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Abstract
Independent or joint control over the amplitude and phase of the complex field by phase-only modulation element is crucial in numerous applications. Existing modulation methods can realize high levels of accuracy but are accompanied by noticeable losses in light-usage efficiency. Here a cascaded modulation method is proposed for the generation of arbitrary complex fields with high efficiency and high fidelity. This approach is based on a gradient descent optimization algorithm that minimizes a customized cost function. The major advantage of our approach over existing modulation methods is that the efficiency is significantly enhanced while ensuring high modulation accuracy. For the generation of Laguerre-Gaussian mode (LG01), with similar high accuracy, the efficiency by our approach can reach 79.5%, which is enhanced by 192% compared with the theoretical maximum efficiency of 41.5% [Opt. Express 25, 11692 (2017) [CrossRef] ]. Furthermore, the efficiency of existing modulation methods deteriorates rapidly as the target field turns more intricate, whereas in our approach it maintains at a relatively high level. The field generation fidelity and energy efficiency of the proposed cascaded modulation method are compared with that of several different single-pass modulation methods in generating a series of typical Hermite-Gaussian and Laguerre-Gaussian modes and an amplitude-only "OSA" pattern. Our proposed method features both high efficiency and high accuracy in the simulation and experiment, which may be of growing interest to applications such as optical manipulation or quantum communication.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The advent of phase-only spatial light modulators (SLMs) allows dynamic step-wise control of phase on a pixel-by-pixel basis, spurring a wide range of applications such as optical trapping and manipulation [1–3], microscopy [4] and quantum information and communication [5]. Nevertheless, in terms of arbitrary complex field modulation, such phase-only devices suffer from a basic limitation: amplitude and phase cannot be independently controlled at each pixel. There are several previous works aiming to surpass the limitation and apply simultaneous control over the phase and amplitude of complex fields. These techniques can generally be divided into two categories: theoretical methods (non-iterative) and optimization-based methods (iterative).
For the theoretical methods, there usually exist analytical solutions for a specific field, and these methods include the use of amplitude-based phase gratings [6–11], double-phase holograms (DPH) [12–15], and random coding techniques [16–18]. The essence of these methods is to design a phase hologram based on the amplitude information of the target field, thus enabling independent control of phase and amplitude by phase-only elements. One can either design the phase gratings based on the amplitude information to retain the desired amplitude and diffract the undesired light into other diffraction orders; or encode the complex information of the target field by macro-pixels, forming a binary checkerboard phase to reconstruct the target field in the center of the Fourier plane and redistribute the unwanted light into high spatial frequencies; or one can multiplex the intended phase hologram carrying the amplitude information with a randomly encoded diverging phase to separate the desired field from noise in the reconstruction plane. For these methods that deliberately remove undesired light from the incident field, the energy efficiency is limited by the overlap of 2D distribution of amplitude between the target field and input field. The amplitude of the target field must not exceed the amplitude of the input field on a pixel-by-pixel basis. This mismatch would in turn limit the overall efficiency of the modulation. For example, as shown in Fig. 1, assume that the input field on SLM is the Gaussian mode with a beam waist of 0.78mm, and the target field to be generated is a set of Hermite-Gaussian (HG) modes with a beam waist of 0.32mm. In this case, for the generation of HG$_{01}$ mode, the maximum intensity is close to 0.9, while for the high-order HG$_{07}$ mode, the peak value would be less than 0.2. The overlap area of HG$_{07}$ mode is noticeably smaller than that of the HG$_{01}$ mode, implying that the efficiency would decrease in the generation of higher-order modes by theoretical methods.
Fig. 1. Overlap of the axial intensity distribution between the input Gaussian mode and the target HG$_{01}$ and HG$_{07}$ mode.
