1. Introduction

The optical fiber operates as a flexible light transmission medium within optical windows from ultraviolet to mid-infrared, enabling broad applications in optical communications and biophotonics since 1966. Currently, almost all fiber-based applications are restricted within the scope of single-mode fibers. Compared to single-mode fibers, multimode fibers (MMFs) have larger core sizes that simultaneously support the propagation of various spatial modes. Unfortunately, this benefit is wiped out by the inter-mode crosstalk, which scrambles carried information. Lots of works have been presented for handling this issue, and a technology termed wavefront shaping is one of the solutions. Wavefront shaping, which was originally developed to overcome optical scattering effects [1], has been referenced to handle inter-mode crosstalk [2,3]. By modeling the transmission process through the MMF as a deterministic input-output relation, the optical fields, in principle, can be synthesized at the distal end of the MMF by properly choosing the input wavefront. As a result, the MMF is a promising endoscopic element for direct image delivery [4–6].

The successful operation of wavefront shaping relies on retrieving the transmission matrix (TM) of the MMF, through optical phase conjugation [7–9], feedback-based optimization system [1,10–12], or direct measurement [13–18]. Up to date, various light patterns, such as a single focal spot (one spatial mode) [8,10,11], multiple focal spots [19,20], and even complex patterns [2,3,21], have been delivered through MMFs. Wavefront shaping has also extended its applicability from monochromatic light to polychromatic light. By measuring the multispectral TM, spatiotemporal focusing through the MMF has also been achieved, enabling the delivery of two-photon images [8,22]. Compared to delivering grey-scale images through MMFs using a single-wavelength laser beam or narrow-bandwidth light, delivering multicolored images directly is more attractive. This desire intuitively requires wavefront shaping to simultaneously function for the entire visible spectrum that covers three hundreds of nanometers. However, this requirement is technically challenging, as the visible spectrum contains too many spectral modes for the current system to handle a typical MMF or a scattering medium [23,24]. An alternative approach is to trick our eyes using the superposition of three primary colors, i.e., red, green, and blue (RGB) colors. In 2012, Conkey et al., showed projecting color images through a piece of eggshell using RGB colors [25]. The genetic algorithm was employed to iteratively optimize the objective function that uniquely mapped to a target image. Although being straightforward, the feedback-based approach requires one to redo the iterative optimization process if the delivered images or the composition of colors need to be changed. To solve this problem, spatial segmentation (SS), i.e., using different pixels of the SLM to modulate light with different wavelengths, has been developed to realize focusing and imaging through scattering media [26]. In a recent work, the performance of different segmentation methods was discussed and compared [27]. By modifying the area ratios of different color components, SS allows one to conveniently alter the colors of the focus.

In this work, we showed the first demonstration of delivering targeted color light through the MMF. Moreover, instead of using SS, we constructed the optimum phase map using field synthesis (FS), which makes every pixel of the SLM to partially account for all three colors. While simultaneously focusing M different light with equal weight to a single spot with SS reduces the focusing contrast by a factor of M², this number reduces to M with FS. With both analytical and numerical tools, we further show that the superiority of FS over SS is universal. Moreover, we experimentally validated the performance of FS by using three lasers emitting RGB light beams. Color focusing through the MMF has been achieved and its color can be switched at video frame rate (60 Hz) through computationally synthesizing fields. This work showed a compact and useful approach for delivering targeted color light through the MMF and can be integrated with an optical endoscopic system in the future.

2. Theoretical and numerical analysis

We start by illustrating the principle of FS. The TMs of a typical MMF for RGB lights are completely uncorrelated. Mathematically, each one can be represented with an $N \times N$ matrix, shown as ${T_a}({j,k} )({a = \textrm{r},\textrm{g},\textrm{b};j,k = 1, \ldots ,N} )$. For simplicity, we describe the principle of FS by focusing light to a single spot. Focusing light on multiple spots can be achieved in the same manner. For monochromatic focusing at the j-th output, the optimum input field can be chosen as the conjugate field ${T_a}^\ast ({j,k} )({a = \textrm{r},\; \textrm{g},\textrm{b};\; k = 1, \ldots ,N} )$ for each color. To form a focus with synthesized colors, ${\alpha _\textrm{r}}$, ${\alpha _\textrm{g}}$ and ${\alpha _\textrm{b}}$ are introduced as the mixing weight for RGB light, respectively. Here, each parameter is a number ranging from 0 to 1, and ${\alpha _\textrm{r}} + {\alpha _\textrm{g}} + {\alpha _\textrm{b}} = 1$ is enforced for the normalization purpose. With these notations introduced, the optical field

