1. Introduction

Implementation of fully complex modulation is an important task in optical information processing [1,2]. It can also be useful for the synthesis of complex optical wave-fields, e.g. high order Bessel beams [3], Laguerre-Gauss beams [4], and other paraxial and non-paraxial beams [5,6], with potential application as optical tweezers [7,8]. A single transmission (or reflection) plate, with arbitrary complex transmittance can not be easily obtained, even with nowadays technology. However, the indirect realization of an arbitrary complex modulation has been provided, for more than a quarter of a century, by optical [9] and computer generated holograms (CGHs) [10–13]. Recently, the possibility of employing different types of programmable spatial light modulators (SLMs) for displaying CGHs has become an attractive option. In particular, electrically addressable pixelated SLMs can provide precise, repeatable, and reconfigurable optical modulation patterns. As a recent tendency, researchers have given preference to the development of CGHs for phase-only [14–18] liquid crystal (LC) SLMs. In a marginal way the encoding of complex fields has been performed with real-only [19], or amplitude-only LC-SLMs [20–22].

We analyze the attributes of CGHs implemented with amplitude LC-SLMs and discuss their optimization. Our analysis in section 2 is based on the observation that the bias term of amplitude CGHs, to be displayed onto a LC-SLM, can be considered as a degree of freedom, only restricted for the requirement of a positive definite hologram transmittance. A similar relaxation in the selection of the bias term has been applied since the early stages of computer holography [23]. We propose two CGH bias functions, different to that of a conventional interferometric CGH. The first proposed bias term, given by a soft envelope of the signal modulus, is appropriate for amplitude-only SLMs. The second proposed bias term, given by a constant function, results appropriate for SLMs whose amplitude transmittance is coupled with a linear phase modulation. We also discuss and implement a method to avoid the signal distortion due to the finite size of SLM pixels. This method is based on a digital pre-filtering of the complex signal to be encoded. The performance of the discussed amplitude CGHs is evaluated by numerical simulations (section 3) and experiments (section 4). Although we performed this evaluation for amplitude CGHs encoding a variety of complex fields with satisfactory results, for brevity we only report CGHs encoding high order Beseel beams. In section 5 we present relevant remarks and conclusions.

2. Amplitude computer-generated hologram optimized for a pixelated SLM

Our purpose is to employ a CGH for the synthesis of a complex field

where the amplitude a(x,y) is a positive normalized function and the phase ϕ(x,y) takes values in the domain [-π,π]. The normalized transmittance of a conventional interferometric hologram encoding the complex signal s(x,y) is analytically expressed as

where c ₀≅1/2 is a normalization constant, (u₀,v₀) are the spatial frequencies of the linear phase hologram carrier, and b(x,y) is the bias function, given by

Generation of the hologram transmittance in Eq. (2) by optical procedures would require the recording of the interference pattern formed by the signal s(x,y) with the off-axis plane wave exp[i2π(u₀x+v₀y)]. However, our aim is to implement a pixelated version of the transmittance h(x,y) onto an electrically addressable amplitude LC-SLM. Remarkable advantages of this approach are the accuracy of the implemented transmittance, and the fact that the signal and reference beams are only required as mathematical entities that are specified with versatility and precision. If an amplitude CGH, with transmittance h(x,y), is displayed onto an amplitude SLM without coupled phase modulation, the signal phase is accurately encoded by the position of the hologram fringes. Thus, the sensitivity of these CGHs to marginal uniformity errors in the SLM modulation [24] is negligible. On the other hand, the main spectral band of amplitude CGHs only presents the off-axis signal term accompanied by the zero-order term and the conjugate of signal, allowing the isolation of signal from noise by means of a spatial filter. Thus, one can expect that the encoding of complex signals with amplitude SLM-CGHs will provide signal reconstruction with high fidelity. The restriction is that the signal bandwidth must be smaller than the SLM bandwidth. This represents an important limitation, common to the different types of CGHs displayed on pixelated LC-SLMs, because the bandwidth of such devices is relatively low.