Compared with the theoretical methods, optimization-based methods can further improve the efficiency and accuracy, which can be broadly categorized into iterative Fourier transform algorithm (IFTA) with its modified versions [19–22] and the minimization of cost function techniques [23–26]. In general, the IFTA methods are computationally efficient but the iteration convergence requires a carefully chosen initial phase pattern and constraints at each step. The minimization of the cost function techniques are inherently more directional than IFTAs, allowing a fast convergence. These techniques efficiently minimize a specified cost function which can be designed to provide a high level of control over the modulated field. Among them, a high-fidelity single-pass modulation method based on gradient descent is proposed [26]. The phase hologram is calculated with the conjugate gradient minimization approach and the accuracy and smoothness surpass that of previous IFTA approaches. For generating a Laguerre-Gaussian (LG$_{01}$) mode, the fidelity and calculated efficiency reach $F=0.999997$ and 41.5$\%$ respectively. Though the accuracy is relatively high, a large part of input energy is still wasted.
In contrast with the previously mentioned single-pass modulation techniques, cascaded modulation methods can provide higher efficiency but at the expense of accuracy [27–31]. In such techniques, the incident field is modulated by two phase-only SLMs located in conjugate Fourier planes, where the first SLM creates the desired amplitude information and the second SLM corrects the undesired phase induced by the previous SLM. In Ref. [31], the energy efficiency of HG$_{09}$ mode is up to 61$\%$. However, the accuracy of the cascaded modulation schemes is still lacking due to the fact that it is impossible to achieve perfect amplitude modulation through the modulation of the first SLM, and the deviation in amplitude cannot be corrected by the second SLM, resulting in a further loss of accuracy.
Here we demonstrate an alternative cascaded modulation approach for the generation of arbitrary complex fields with high efficiency and high accuracy simultaneously. It is based on the minimization of a customizable cost function with an optimization strategy inspired by a non-convex gradient descent algorithm [32–34]. The cost function is defined as the difference between the modulated field and the target field. It also incorporates a power scaling factor that allows a trade-off between high fidelity and high efficiency. The gradient of the cost function versus the cascaded phase profile is deduced and applied to a quasi-Newton gradient descent procedure that converges faster but less computer memory is required [35]. By sequentially modulating the incident field, the whole incoming field is redistributed and contributes to the final generation of the target complex field. Therefore, it is possible to significantly increase the efficiency with a high degree of fidelity.
2. Cascaded modulation method
In this section, firstly the theoretical model of the cascaded modulation method is described. Then the definition of cost function and calculation of the gradient is presented, and the optimization algorithm is introduced.
2.1 Concept of the cascaded modulation method
The purpose is to generate an arbitrary complex field ${f_T}({x_T},{y_T}) = {A_T}({x_T},{y_T})\exp (i{\phi _T}({x_T},{y_T}))$ usually in the Fourier plane of a single focusing element, where ${A_T}({x_T},{y_T})$ and ${\phi _T}({x_T},{y_T})$ are the required amplitude and phase profile. The general scheme of single-pass modulation methods are as described in Fig. 2(a). The incident Gaussian field, ${f_0}({x_0},{y_0})$, propagates after a distance $z$ onto a phase-only SLM, which carries a phase function of $\exp (i{\phi _{SLM}}(x,y))$ and modulates the incoming field ${f_1}({x_1},{y_1})$. Then the reconstructed complex field ${f_R}({x_R},{y_R})$ is generated on the Fourier plane, where $({x_j},{y_j})$ with $j = 0,{{\ }}1$ and ${{\ }}R$ stands for the transverse coordinate in each plane. Under the paraxial approximation, the complex fields at each plane can be described as