Having introduced the operation principle, we performed numerical simulations to examine the performance of FS. For the input light, the RGB components were kept to have the same intensity. Based on the uncorrelated transmission coefficients model, each element of the TM was drawn from a circular Gaussian distribution. N = 250 was chosen and 200 times averaging was performed to minimize statistical fluctuations. To begin with, we examined the performance of mixing two colors by temporally setting ${\alpha _\textrm{b}} = 0$. In wavefront shaping, although full-field modulation is ideal, phase-only modulation is the most commonly used for practice reasons. As a result, previous works retrieved only the phase values rather than the complete optical fields of the elements in the TM [13]. To examine how this practice affects the performance of FS, we compared focusing effects under different modulation schemes. Figure 1(a) plots the normalized focusing intensity (to the corresponding monochromatic case) of red and green components as a function of weight ${\alpha _\textrm{r}}$ for both full-field modulation (dashed lines) and phase-only modulation (solid lines). For full-field modulation, the normalized intensities for the red and green components follow two straight lines $1 - {\alpha _\textrm{r}}$ and ${\alpha _\textrm{r}}$, which matches the theoretical prediction (see Eqs. (17) and (18) in the Appendix). It is worth noting that when ${\alpha _\textrm{r}} = 0.5$, the focusing intensity is reduced by a factor of 2. In contrast, the red and green components obtained with phase-only modulation follow the trends described by complete elliptic integrals (see Eqs. (25) and (26) in the Appendix). We note here that curves with similar shapes have been observed to form multiple focal spots by synthesizing fields [2]. When ${\alpha _\textrm{r}} = 0.5$, the focusing intensity becomes $4/{\pi ^2}$, which is slightly smaller than 0.5. With phase-only modulation, a typical example of color light focusing when (${\mathrm{\alpha }_\textrm{r}}$, ${\mathrm{\alpha }_\textrm{g}}$, ${\alpha _\textrm{b}}$) = (0.50, 0.50, 0) is shown in Fig. 1(b), exhibiting a yellow (255, 255, 0) focal spot at the center of the image. While interpreting colors, the rule we adopted here is to normalize the largest component to 255. As a comparison, Fig. 1(c) also plots the normalized focusing intensity obtained with SS as a function of the area ratio of red light ${A_\textrm{r}}$. As shown in the figure that, the red and green components follow the trend of ${A_\textrm{r}}^2$ and ${({1 - {A_\textrm{r}}} )^2}$, respectively (see Eqs. (6) and (7) in the Appendix), regardless of the modulation scheme. Notably, when ${A_\textrm{r}} = 0.5$, the focusing intensity is 0.25, which is smaller than the value achieved with FS when ${\alpha _\textrm{r}} = 0.5$. To make a fair comparison between FS and SS, Fig. 1(d) plots the focusing efficiency, defined as the summation of the normalized focusing intensity of both colors, as a function of the intensity ratio of the red component in the final focus. This physical quantity represents the total intensity of the delivered focus. Remarkably, FS with full-field modulation has the largest focusing efficiency, being a straight line at 1. FS with phase-only modulation has slightly degraded performance, but still outperforms SS in all circumstance. The same conclusion still holds when mixing three colors. For a representative situation when the intensity ratio of RGB is 1:1:1, the focusing efficiencies achieved with FS (full field), FS (phase only), and SS are 1, 0.8028, and 1/3, respectively.

 

Fig. 1. Numerical results of delivering red and green light through the MMF. (a) Performance of full-field and phase-only modulation as a function of weight ${\mathrm{\alpha }_\textrm{r}}$, when FS was employed. (b) A representative color image when (${\mathrm{\alpha }_\textrm{r}}$, ${\mathrm{\alpha }_\textrm{g}}$, ${\alpha _\textrm{b}}$) = (0.50, 0.50, 0). A yellow focus is delivered at the center. (c) Performance of SS as a function of the area ratio ${A_\textrm{r}}$. (d) Comparison of the focusing efficiency of FS (black lines) and SS (red line).