It is interesting to note that for amplitude CGHs implemented with a LC-SLM, the bias function b(x,y) [in Eq. (2)] can be chosen from the set of functions making positive the transmittance h(x,y), instead of restricting its definition to Eq. (3). Next we propose two bias terms, different to the conventional one, that may offer advantages in specific situations.

2.1. Bias function for amplitude hologram without coupled phase modulation.

We assume firstly that the CGH transmittance in Eq. (2) is implemented with a SLM providing positive transmittance, free of a coupled phase modulation [25]. In this case, the CGH Fourier spectrum is given by

where S and S ^* denote the Fourier transform of the encoded signal and its complex conjugate, respectively. The complex signal is recovered by applying a band-pass filter to the CGH spectrum, centered at frequency coordinates (-u₀ ,-v₀). A high SNR is possible by reducing the contribution of the bias spectrum B(u,v) within the band-pass filter. This condition can be attained reducing the energy and the bandwidth on the bias b(x,y). To ensure the positive nature of the CGH transmittance we assume the restriction b(x,y)≥a(x,y). Thus, in order to reduce the energy and bandwidth of b(x,y), this function is proposed as a soft envelope of the signal modulus a(x,y). A simple method that we employ to obtain a soft envelope of a(x,y) is by performing a spatial filtering of this function, employing a Gaussian low-pass filter. In this case, the bias spectrum is given by

where A(u,v) is the Fourier transform of the signal modulus a(x,y), and ρ is the radial coordinate in the signal Fourier domain [ρ ²=u ²+ v ²]. The width α of the Gaussian filter is approximately (or in the order of) the signal bandwidth, and A ₀ is the minimum constant ensuring the condition b(x,y)≥a(x,y). The bias is obtained as the inverse Fourier transform of the spectrum B(u,v). The conventional bias function, in Eq. (3), and the modified one, determined by Eq. (5), will be referred to as bias functions 1 and 2 respectively. The CGHs corresponding to these bias functions are referred to as holograms of types 1 and 2.

2.2 Bias function for amplitude hologram with coupled phase modulation

A simple setup to produce programmable gray-level amplitude modulation employs a twisted nematic (TN) LC-SLM with two linear polarizers. However, the amplitude modulation provided by this simple setup, is usually coupled with some phase modulation [26], and does not display exactly the amplitude CGH transmittance in Eq. (2). A significant reduction of the coupled phase modulation in the SLM can be obtained by adding two quarter wave plates to the basic SLM setup [25]. Another approach, that we develop next, is based on the selection of an appropriate bias term in the CGH design, which minimizes the distortion of the reconstructed signal due to the SLM coupled phase.

To analyze the influence of the coupled phase in amplitude CGHs displayed with a LC-SLM we rewrite the desired hologram transmittance in Eq. (2) as

where t_s(x,y)=(1/2)s(x,y)exp[-i2π(u₀x+v₀y] is the CGH signal term. The CGH transmittance, modified by the SLM coupled phase, can be expressed as

where the coupled phase ϕ_c(x,y) is a function of the amplitude h(x,y). The exponential in Eq. (7) is given by its power series

In general, the coupled phase ϕ_c(x,y) can be expressed by a polynomial expansion C₀ +C₁ h(x,y)+…C_N h^N(x,y). However, for brevity we only discuss the case of a linear expansion ϕ_c(x,y) = γh(x,y), with slope γ. In this case, the modified CGH transmittance is given by

Considering Eq. (6), and the Newton’s binomial expansion, we obtain

where we omitted the variables (x,y) in functions b, t_s and $t_s^{*}$. By expanding the expression (b+t_s) ^n+1-q it is found that the right side in Eq. (10) contains mixed powers of b, t_s and $t_s^{*}$. The main contribution to the signal in these mixed powers, is given by the term (c₀) ⁿ⁺¹ (n+l) bⁿt_s . Considering this result, we obtain that the main contribution to the signal in the series of Eq. (9) is given by