The normalized fidelity $F$ quantifies the accuracy of the generated complex field [7]. This can be calculated from the overlap integral of the target complex field ${f_T}(x,y)$, and the reconstructed complex field ${f_R}(x,y)$ as
Here $(x,y)$ are the coordinates in the region of interest, the superscript ${{( \cdot )}^{\text {*}}}$ denotes the complex conjugate. And the energy efficiency is calculated by comparing the power in reconstructed field ${f_R}$ and incident field ${f_0}$ by using :
Here ${f_0}$ is normalized as ${\iint {\left | {{f_0}(x,y)} \right |} ^2}dxdy = 1$. For the proposed cascaded modulation method, the general design principle is described in Fig. 2(b). The complex fields at each plane can be expressed by using notations ${\mathcal {T}_z}$ and ${\mathcal {T}_f}$ as
2.2 Optimization algorithm
For the proposed cascaded modulation approach, the number of SLM pixels to be considered is $X \times Y \times N$. $X$ and $Y$ is the number of pixels along x and y direction and $N$ represents the number of phase holograms. The calculation of gradient and optimization procedure must be conducted in high-dimensional space [32,33]. In this case, the choice of cost function and optimization procedure becomes critical. The cost function is proposed as
where $\phi$ denotes the phase holograms with the size of $X \times Y \times N$, ${f_R}(\phi )$ is the reconstructed field calculated by the phase $\phi$ according to Eq. (5), $\alpha$ is a positive constant smaller than 1 and ${f_T}$ is normalized as ${\iint {\left | {{f_T}(x,y)} \right |} ^2}dxdy = 1$. According to Eq. (4), the expected efficiency would be ${\alpha ^2}$. Now the optimization problem can be concluded byThe challenge is to find the phase hologram ${\phi _{{\text {opt}}}}$ that minimizes the cost function $L(\phi )$, i.e., the field generated by cascade phase-only modulation is closest to the target field. The optimization problem of Eq. (7) is non-convex, for which it is hard to find the global optimal solution directly. However, the cost function has a well-defined derivative, which makes it possible to solve the problem using gradient-descent optimization methods. For ease of expression, the proposed cascade modulation method is termed as cascaded modulation method by non-convex optimization (CMNO). The generated field ${f_R}(\phi )$ is computed by a recursive sequence as illustrated in Eq. (5). The gradient is then calculated in each plane of the phase profile ${\phi _j}{\text {, }}j = 1,{\text { }} \cdots,{\text { }}N$. In the $j$-th plane, Eq. (6) can be described in a matrix style as
where ${S_j}(\phi _j) \triangleq ({{\mathbf {A}}_j}{f_0}\exp (i{\phi _j}) - \alpha {f_T})$ is the difference between the generated field and target field. ${{\mathbf {A}}_j}$ is the transform matrix with size $M \times N \times j$ which stands for the linear operation from the input field ${f_0}$ to the current field ${f_j}$, satisfying the relation of ${{\mathbf {A}}_j}{f_0} = {f_j}$. The quadratic term in Eq. (8) can be expanded as where ${( \cdot )^{\text {H}}}$ denoted Hermitian operator. According to Ref. [33], the partial derivative of $L({\phi _j})$ with respect to the phase ${\phi _j}$ is described asConsidering that ${S_j}(\phi ) \triangleq ({{\mathbf {A}}_j}{f_0}\exp (i{\phi _j}) - \alpha {f_T})$, so $\frac {{\partial {S_j}(\phi )}}{{\partial {\phi _j}}} = i{{\mathbf {A}}_j}{f_0}\exp (i{\phi _j})$. Therefore the gradient at $j$-th plane is expressed as
According to Eq. (12) and Eq. (13), a local optimal solution for Eq. (7) can be obtained by first or second-order optimization algorithms. Here, a quasi-Newton gradient descent approach is used as the optimization procedure that converges to a stationary point and only requires limited amounts of computation memory [35]. Once the cost function $L(\phi )$ is determined and the corresponding gradient ${\nabla _{{\phi _j}}}L$ is calculated, a Matlab function $fmincon( \cdot )$ is adopted to perform the optimization, which is easy to implement and fast in computation.
3. Numerical simulations
In this section, numerical simulations are performed to compare the efficiency and fidelity of CMNO and several widely-used single-pass modulation methods.