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Next, we consider the performance of FS when synthesizing RGB colors with phase-only modulation. Figures 2(a)–2(c) show the normalized focusing intensity of each component for different combinations of $({{\alpha_\textrm{r}},{\alpha_\textrm{g}},{\alpha_\textrm{b}}} )$. In these figures, the horizontal and vertical axes are used to represent ${\mathrm{\alpha }_\textrm{r}}$ and ${\mathrm{\alpha }_\textrm{g}}$, and ${\alpha _\textrm{b}}$ is determined through the normalization condition. As a result, the effective area has a right triangular shape. Figure 2(a) indicates that the red component reaches the maximum value when ${\alpha _\textrm{r}} = 1$ and gradually decreases as ${\alpha _\textrm{r}}$ approaches 0. The same trend can be observed for the green and blue components as well in Figs. 2(b) and 2(c), respectively. As typical examples, three representative color focusing (white, rose, and purple) with different combinations of $({{\alpha_\textrm{r}},{\alpha_\textrm{g}},{\alpha_\textrm{b}}} )$ are shown in Figs. 2(d)–2(f).

 

Fig. 2. Numerical results of delivering RGB light through the MMF using FS. (a-c) Normalized intensity of each component. (d-f) Three representative color foci achieved by using FS.

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3. Experimental setup and experimental results

Having explored the performance of FS through numerical approaches, we built an experimental setup, shown in Fig. 3, to realize targeted color light delivery through the MMF. Three lasers that operated at 642 nm, 532 nm, and 473 nm were used as the light source. Each laser was followed by a half-wave plate and a polarizing beam splitter for keeping output power equaling. This configuration also kept the output polarization state of light to be horizontal. Then, the emitted light from lasers was combined in order and subsequently magnified to a 1-inch by a pair of lenses. An SLM (PLUTO-2-NIR-013) that refreshes 1920 × 1080 pixels at a 60-Hz video rate was used for phase modulation. The active area was uniformly divided into 1024 segments. Each segment has 60 × 33 pixels, and the unused pixels were set to zero phases during the entire experiment. After being modulated and reflected by the SLM, the light was aligned to an MMF with a core diameter of 200 $\mathrm{\mu }\textrm{m}$ and a numerical aperture of 0.37. After passing through the MMF, The delivered light was captured by a camera (Grasshopper GS3-U3-32S4C).

 

Fig. 3. Experimental setup of the system. HWP, half-wave plate; PBS, polarization beam splitter; BB, beam block; M, mirror; BS, beam splitter; L, lens; SLM, spatial light modulator; OBJ, objective lens; MMF, Multimode fiber; C, Camera.

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When using a single SLM to modulate wavefront for RGB light simultaneously, a subtlety here is that light with different wavelengths acquired different phase values. For each color, the phase response of the SLM for a given input was measured using the strategy developed in Ref. [28], by employing Michelson interferometry (isoclinic interferometry) with two arms formed by the SLM and a tilted mirror. Figure 4(a) plots the measured phase values for different colors as a function of the desired phase value. The circles represented measured data points from experiments, while the solid lines were obtained from linear fitting. Since the SLM had been previously calibrated for the red, the measured phase values of the red match with the desired ones. In contrast, the measured phase values of the green and blue are significantly larger than the desired phase values. Physically, the imposed phase value by the SLM is wavelength ($\mathrm{\lambda }$) dependent $\varphi (\mathrm{\lambda } )= 2\pi n(\mathrm{\lambda } )d/\mathrm{\lambda }$, where d is the thickness of the twisted liquid crystals and $n(\mathrm{\lambda } )$ is the wavelength-dependent refractive index. Through numerical fitting, we found that the ratio of the slopes for RGB was 1:1.28:1.49. Possibly due to the material dispersion, this relation slightly deviates from the inverse ratio of their corresponding wavelengths 642⁻¹:532⁻¹:473⁻¹ (1:1.21:1.36)_. When using FS to synthesize the field, these calibration curves were incorporated to transform measured phase values into real phase values.

 

Fig. 4. Experimental results of delivering RGB light through the MMF using FS. (a) Calibration curves of the SLM for RGB light. (b) Normalized intensity as a function of weight ${\alpha _\textrm{r}}$. Experimental data and linear fittings are represented using circles and solid lines. (c) Three representative images of the delivered color foci (white, yellow, and purple).