Function s_m(x,y) represents the modified signal, reconstructed from the CGH transmittance h_m(x,y) [Eq. (7)], which is affected by the coupled phase modulation. An interesting result is that the modified signal s_m(x,y) becomes identical to the desired signal t_s(x,y), except by a constant complex factor, if we adopt the constant bias function

Although the above analysis assumed the linear relation ϕ_c(x,y) = γh(x,y), between the CGH amplitude modulation and the coupled phase, we have found that the constant bias function is also appropriate for arbitrary coupled phase, with moderate phase range. The constant bias function in Eq. (12) is referred to as bias 3, and its corresponding CGH is named as type 3 CGH.

2.3 Correction for pixelated SLM structure

We assume that the SLM employed to implement amplitude CGHs has rectangular pixels of dimensions a and b, and pixel pitch δx (in both the horizontal and vertical axes). To implement a CGH, uniformly sampled values of h(x,y) are displayed onto the SLM. In addition, assuming that the CGH support is a window defined by p(x,y), the signal term in the CGH Fourier spectrum ia given by

where S(u,v) and P(u,v) denote the Fourier spectra of the signal s(x,y), and its support p(x,y), respectively, and Δu=1/δx is the SLM bandwidth. The weighting factor W(u,v), given by the Fourier transform of the pixel window, is

where sinc (ξ)=sin (πξ)/(πξ). The signal can be recovered by applying a band pass filter to the CGH spectrum, transmitting the light from the signal spectrum term S_m(u,v). According to Eq. (13), this signal spectrum term is modulated by the factor W(u,v), that can affect the quality of the recovered signal. If the influence of the envelope W(u,v) were disregarded, the reconstructed signal, given by the inverse Fourier transform of S_m(u,v), would be s(x,y) p(x,y) exp[-i2π(u₀ x+v₀ y)], which corresponds to the encoded complex signal s(x,y), within the finite support p(x,y), modulated by the linear phase carrier.

Now we describe a method to avoid the signal distortion due to the factor W(u,v). This method is based on the substitution of the desired complex signal s(x,y) by an appropriately modified complex signal s′(x,y), which represents a pre-filtered version of s(x,y). For the modified signal s′(x,y), the signal spectrum term in Eq. (13) is rewritten as

In order to recover the desired signal s(x,y) from this spectrum S′_m , the convenient definition of the pre-filtered signal is given by the relation

This function T(u,v) can be defined within the off axis signal spectral band, making the reasonable assumption that the weighting function W(u,v) has no zeros within this band. After that function T(u,v) in Eq. (16) is computed from the desired signal s(x,y) and the function W(u,v), we can digitally obtain its inverse Fourier transform t(x,y)=s′(x,y)exp[-i2π(u₀x+ v₀y)]. The pre-filtered signal itself is obtained as s′(x,y)=t(x,y) exp[i2π(u₀x+ v₀y)]. Finally, this function is renormalized in order to fulfill the condition ∣s′(x,y)∣≤1. For the appropriate synthesis of the desired field s(x,y), instead of generating the CGH for this field the CGH is implemented for the modified function s′(x,y). The modified hologram is referred to as compensated CGH, while previously discussed holograms are referred to as non-compensated CGHs.

2.4 Signal to noise ratio and efficiency of holograms

For the quantitative evaluation of the holograms performance we employ the SNR definition

where s(x,y) and s_t(x,y) denote the ideal signal and the signal under evaluation, respectively, D_S is the domain where evaluation is performed, and the constant β is

where Re{‥} represents the real part of the function within brackets.