3.1 Comparison of different modulation methods
The analytical expressions for the phase hologram ${\phi _{SLM}}(x,y)$ of eight single-pass modulation techniques are listed in Table 1. For methods S1 to S8, the reconstructed fields are spatially separated in the first diffraction order by multiplexing an extra grating phase given as ${\phi _g}(x,y) = {\text {Mod}}(2\pi x/\Lambda,2\pi )$, where $\Lambda$ is the grating period. In our simulations and corresponding experiments $\Lambda$ is taken to be 8 pixels. The first six methods labelled from S1 to S6 are described in detail in Ref. [6]. In addition, a random encoding modulation method labelled as S7 and a DPH modulation method based on macro-pixels labelled as S8 are also taken into account.
Table 1. Phase hologram of different single-pass modulation methodsa
The parameters of the simulation are consistent with the experiment setup in the next section. The input field ${f_0}$ is the fundamental Gaussian mode with a beam waist radius of 0.42mm located at a distance of $z = 650{\text {mm}}$ from the first plane of the phase modulation. The incident angle is $3^\circ$. The focal length of the lens is $f = 300{\text {mm}}$ and the wavelength is $\lambda = 1064{\text {nm}}$. The sampling number is 636$\times$636 and the size of each sampling unit is $12.5\mu {\text {m}}$ $\times$ $12.5\mu {\text {m}}$. For CMNO method, the distance between adjacent phase holograms is $l = 152{\text {mm}}$. The target fields are a series of HG and LG beams with a beam waist radius of 0.25mm. The target ${\text {H}}{{\text {G}}_{mn}}$ modes are parameterized by their mode order numbers $m$ and $n$, indicating mode orders along the $x$ and $y$ directions in the transverse plane. The target ${\text {LG}}_{pl}^*$ modes are selected from ${\text {LG}}_{01}^*$ to ${\text {LG}}_{08}^*$, where the $p$ and $l$ represent the radial and azimuthal degree of freedom and the superscript ${( \cdot )^*}$ denotes the petal-like beam from the superstition of ${\text {L}}{{\text {G}}_{p,+l}}$ and ${\text {L}}{{\text {G}}_{p,-l}}$ mode as ${\text {LG}}_{pl}^*={\text {L}}{{\text {G}}_{p,+l}}+{\text {L}}{{\text {G}}_{p,-l}}$. For CMNO method, there is a trade-off in the choice of the power scaling factor $\alpha$. A large $\alpha$ means that more energy is restricted into the region of interest in the reconstruction plane, which makes it hard to obtain a high fidelity. While for a small $\alpha$, the fidelity can be significantly improved, whereas the efficiency may be affected since the ideal efficiency is equal to ${\alpha ^2}$. For simplicity, we test different power scaling factors for different target modes, ensuring that for each target mode, the fidelity is adequately high ($F \geqslant 0.9999$).
The calculated efficiency and fidelity of the different modulation methods are shown in Fig. 3. In the bottom legend, S1 to S8 denotes the single-pass modulation methods as listed in Table 1. CMNO-1 represents the proposed cascade modulation method with one plane of modulation, while CMNO-2 represents two planes of modulation. In general, the fidelity $F$ is very close to 1. For clarity, the fidelity data is plotted logarithmically as $- {\log _{10}}(1 - F)$.
Fig. 3. Light-usage efficiency and fidelity of each modulation method in generating a series of HG and LG modes.
As depicted in Fig. 3(a) and Fig. 3(c), the energy efficiency of CMNO-2 approach significantly exceeds that of the other modulation methods. It can also be seen that, as the order of the target mode increases, the efficiency of all methods tends to decrease noticeably. This is because for high-order target modes, the transverse size enlarges and the overlap between the desired field and the input field imposes further limits to the amplitude modulation, which will degrade the overall energy efficiency. Nevertheless, the efficiency of CMNO-2 approach in generating high-order modes still prevails that of the other methods. For example, in the generation of HG$_{44}$ mode as shown in Fig. 3(a), the efficiency of CMNO-2 method is close to 20$\%$ while the other methods are no more than 5$\%$. In the special case such as CMNO-1 method in Fig. 3(c), the efficiency in the generation of ${\text {LG}}_{02}^*$ mode, is somewhat higher than that of the ${\text {LG}}_{01}^*$ mode, which can also be attributed to the spatial overlap of the input field and generated field. Under the current simulation parameters, the overlap area of ${\text {LG}}_{02}^*$ mode is slightly larger and therefore the potential efficiency is higher. The fidelity of CMNO approach with either 1-plane or 2-planes modulation is noticeably higher than that of the other methods as demonstrated in Fig. 3(b) and Fig. 3(d), since the cost function of the minimization can be considered to be directly related to fidelity. Theoretically, each method has a high degree of fidelity, even for the generation of the higher order mode such as ${\text {LG}}_{08}^*$ mode ($F{\text { > }}0.95$).