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As a proof of concept, we adopted feedback-based wavefront shaping with the previously developed Hadamard-encoding algorithm [10] to retrieve one row of the TMs for all three colors. These TMs are completely uncorrelated as expected. When light with different colors was independently shaped, monochromatic focusing through the MMF can be achieved. Experimentally, the enhancements, defined as the intensity ratio after and before wavefront shaping at the target position, were quantified to be 115, 121, and 130 for RGB light, respectively. Given that N = 1024, we achieved roughly 15% of the theoretical value (∼804). Before proceeding, the camera readings of the peak intensities of the RGB light were equalized by adjusting the half-wave plates. The first validation of FS was to reproduce the curves in Fig. 1(a). Figure 4(b) plots the normalized focusing intensity as a function of ${\alpha _\textrm{r}}$ when light emitted from the blue laser was blocked. As shown in the figure that, the experimental data (circles) match with the theoretical curves (solid lines), indicating the validity of FS. Then, we demonstrated delivering color focus through the MMF and switched the color at video framerate (60 Hz) by computationally adjusting the combinations of $({{\alpha_\textrm{r}},{\alpha_\textrm{g}},{\alpha_\textrm{b}}} )$. Using a personal computer with a CPU (AMD Ryzen 7 1800X Eight-Core Processor), it generally took 1∼2 milliseconds to synthesize the optical field. Three representative color foci are shown in Figs. 4(c)–4(e), corresponding to white, yellow, and purple.

In the current work, a monochrome camera was used to measure the light intensity of different components by orderly measuring different color components while blocking the others. We note that a color camera can be directly employed to parallel this process in future works. In that scenario, the finite spectrum response of the sensor and the built-in data processing software of the camera needs to be carefully handled. Although only focusing color light through the MMF was experimentally demonstrated, we emphasize here that FS can be directly extended to multisport color focusing [2], as well as delivering color images through the MMF. Moreover, as we do not intentionally optimize the system speed, the current system took roughly 30 minutes for the HEA to retrieve one row of the TM. In practice, MMFs are also extremely sensitive to bending and movement, limiting its broad applications. Therefore, future explorations of our work should be combined with previous works on addressing this issue [17,29–32].

4. Conclusions

In conclusion, we demonstrated delivering targeted color light through the MMF by using FS. The validity of FS was confirmed by both numerical and experimental approaches. Moreover, we showed that by computationally adjusting the combinations of $({{\alpha_\textrm{r}},{\alpha_\textrm{g}},{\alpha_\textrm{b}}} )$, desired colors can be delivered through the MMF to the target position and switched at a video framerate. We anticipate that this work paves a way for delivering color images through the MMF in endoscopy.

Appendix

1. Analytical analysis of mixing two colors with spatial segmentation

Full-field modulation

For spatial segmentation (SS), ${A_\textrm{r}}$ and ${A_\textrm{g}}$ $({{A_\textrm{r}} + {A_\textrm{g}} = 1} )$ are used to represent the area ratio of the pixels that accounts for red and green, respectively. Mathematically, ${T_\textrm{r}}({j,k} )$ and ${T_\textrm{g}}({j,k} )$ are used to represent the elements of the transmission matrices for red and green, which connects j-th output and k-th input. For simplicity, the total number of independent elements is set to be N for both the input and the output fields. With the notations introduced, the j-th output element for red can be computed as [33]

(2)$$\begin{array}{c} {{{|{E_\textrm{r}^{\textrm{out}}(j )} |}^2} = {{\left|{\mathop \sum \nolimits_k^{{N_\textrm{r}}} {T_\textrm{r}}({j,k} )T_\textrm{r}^\ast ({j,k} )+ \mathop \sum \nolimits_k^{{N_\textrm{g}}} {T_\textrm{r}}({j,k} )T_\textrm{g}^\ast ({j,k} )} \right|}^2}} \end{array}$$

here, ${N_\textrm{r}} = {A_\textrm{r}}N$ and ${N_\textrm{g}} = {A_\textrm{g}}N$. are the number of independent elements for red and green, respectively. Since the transmission matrices for different color are completely uncorrelated,