Now, let us discuss the reconstruction efficiency of amplitude CGHs. Considering that constant c₀ in Eq. (2) is approximately ½, the signal term in the continuous amplitude CGH is given by s_i(x,y) = (¼)s(x,y) exp[-i2π(u₀ x+v₀ y)]. It is reasonable to define the efficiency of the continuous CGH in Eq. (2) as the power of function s_i(x,y), normalized with the power of signal s(x,y), in a given signal support. Under this definition, the efficiency of the continuous CGH in Eq. (2) is η_i =1/16. For a pixelated and compensated CGH, we denote the reconstructed signal as s′_m(x,y), which is computed by the inverse Fourier transform of spectrum S′_m(u,v), defined in Eq. (15). Thus, the efficiency of the pixelated CGH, normalized by the efficiency η_i , of the continuous CGH, is given by

This normalized efficiency represents the power reduction in the CGH reconstructed signal due to pixelated SLM structure. It can be shown that the upper limit of the normalized efficiency in Eq. (19), if both the signal bandwidth and the CGH carrier frequency tend to zero, is η_lim =a²b² /δx⁴ , for the SLM pixel parameters a, b and δx. In the next section, the above expressions for SNR and efficiency will be employed to evaluate the quality of the discussed amplitude CGHs.

3. Performance of amplitude CGHs encoding high order Bessel beams.

To illustrate the performance of amplitude CGHs displayed with a pixelated SLM, we will consider that the encoded complex field is a high order Bessel beam, with complex amplitude

expressed in polar coordinates (r,θ). In Eq. (20) J_w is the Bessel function of real-valued order w, ρ₀ is the radial spatial frequency of the beam, a₀ is a normalization constant, and circ(r/R) represents a circular support of radius R. For the implementation of a CGH with a pixelated SLM it is necessary to employ a uniformly sampled version of the signal s(r, θ). Basic parameters of the SLM are the sampling period (or pixel pitch) δx and the bandwidth Δu=1/δx. For numerical simulations we consider the SLM pixel parameters δx=33 μm, a=24 μm, and b=28 μm, corresponding to the SLM set “HoloEye LC2000”, from HOLOEYE Photonics AG, which will be employed in the experimental section. The active pixels in this SLM form an array of 600 rows and 800 columns. Since usual SLMs provide 256 modulation levels we disregard gray level quantization for our simulations.

3.1 Amplitude holograms without coupled phase modulation

We first evaluate amplitude CGHs with no coupled phase, whose transmittance is defined by Eq. (2). Let us start considering a CGH encoding a first order Bessel beam with spatial frequency ρ₀ =Δu/20. The modulus and phase of this complex function are shown in the false color images of Fig. 1. All the encoded signals and CGHs to be considered present a circular support with radius R=100δx.

 

Fig. 1. (a) Modulus and (b) phase of a first order Bessel beam.

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For every type 2 CGH encoding Bessel beams, we compute the bias 2 employing α=ρ₀ in Eq. (5). The profiles of bias functions 1 and 2, at the central row of the signal domain, are plotted in Fig. 2. The profile of the signal modulus a(x,y) is included as a reference.

 

Fig. 2. Profiles of the bias functions (of type 1 and 2) and the signal modulus a(x,y), for a first order Bessel beam, at the central row of the 2-D signal domain.

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For a first qualitative comparison of CGHs of types 1 and 2, we encoded the first order Bessel beam considered above, using carrier frequencies u₀ =v₀ =Δu/9. The computed normalized modulus of the signal spectra for these holograms, displayed in Fig. 3, shows that the CGH of type 2 is less noisy than the CGH of type 1. To obtain an appropriate display of signal spectra, the images in Fig. 3 do not include the peaks of the bias contributions, found close to the top-right corners of the displayed domains.

 

Fig. 3. Signal spectra for the amplitude CGHs of types (a) 1 and (b) 2 encoding a first order Bessel beam, with carrier frequencies u₀ =v₀ =Δu/9.

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We complete the numerical synthesis of the complex signal s(x,y) by introducing a band pass filter centered at the frequency coordinates (-u₀ ,-v₀). For our simulations the filter is a circular pupil of radius ρ_s =2ρ₀ . The signal reconstruction is generated by the inverse Fourier transform of the field transmitted by the filter. The moduli of the signals recovered from the designed CGHs of types 1 and 2, are shown in Fig. 4. Comparison of the reconstructed beams with the original signal, shows the better performance of the CGH of type 2.