To better illustrate the performance of each method, the ${\text {LG}}_{08}^*$ mode generated by different modulation methods is presented in detail as shown in Fig. 4. The target intensity and phase of the ${\text {LG}}_{08}^*$ mode are shown in Fig. 4(a). The efficiency and fidelity of different modulation methods are depicted in Fig. 4(b), where CMNO-2 method exhibits the highest fidelity and efficiency among all methods. The axial intensity distribution of the generated ${\text {LG}}_{08}^*$ mode by each method is presented in Fig. 4(c). The axial distributions are chosen at the white dashed line in the middle of the 2D intensity profile respectively. As shown in the zoomed intensity distribution at the main lobe in Fig. 4(c), the intensity distribution of the reconstructed field generated by CMNO-2 method coincides almost perfectly with that of the target field. While the other methods are more or less deviated, especially the S4 method and S7 method, the intensity distribution of the main lobe differs considerably from the target field, indicating a relatively low fidelity. This is in accordance with the fidelity distribution in Fig. 4(b) where CMNO-2, CMNO-1 and S6 methods rank the top three methods in the generation of ${\text {LG}}_{08}^*$ mode respectively.
Fig. 4. (a) Target phase and intensity of the ${\text {LG}}_{08}^*$ mode. (b) Simulated efficiency and fidelity of different modulation methods. (c) From left to right: The calculated 2D intensity profile and phase (in inset), the axial intensity distribution at the white dashed line, and the zooming of the intensity distribution in the red box.
3.2 Influence of the number of modulation planes for CMNO method
The 2-planes CMNO method has demonstrated higher efficiency and fidelity among all the modulation methods. However, as the complexity of the target mode increases, the achievable efficiency degrades considerably. The following simulations show that this can be resolved by increasing the number of modulation planes and the efficiency can be further improved. The simulation results are as shown in Fig. 5. The efficiency of CMNO-3 method is increased by $\sim$16$\%$ on average compared to CMNO-2 method while maintaining the same level of fidelity ($F{\text { > }}0.9999$). When the number of modulation planes comes to 4, in the generation of HG modes, it is observed that the efficiency maintains at a high level for the first 7 orders of HG modes (all around 90$\%$). For the generation of high-order LG modes as seen in Fig. 5(c), the efficiency of generating ${\text {LG}}_{06}^*$, ${\text {LG}}_{07}^*$ and ${\text {LG}}_{08}^*$ mode is enhanced by 158$\%$, 173$\%$ and 150$\%$ respectively compared to that of CMNO-3 method. The general tendency shows that the efficiency of CMNO method can be further improved as the number of modulations increases. Indeed, it has been theoretically proven that a finite succession of modulations can realize an arbitrary unitary transform of light, i.e. lossless generation of arbitrary complex fields with high fidelity [36].
Fig. 5. Light-usage efficiency and fidelity of CMNO method with 1-4 modulation planes in generating a series of HG and LG modes.
4. Experimental verification
In this section, the efficiency and fidelity of CMNO-2 method are experimentally measured and compared with the single-pass modulation methods in the generation of a series of HG and LG modes as well as an amplitude-only "OSA" pattern.
