which is provided by a coupledamplitude and phase SLM) by the formula h�x, y�� A0 c�x, y� � e�x, y�. In this relation the signal isimplicitly generated on axis. On the other hand, error
Fig. 7. Performance of the CGH designed for the synthesis of two central rings of a first-order Bessel field AJ1�2��0r�exp�i��circ�r�R�(numerical evaluation). Modulation of the encoded complex field appears at column 1; modulation of the CGH and the reconstructed fieldappear, respectively, at columns 2 and 3. Intensity modulation [(a)–(c)] appears at row 1 and phase modulation [(d)–(f)] at row 2.
e�x, y� has been established as the product of a high-frequency off-axis binary carrier grating with a lowbandwidth on-axis function. Because of this feature,the light energy that corresponds to error field e�x, y�propagates off axis.
In CGH designs that use the signal phase as a freeparameter, it is customary to introduce a phase dif-fuser in the signal domain to minimize the amplitudevariations in the CGH plane.16,17 Since our discussedCGH requires a signal whose amplitude and phaseare prespecified, it is not possible to employ suchdiffusers. For example, if a CGH is designed for thesynthesis of a Bessel beam, multiplication of the sig-nal by a random phase modulation would seriouslyaffect the invariance of the beam under propagation.
We have determined [Eq. (17)] the constrainedcomplex modulation of the optimum on-axis CGHbased on a TNLCD that encodes the arbitrary com-plex modulation c�x, y�. The minimum spatial com-plexity of the CGH [in Eq. (2)], together with thespecified CGH modulation error, provides an opti-mum signal bandwidth. The modulation error isformed by the factors ��1�n�m and gnm [at the right-hand side of Eq. (9)]. Because of the first factor, theerror contribution in the CGH Fourier domain ismainly formed by four off-axis bands (Fig. 4) contain-ing shifted images of the Fourier transform of g�x, y�[Eq. (7)] whose distance to the zero frequency hasbeen maximized. The presence of a similar noise car-rier grating with modulation ��1�n�m has also been
Fig. 9. Performance of the CGH designed for the synthesis of two central rings of the enhanced first-order Bessel fieldAr J1�2��0r�exp�i��circ�r�R� (numerical evaluation). Modulation of the encoded complex field appears at column 1; modulation of the CGHand the reconstructed field appear, respectively, at columns 2 and 3. Intensity modulation [(a)–(c)] appears at row 1 and phase modulation[(d)–(f)] at row 2.
Fig. 10. Experimental intensity distribution of the field gener-ated with the CGH in Fig. 7 (a) at 5 cm and (b) at 90 cm from thesecond transforming lens of the 4�f spatial filtering setup.
found in double-phase CGHs.3,6 However, in this casethe period of the noise carrier grating along the ori-entation axis of the double-pixel cells (which we as-sume to be the x axis) is equal to 4p (p is the SLMpixel pitch in the x axis). Thus, the horizontal sepa-ration of noise bands in the spectrum of double-phaseholograms is half of the separation of such bands inthe case of the optimum CGH analyzed here. Anotherfact that contributes to a large bandwidth and highSNR is the minimized bandwidth of G�u, v� [the Fou-rier spectrum of g�x, y�] by the appropriate selectionof discrete modulation dnm [see Eq. (16)]. As a partic-ular case, Eq. (21) gives the convenient function dnm
for the family of complex functions cnm defined by Eq.(20).
The CGH modulation in Eq. (17) represents a gen-eralization of the CGH modulation obtained in Ref. 4for a perfect phase SLM. In this ideal case, the twopossible values of CGH modulation Mnm are given byMnm
R � exp�nm � arccos��cnm��� and MnmL
� exp�nm � arccos��cnm���. In this case, it is notedthat both vectors Mnm
R and MnmL belong to a SLM
modulation curve, identical to the unitary complexcircle, and that the middle point of these vectors is theencoded complex modulation cnm.