 

Fig. 4. Moduli of reconstructed signals for the CGHs of types (a) 1 and (b) 2, encoding the first order Bessel beam shown in Fig. 1.

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Computation of the SNR with Eq. (17), considers both the modulus and the phase of the synthesized fields. Figure 5 displays the SNR versus the carrier frequency u₀ =v₀ and versus the signal spatial frequency ρ₀ , for non-compensated CGHs encoding a first order Bessel beam. According to these plots, the CGHs of type 2 show higher SNR than the CGHs of type 1. In the domain for SNR evaluation, we excluded the last ring of the generated beams. Figure 5(a) shows maximum SNR around the frequency u₀ =0.15Δu=3ρ₀ . For u₀>0.15Δu the SNR decrease, for both CGH types, is explained in the distortion of the signal spectrum [Eq. (13)] due to the factor W(u,v). The SNR decrease versus the signal frequency [Fig. 5(b)] is an expected result that confirms the consistency of our simulations. The influence of the envelope W(u,v) is illustrated in Fig. 6, which displays the signal spectra for non-compensated CGHs of type 2 encoding a first order Bessel beam with radial frequency ρ₀ =Δu/10, using carrier frequencies u₀ =v₀ =Δu/6 and u₀ =v₀ =Δu/3, respectively. The influence of the spectrum envelope W(u,v) is more significant for the CGH employing the higher carrier frequency.

 

Fig. 5. SNR for non-compensated CGHs of types 1 (blue) and 2 (green) encoding a first order Bessel beam. (a) SNR versus u₀ (with ρ₀ =Δu/20), and (b) SNR versus ρ₀ (with u₀ =Δu/6).

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Fig. 6. Modulus of signal spectrum for a non-compensated type 2 CGH encoding a first order Bessel beam with radial frequency ρ₀ =Δu/10 and carrier frequencies (a) u₀ =v₀ =Δu/6 and (b) u₀ =v₀ =Δu/3.

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The first order Bessel beam considered above was also encoded by compensated CGHs of types 1 and 2, described in section 2.3. The SNR dependence on the carrier frequency for compensated CGHs is shown in Fig. 7(a). Plots in this Fig. indicate that the SNR reduction, beyond the critical carrier frequency u₀ =0.15Δu [in Fig, 5(a)], is effectively avoided in compensated CGHs. Additional SNR plots versus carrier frequency (u₀ =v₀ ) for compensated holograms, encoding Bessel beams of several orders (with ρ₀ =Δu/20), are shown in Fig. 7(b).

 

Fig. 7. (a) SNR versus u₀ for compensated CGHs of types 1 (blue) and 2 (green) encoding a first order Bessel beam. (b) SNR versus u₀ for compensated CGHs of type 2 encoding Bessel beams of orders 2 (green), 3 (pink), 4 (black), and 5 (blue).

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3.2 Amplitude holograms with coupled phase modulation

Now we design and evaluate type 3 amplitude CGHs, whose bias function is b(x,y)=1. It is assumed that the CGHs are displayed with a SLM providing amplitude modulation coupled with a linear phase modulation. The CGH transmittance is given by Eq. (7), with the coupled phase ϕ_c(x,y)= γh(x,y), where h(x,y) is the CGH amplitude modulation defined in Eq. (2). For simulations we assume that the slope of the coupled phase is γ=π/4. It is also assumed that the CGH design is of the compensated type. Other conditions for signal reconstruction are equal to those employed in subsection 3.1.