4.1 Experimental setup
The experimental setup is presented in Fig. 6. A fundamental Gaussian output from a Nd:YAG laser at a wavelength of 1064nm is split into two beams using a beam splitter (BS). One path is incident on the phase-only liquid crystal SLM (Hamamatsu X15213 series) at an angle of $3^\circ$ and focused on a CCD camera (Spiricon SP928) by a plano-convex lens with a focal length of 300mm (Thorlabs LA1484-C). The other path acts as a reference beam and interferes with the modulated beam to produce fringes, which are used to recover the phase of the modulated field via a fringe deciphering technique [37].
Fig. 6. Experiment setup of (a) interference scheme for phase recovery and (b) cascaded configuration for CMNO-2 method. (c) Detailed schematic of the cascaded modulation.
A single SLM (15.9mm$\times$12.8mm) and a mirror (12.7mm$\times$12.7mm) constitute the cascaded scheme of the proposed approach as seen in Fig. 6(b). The SLM has a pixel number of 1272$\times$1024, a pixel pitch of $12.5{\mu \text {m}}$. Consistent with the parameters of the simulation, the SLM is separated into two parts, each of which has a pixel number of 636$\times$636. For the realization of single-pass modulation methods including CMNO-1 method, one can simply load the phase hologram on the left half of the SLM as shown in Fig. 6(c) while the other half of SLM carries the phase of flat mirror, without other extra adjustments of the experiment configuration.
4.2 Results and discussions
The measured efficiency and fidelity of different modulation methods are displayed in Fig. 7. The efficiency $\eta$ is the fraction of power between the reconstructed field and the input field measured by a power meter (Thorlabs PM100D). The fidelity is calculated by Eq. (3) according to the intensity measured by a CCD camera and the phase recovered by interference fringes.
Fig. 7. Experimentally measured light-usage efficiency and fidelity of different methods in the generation of a series of HG and LG modes.
As depicted in Fig. 7(a) and Fig. 7(c), the energy efficiency of CMNO-2 approach significantly exceeds that of the other methods, which is consistent with the simulation results. For example, in the generation of HG$_{10}$ mode, the efficiency of CMNO-2 method is 70$\%$, while the other single-pass modulation methods exhibit an efficiency lower than 30$\%$. The enhancement of CMNO-2 method is up to 233$\%$. In general, the measured efficiency is lower than that of theoretical simulations due to the fact that the SLM has a certain reflectivity (97$\%$ at 1064nm) and an incomplete filling factor (96$\%$). But the overall tendency of efficiency is consistent with simulation results in which the energy decreases as the order of target modes increases. The fidelity of CMNO-2 method is the highest among all methods as shown in Fig. 7(b) and Fig. 7(d). For the single-pass modulation methods, the fidelity degrades rapidly as the order of target modes increases. This is because relatively low efficiency means higher noise levels, or more precisely, reduced fidelity. Moreover, in simulations, the use of the grating phase allows complete separation of the light in different diffraction orders. However, in the experiment, the SLM suffers from imperfect phase modulation calibration, nonlinear phase responses, and deviation from perfect flatness. The crosstalk between the desired field and the undesired field in other diffraction orders, especially 0th-order, becomes an important factor that affects the accuracy of single-pass modulation methods. The unmodulated zeroth-order light will inevitably be superimposed on the reconstructed field, which deteriorates the accuracy of field generation. This can be avoided in CMNO-2 approach because no grating phase is used and all the incident light is redistributed into the region of interest in the reconstructed plane, which is one of the major advantage of our approach in improving the fidelity of modulation.