The discussed CGH can be compared with otherCGHs that also employ single-pixel cells and are de-signed to encode fully complex signals. PreviousCGHs in this category, mentioned in Refs. 1, 2, and 4,are implemented with phase-only SLMs, whereas the
CGH discussed here can be implemented with aphase-mostly SLM (e.g., a TNLCD), in which the am-plitude modulation variance is not null. A commonfeature of the CGH discussed here and the one de-scribed in Ref. 4 is a Fourier spectrum formed by arelatively wide signal band centered on axis and nar-row noise bands centered on the off-axis diffractionorders of a binary phase carrier grating. For the CGHin Ref. 2, the signal can also be placed on axis, but thenoise appears, sharing the signal domain, as a ran-dom and diluted light background. The CGH in Ref.1 can employ an off-axis signal carrier, in which casea significant portion of the SLM bandwidth is sacri-ficed. As an option, the encoded complex function canbe multiplied by a quadratic phase function thatbrings about the Fourier transformation of the en-coded field. In this case, the noise contributions ap-pear in a diluted form, sharing the signal domain. Amore detailed comparative study of these CGHs (andthe one discussed here) deserves a special treatment,which is out of the scope of this paper.
Appendix
The aim here is to determine the appropriate valuesfor binary function dnm in Eq. (16), corresponding tothe encoded complex function in Eq. (20). As indi-cated in the text [below Eq. (17)] dnm must be deter-mined to obtain the values of gnm that minimize thebandwidth of g�x, y�. To analyze gnm we first note that,if the modulus of each complex point in the SLMmodulation plot were equal to unity, the phases ofpoints Mnm
R and MnmL would be nm
R � nm
� arccos��cnm�� and nmL � nm � arccos��cnm��. This is
corroborated by use of Eq. (15) with A0 � 1. For thetrue modulation plot the moduli and phases of suchpoints obey the relations |Mnm
R| � |MnmL| � 1,
nmR � nm � arccos��cnm��, and nm
L � nm
� arccos��cnm��. From Eq. (20) we determined thatthese relations can be expressed as
nmR ��nm
0 � arccos(|cnm|) rnm � 0
nm0 � � � arccos(|cnm|) rnm � 0, (A1)
nmL ��nm
0 � arccos(|cnm|) rnm � 0
nm0 � � � arccos(|cnm|) rnm � 0. (A2)
Note that for the encoded complex function in Eq. (20)both |cnm| and nm
0 are soft functions. Thus, accord-ing to relations (A1) and (A2), for a set of pixels �1
with rnm � 0, nmR, and nm
L are soft functions. As aconsequence, their corresponding discrete modula-tions Mnm
R and MnmL are also soft functions, and gnm
[in Eq. (16)] becomes a soft function if dnm is constantfor all the pixels within the �1 set. We adopted thevalue dnm � 1 for any set of pixels in which rnm � 0.
We now consider another set of pixels �2 with rnm
� 0, which is adjacent to the �1 set. With argumentssimilar to those used above, we can establish that gnm
is a soft function within �2, making dnm constant forall the pixels in this domain. Next we show that, to
Fig. 11. Experimental intensity distribution of the field gener-ated with the CGH in Fig. 9 (a) at 5 cm and (b) at 90 cm from thesecond transforming lens of the 4�f spatial filtering setup.
avoid discontinuities of gnm in the border between �1and �2, it is convenient to set dnm � �1 for the pixelsin �2. Let us consider two adjacent pixels with indices�i, j� and �q, l� that belong to the �1 and �2 domains,respectively. Since rnm is a soft function and rij � 0and rql � 0, we established that |cij| � |cql| � 0.Thus, relations (A1) and (A2) give the relations ij
R
� ij0 � ��2, ij
L � ij0 � ��2, ql
R � ql0 � ��2, and
qlL � ql
0 � ��2. In addition, since nm0 is a soft
function, we determined that MqlR � Mij
L and MqlL
� MijR. Using these results in Eq. (16) and the iden-
tity dij � 1 (valid for rij � 0), we obtained
gij � (MijR � Mij
L)�2, (A3)
gql � dql(MijL � Mij
R)�2. (A4)
From Eq. (A3) and relation (A4) we note that to avoida discontinuity in function gnm between adjacent pix-els �i, j� and �q, l�, identity dql � �1 must be estab-lished. Since dnm is constant for all the pixels indomain �2 (as stated above), we set dnm � �1 for allthe pixels in �2. This result completes the proof of Eq.(21) and is valid for the encoded complex function inEq. (20).
For the particular case in which rnm � |cnm| � 0 foreach pixel of the encoded function [in Eq. (20)], wedetermined that dnm � 1 for all the CGH pixels. Thisresult shows that the CGH modulation given in Eq.(19) is appropriate for a discrete modulation cnm
� |cnm|exp�inm� in which both the modulus |cnm|and the phase nm are soft (or quasi-continuous) func-tions.
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