We designed compensated type 3 CGHs, encoding the first order Bessel beam already shown in Fig. 1. We first computed the phase of the reconstructed field for the type 3 CGH without coupled phase. This phase [shown at Fig. 8 (a)] is essentially equal to that of the encoded signal [in Fig. 1 (b)]. The phase of the signal reconstructed from the CGH with coupled phase is shown at Fig. 8 (b). Thus, the CGH with coupled phase produces a reconstructed signal with a phase modulation which is equivalent to that of the encoded signal, except by a constant phase shift. According to the analysis in subsection 2.2, it is expected that this phase shift correspond to the phase of the factor under brackets in Eq. (11). We computed the phase of this factor, considering c₀ =1/2, γ=π/4, and b(x,y)=1. The resulting phase, Δ(ϕ)≅0.244π, corresponds closely to the phase shift between the encoded signal and the reconstructed signal.

 

Fig. 8. (a) Phase modulation of first order Bessel beams synthesized with compensated type 3 CGHs (a) without and (b) with coupled phase.

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We also computed the SNR versus the carrier frequency for compensated type 3 CGHs, with and without coupled phase, encoding high order Bessel beams. In the first case, for a fair comparison of the reconstructed signal with the desired signal, the first one is multiplied by the phase shift exp(-iΔϕ), where Δϕ is the phase of the factor under brackets in Eq. (11). The SNR plots for the type 3 CGHs, versus the carrier frequency (u₀ =v₀ ) are shown in Fig. 9.

 

Fig. 9. SNR versus u₀ for compensated CGHs of type 3 (a) without and (b) with coupled phase. The signal spatial frequency is ρ₀ =Δu/20. The orders of encoded Bessel beams are 1(blue), 2 (green), 3 (red), 4 (cyan), and 5 (magenta).

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Comparison of Figs. 7 and 9 indicate that type 3 CGHs present lower SNR than CGHs of type 2. This result is partially explained considering that a type 3 bias function shows larger bandwidth and power than a type 2 bias function. It is also noted in Fig. 9 that the coupled phase in type 3 CGHs produces a further decrease in the SNR. This noise increment can be explained by the generation of multiple mixed powers of the signal t_s and its conjugate, due to the CGH coupled phase, as established in the analysis of section 2.2. In other simulations we proved that type 2 CGHs with coupled phase show much smaller SNR than type 3 CGHs with coupled phase. This result justifies the necessity of employing type 3 CGH designs if the amplitude SLM presents a coupled phase modulation.

3.3 Numerical evaluation of efficiency

An important drawback of an amplitude CGH is its low efficiency. The normalized efficiency in Eq. (19) represents the power of the reconstructed signal for the pixelated CGH, normalized by the power of the CGH signal term s_i(x,y)= (¼)s(x,y) exp[-i2π(u₀ x+v₀ y)]. Thus, this normalized efficiency only represents the reduction in reconstruction signal power due to the pixelated structure of the SLM. The limit value of the normalized efficiency (η_lim =a²b² /δx⁴ ) for the assumed SLM pixel parameters in our simulations (a=24μm b=28μw, and δx=33μm) is η_lim ≅0.38. To consider non-compensated CGHs, the signal s′_m(x,y) [in Eq. (19)] is replaced by s_m(x,y), which represents the inverse Fourier transform of the non-compensated signal spectrum S_m(u,v) [in Eq. (13)]. The efficiency versus carrier frequency (u₀ =v₀ ) for type 2 CGHs encoding Bessel beams of several orders (with ρ₀ =Δu/20), is shown in Figs. 10(a) and 10(b) (for non-compensated and compensated cases). According to these results, compensated CGHs show slightly smaller efficiencies than non-compensated CGHs. This difference is understood as a consequence of the final renormalization imposed to the pre-filtered signal s′(x,y) to fulfill the restriction ∣s′(x,y)∣≤1.

 

Fig. 10. Normalized efficiency versus u ₀ for (a) non-compensated and (b) compensated type 2 CGHs encoding Bessel beams of orders 2 (green), 3 (pink), 4 (black), and 5 (blue).