(3)$$\begin{array}{c} {{{|{E_\textrm{r}^{\textrm{out}}(j )} |}^2} = {{\left|{\mathop \sum \nolimits_k^{{N_\textrm{r}}} {{|{{T_\textrm{r}}({j,k} )} |}^2}} \right|}^2} + 2\mathop \sum \nolimits_k^{{N_\textrm{r}}} {{|{{T_\textrm{r}}({j,k} )} |}^2}\mathop \sum \nolimits_{k^{\prime}}^{{N_\textrm{g}}} \textrm{Re}({{T_\textrm{r}}({j,k^{\prime}} )T_\textrm{g}^\ast ({j,k^{\prime}} )} )+ {{\left|{\mathop \sum \nolimits_k^{{N_\textrm{g}}} {T_\textrm{r}}({j,k} )T_\textrm{g}^\ast ({j,k} )} \right|}^2}} \end{array}.$$

Taking the ensemble average on both sides of Eq. (3), we get

(4)$$\langle I_\textrm{r}^{\textrm{out}}(j )\rangle = N_r^2{\langle|T |^2\rangle}^2 + {N_\textrm{g}}{\langle|T |^2\rangle}^2$$

Here, ${\langle|T |^2\rangle}$ is the ensemble average of the absolute square the elements in the transmission matrix. While deriving Eq. (4), we assumed that the elements of the transmission matrix for different colors share the same statistical property. Since in the monochromatic situation, the j-th output element of red is given by

(5)$$\langle I_\textrm{r}^{\textrm{mono}}(j )\rangle = \langle{\left|{\mathop \sum \nolimits_k^N {T_\textrm{r}}({j,k} )T_\textrm{r}^\ast ({j,k} )} \right|^2}\rangle = {N^2}{\langle|T |^2\rangle}^2$$

The normalized focusing intensity becomes

(6)$${\langle I_\textrm{r}}{(j )\rangle_{\textrm{nor}}} = \langle I_\textrm{r}^{\textrm{out}}(j )\rangle/\langle I_\textrm{r}^{\textrm{mono}}(j )\rangle = ({N_\textrm{r}^2 + {N_\textrm{g}}} )/{N^2} = {A_\textrm{r}}^2 + ({1 - {A_\textrm{r}}} )/N$$

When N is large, we can approximately have

(7)$${\langle I_\textrm{r}}{(j )\rangle_{\textrm{nor}}} = {A_\textrm{r}}^2$$

Similarly, for the green component, we have

(8)$${\langle I_\textrm{g}}{(j )\rangle_{\textrm{nor}}} = {A_\textrm{g}}^2 = {({1 - {A_\textrm{r}}} )^2}$$

Phase-only modulation

For phase-only modulation, Eq. (3) needs to be modified as

(9)$$\begin{array}{c} {{{|{\tilde{E}_\textrm{r}^{\textrm{out}}(j )} |}^2} = {{\left|{\mathop \sum \nolimits_k^{{N_\textrm{r}}} {T_\textrm{r}}({j,k} )T_\textrm{r}^\ast ({j,k} )/|{{T_\textrm{r}}({j,k} )} |+ \mathop \sum \nolimits_k^{{N_\textrm{g}}} {T_\textrm{r}}({j,k} )T_\textrm{g}^\ast ({j,k} )/|{{T_\textrm{g}}({j,k} )} |} \right|}^2}} \end{array}$$

After mathematical simplification, it becomes

(10)$$\langle \tilde{I}_\textrm{r}^{\textrm{out}}(j )\rangle = N_\textrm{r}^2{\langle |T |\rangle}^2 + {N_\textrm{g}}{\langle |T |^2\rangle}$$

Here, $\langle |T|\rangle $ is the ensemble average of the absolute of the elements in the transmission matrix. For circular Gaussian distribution, ${\langle |T |\rangle^2}/{\langle |T |^2\rangle} = \pi /4$. In contrast, the j-th output element for red in the monochromatic situation is

(11)$$\langle \tilde{I}_\textrm{r}^{\textrm{mono}}(j)\rangle = \langle {\left|{\mathop \sum \nolimits_k^N {T_\textrm{r}}({j,k} )T_\textrm{r}^\ast ({j,k} )/|{{T_\textrm{r}}({j,k} )} |} \right|^2}\rangle = {N^2}{\langle |T |\rangle^2}$$

The normalized focusing intensity becomes

(12)$${\langle \tilde{I}_\textrm{r}}{(j)\rangle_{\textrm{nor}}} = \langle \tilde{I}_\textrm{r}^{\textrm{out}}(j)\rangle /\langle \tilde{I}_\textrm{r}^{\textrm{mono}}(j )\rangle = \left( {N_\textrm{r}^2 + \frac{4}{\pi }{N_\textrm{g}}} \right)/{N^2} = {A_\textrm{r}}^2 + \frac{4}{\pi }({1 - {A_\textrm{r}}} )/N$$

When N is large, we found that the results obtained from phase-only modulation are the same as the ones obtained from full-field modulation.