To better illustrate the fidelity of different modulation methods, the generation of an HG mode (HG$_{22}$), an LG mode (${\text {LG}}_{08}^*$), and an amplitude-only "OSA" pattern is selected for a detailed demonstration. The measured 2D intensity profile, the recovered phase, and the axial intensity distribution of each mode generated by each method are presented in Fig. 8 to Fig. 10. Consistent with Fig. 3, the fidelity data is plotted logarithmically. In Fig. 8, for generating HG$_{22}$ mode, the efficiency and fidelity of CMNO-2 approach both exceed that of the other methods as seen in Fig. 8(b). The fidelity of the different modulation methods can be observed more directly in the axial intensity distribution as shown in Fig. 8(c). The axial intensity distribution of the HG$_{22}$ mode generated by methods S4 and S7 differs from that of the target field to a relatively large extent. While the intensity distribution produced by methods S1, S2, S3, and S5 differ slightly from the target one. The modulated field by methods S6, S8, CMNO-1, and CMNO-2 method corresponds well with the target mode, wherein CMNO-1 method and CMNO-2 method produces near-perfect axial intensity distribution, the curve of axial intensity almost overlaps entirely with the target field. For the generation of ${\text {LG}}_{08}^*$ mode as presented in Fig. 9, the performance of each method is similar as in the HG$_{22}$ case. CMNO-2 method features the highest efficiency and high fidelity simultaneously. The efficiency of CMNO-2 method is 18$\%$ while that of the single-pass modulation methods is no more than 12$\%$. Meanwhile, the axial intensity distribution shows that the reconstructed field generated by CMNO-2 method fits best with the target field and has the highest fidelity.
Fig. 8. (a) Target phase and intensity of the HG$_{22}$ mode. (b) Experimental efficiency and fidelity of different modulation methods. (c) The recovered phase, measured 2D intensity profile, and the axial intensity distribution at the white dashed line.
Fig. 9. (a) Target phase and intensity of the ${\text {LG}}_{08}^*$ mode. (b) Experimental efficiency and fidelity of different modulation methods. (c) The recovered phase, measured 2D intensity profile, and the axial intensity distribution at the white dashed line.
Fig. 10. (a) Target intensity of the amplitude-only "OSA" pattern. (b) Experimental efficiency and fidelity of different modulation methods. (c) The measured 2D intensity profile, the axial intensity distribution at the white dashed line, and a zooming of the axial intensity in the red square. “OSA” logo used with permission from Optica–formerly OSA.
For the generation of an amplitude-only "OSA" pattern as seen in Fig. 10, the fidelity of the single-pass modulation methods from S1 to S8 exhibits a similar level while only the efficiency differs slightly. The axial intensity distribution in the middle of the letter "O" is zoomed in to observe the noise in the background as shown in Fig. 10(c). CMNO-2 method produces the least noise level in the middle of the letter "O", and the second lowest noise is produced by CMNO-1 method while the other single-pass modulation methods produce relatively high noise. This corresponds with the fidelity distribution in Fig. 10(b) that the fidelity of CMNO-2 method is the highest, thereby a smooth, high contrast amplitude-only field can be generated.
5. Conclusion
In conclusion, we have proposed an effective cascaded modulation method for the generation of arbitrary complex fields with high efficiency and high fidelity. The performance of the proposed cascaded modulation approach is compared with different single-pass modulation methods both in simulation and in the experiment, exhibiting high efficiency and high fidelity at the same time. This approach allows for accurate control over amplitude and phase, which is essential for a wide range of applications.
In the proof-of-principle experiments, the proposed cascaded modulation approach with only two planes of modulation is conducted. It is expected that as the number of modulations increases, the light-usage efficiency can be further enhanced with a fidelity level at $F{\text { > }}0.9999$. A theoretically lossless generation of arbitrary complex fields with high fidelity can be realized by a succession of cascaded modulations. Although the proposed method features high accuracy and efficiency, there is still room for improving the computational efficiency. Even so, the proposed cascaded approach with two planes of modulation still outperforms the existing single-pass modulation methods. We believe that the proposed approach may find its applications where high efficiency and high fidelity are both required. Potential applications include the generation of LG beams for optical manipulation, the generation of HG beams for optical parametric oscillator, and the generation of flat-top or multi-foci beams for material processing.
Funding
Key Research and Development Program of Hubei Province (2022BAA009); Major Science and Technology Projects of Wuhan (2021012002023424); Major Science and Technology Projects of Hubei Province (2021AAA009).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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