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4. Experimental implementation of amplitude holograms

For implementation of amplitude CGHs encoding arbitrary complex beams we employ the TNLC-SLM set “LC 2000”, from HOLOEYE Photonics AG. To provide amplitude modulation the SLM employs two external polarizers, as indicated in Fig. 11. The electronic driver of the SLM polarizes each SLM pixel with different voltages, related with the so called gray level g that takes integer values from 0 to 255. The optical modulation of the SLM setup will depend on the orientation of the transmission axes of the polarizer and the analyzer, respect to the LC director axis at the SLM input plane [26]. According to the discussion in section 2, for appropriate realization of amplitude CGHs with the SLM, the range of the phase modulation, coupled to the amplitude modulation, must be minimized (for type 2 CGHs) or linearized (for type 3 CGHs). We have found that the setup in Fig. 11, with the polarizer perpendicular to the input plane director axis, provides a relatively reduced and quasi-linear phase modulation. Our experimental setup for the TNLC-SLM employs a collimated He-Ne laser beam (632.8 nm), and angles θ₁=90° (for polarizer) and θ₂=0° (for analyzer), measured respect to the input SLM director axis. The experimentally measured amplitude modulation and the coupled phase modulation, versus g, are shown in Fig. 12 (a,b). The phase modulation takes values in a range of size Δϕ<0.327π. The phase modulation versus the amplitude modulation is shown in Fig. 12 (c).

 

Fig. 11. Arrangement of a TNLC-SLM with two linear polarizers to obtain amplitude modulation.

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Fig. 12. Experimental (a) amplitude and (b) phase modulations versus gray level g, and (c) phase versus amplitude modulation, measured for the TNLC-SLM with polarizers oriented at angles θ₁=90°, θ₂=0°.

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Considering the quasi-linear relation between the coupled phase and the amplitude modulation [in Fig. 12(c)], we employed the amplitude modulation [in Fig. 12 (a)] to design type 3 amplitude CGHs for the synthesis of a variety of complex fields. It is noted that the minimum amplitude in Fig. 12(a) is B₀ ≅0.02. This fact is considered by displaying into the SLM the modified CGH h′(x,y)=B₀ +C₀ h(x,y), where h(x,y) is the original normalized CGH defined in Eq. (2), and C₀ =0.98. The experimental setup for the synthesis of complex beams, depicted in Fig. 13, is a spatial filtering processor, where the SLM (CGH) is placed at the input plane. A band-pass circular pupil, at the Fourier plane, transmits the light from the signal spectrum, generating the encoded signal at the output plane of the setup.

 

Fig. 13. Spatial filtering setup for the synthesis of arbitrary complex fields with amplitude CGHs.

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We implemented several type 3 compensated CGHs, using a radius support R=100δx and carrier frequencies u₀ =v₀ =Δu/4. As an initial task, we designed amplitude CGHs encoding first-order Bessel beams with spatial frequencies ρ₀ =Δu/Q, for Q=12, 16, 20 and 24. The intensity distributions of the beams generated by these CGHs, registered with a CCD camera, are shown in Fig. 14. In a second experiment, we designed CGHs to synthesize Bessel beams of orders w=2, 3 and 4, with spatial frequency ρ₀ =Δu/20. The images of the recorded output fields in the second experiment are shown in Fig. 15.

 

Fig. 14. Experimentally recorded first order Bessel beams generated with type 3 compensated amplitude CGHs, employing carrier frequencies u₀ =v₀ =Δu/4. The beam spatial frequencies are (a) ρ₀ =Δu/12, (b) ρ₀ =Δu/16, (c) ρ₀ =Δu/20, and (d) ρ₀ =Δu/24.

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Fig. 15. Experimentally recorded Bessel beams of orders (a) 2, (b) 3, and (c) 4, generated with type 3 compensated amplitude CGHs employing carrier frequencies u₀ =v₀ =Δu/4. The beam spatial frequency is ρ₀ =Δu/20.

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To visualize the azimuthal phase of the experimentally generated Bessel beams, we introduced a reference plane wave at the output of the setup (in Fig. 13) by adding a pinhole at the zero frequency position in the spatial filter. This pinhole transmits the central portion of the CGH zero order spectral spot. The experimental interference patterns obtained at the output plane proved that the azimuthal phases of the synthesized beams were appropriately encoded by the CGHs. Figure 16 shows close views of the interference patterns obtained for the synthesized Bessel beams of orders 1, 2, 3 and 4, with spatial frequency ρ₀ =Δu/20. The system of fringes, due to the linear phase factor associated to off-axis reconstruction, shows the expected dislocations of π radians between adjacent rings. On the other hand, the characteristic fork shapes, for azimuthal phases of orders w=1-4, are observed at the centers of images.

 

Fig. 16. Interferograms of the experimentally generated Bessel beams of order (a) 1, (b) 2, (b) 3, and (b) 4, with spatial frequency ρ₀ =Δu/20, employing a on-axis reference plane wave.

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In another experiment we encoded the real valued signal

corresponding to the superposition of two Bessel beams of orders w=4 and w=-4, respectively. The real modulation in Eq. (21) allows the generation of an azimuthally modulated ring shaped field. This field, obtained as the Fourier transform of the signal s(r, θ), in Eq. (21), can be rotated by introducing a variable phase shift Δϕ. We implemented such a rotating field by displaying a sequence of compensated, type 3, CGHs in the amplitude TNLC-SLM. The radial frequency of the Bessel function J₄ was ρ₀ =Δu/16, and the CGH carrier frequencies were u₀ =v₀ =Δu/4. The experimentally recorded rotating signal spectrum is shown in Fig. 17.

 

Fig. 17. (393 KB) Rotating ring shaped field with azimuthal intensity modulation, generated by a sequence of type 3 compensated amplitude CGHs. The CGHs encode the complex field J₄(2πρ₀r) cos(4θ-Δϕ) in a finite circular support, with increasing values of the phase shift Δϕ.

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5. Conclusions and remarks

We analyzed the attributes of amplitude CGHs displayed on amplitude LC-SLMs and discussed their optimization, for the synthesis of fully complex optical fields. Since the CGH transmittance is displayed into a SLM, and not by employing an optical holographic setup, it was possible to substitute the bias term in the expression for the hologram transmittance by modified bias functions. Specifically, for CGHs displayed into amplitude SLMs with no coupled phase modulation we obtained that the convenient bias is a soft envelope of the signal modulus. On the other hand, if the CGH is displayed into a SLM with coupled phase modulation, the convenient bias is a constant function.

Performing numerical simulations we proved that amplitude CGHs show remarkably high SNR. In general, this high SNR is possible since the transmittance of an amplitude CGH has the structure of an interferogram, whose main spectral band only presents the off-axis signal term accompanied by the zero-order term and the conjugate of signal. This structure of the CGH spectrum enables a high SNR reconstruction by employing a band-pass spatial filtering process. An obvious restriction is that the bandwidth of the encoded complex field must be smaller than the bandwidth of the SLM displaying the amplitude CGHs.

A further increment in the SNR of amplitude CGHs was attained by imposing a pre-filtering to the encoded complex signals. The aim of this process was to avoid the distortion of the off-axis signal spectrum due to the factor W(u,v), which represents the Fourier transform of the SLM pixel window.

As a carrier signal we employed a plane wave exp[-i2π(u₀ x+v₀ y)], with carrier spatial frequencies u₀ =v₀ =Δu/4, which provided reasonable SNR. The experimental synthesis of several high-order Bessel beams was performed in a spatial filtering setup. The phase of the synthesized beams was qualitatively evaluated by performing the interference of the beams with a reference on-axis plane wave.

In order to obtain appropriate reconstruction, the bandwidth of the encoded signal must be smaller than the bandwidth Δu of the employed SLM. In the numerically and experimentally implemented CGHs, we encoded signals with spatial frequencies smaller or equal to Δu/12.

The main drawback of an amplitude CGH is its low efficiency. However, the high SNR provided by this CGH and the simplicity of the setup required for its implementation, justify its application in cases where the efficiency is not the main concern.