2. Analytical analysis of mixing two colors with field synthesis

Full-field modulation

For field synthesis (FS), we defined mixing weight ${\alpha _\textrm{r}}$ and ${\alpha _\textrm{g}}$ $({{\alpha_\textrm{r}} + {\alpha_\textrm{g}} = 1} )$ for red and green, respectively. The synthesized input field is given by

(13)$${E_{FS}}({j,k} )= \sqrt {{\alpha _\textrm{r}}} {T_\textrm{r}}^\ast ({j,k} )+ \sqrt {{\alpha _\textrm{g}}} {T_\textrm{g}}^\ast ({j,k} )$$

Here, the square root operation of ${\alpha _\textrm{r}}$ and ${\alpha _\textrm{g}}$ when performing field superposition in Eq. (13) is designed for the intensity normalization defined above. After passing through the scattering medium, the j-th output element for red can be computed as

(14)$$\begin{array}{c} {{{|{E_\textrm{r}^{out}(j )} |}^2} = {{\left|{\mathop \sum \nolimits_k^N {T_\textrm{r}}({j,k} ){E_{FS}}({j,k} )} \right|}^2} = {{\left|{\sqrt {{\alpha_\textrm{r}}} \mathop \sum \nolimits_k^N {T_\textrm{r}}({j,k} )T_\textrm{r}^\ast ({j,k} )+ \sqrt {{\alpha_\textrm{g}}} \mathop \sum \nolimits_k^N {T_r}({j,k} )T_\textrm{g}^\ast ({j,k} )} \right|}^2}} \end{array}$$

Following the same strategy used in the above section, we get

(15)$$\begin{array}{c} {\langle I_\textrm{r}^{\textrm{out}}(j )\rangle = {\alpha _\textrm{r}}{N^2}{{\langle|T |}^2\rangle}^2 + {\alpha _\textrm{g}}N{{\langle|T |}^2\rangle}^2} \end{array}$$

The normalized focusing intensity becomes

(16)$${\langle I_\textrm{r}}{(j)\rangle_{\textrm{nor}}} = \langle I_\textrm{r}^{\textrm{out}}(j )\rangle/\langle I_\textrm{r}^{\textrm{mono}}(j )\rangle = ({{\alpha_\textrm{r}}{N^2} + {\alpha_\textrm{g}}N} )/{N^2} = {\alpha _\textrm{r}} + {\alpha _\textrm{g}}/N$$

When N is large, we can approximately have

(17)$${\langle I_\textrm{r}}{(j)\rangle _{\textrm{nor}}} = {\alpha _\textrm{r}}$$

Similarly, for the green component, we have

(18)$${\langle I_\textrm{g}}{(j)\rangle_{\textrm{nor}}} = {\alpha _\textrm{g}} = 1 - {\alpha _\textrm{r}}$$

Phase-only modulation

In this situation, the synthesized input field is given by

(19)$${\tilde{E}_{FS}}({j,k} )= \textrm{exp}({i{{\tilde{\phi }}_{FS}}({j,k} )} )$$

where

$${\tilde{\phi }_{FS}}({j,k} )= \textrm{Arg}\left( {\sqrt {{\alpha_\textrm{r}}} \textrm{exp}({i{\phi_\textrm{r}}^\ast ({j,k} )} )+ \sqrt {{\alpha_\textrm{g}}} \textrm{exp}({i{\phi_\textrm{g}}^\ast ({j,k} )} )} \right)$$

or

(20)$$\textrm{Arg}\left( {\textrm{exp}({i{\phi_\textrm{r}}^\ast ({j,k} )} )+ \left( {\sqrt {{\alpha_\textrm{g}}} /\sqrt {{\alpha_\textrm{r}}} } \right)\textrm{exp}({i{\phi_\textrm{g}}^\ast ({j,k} )} )} \right)$$

Here, ${\phi _\textrm{r}}^\ast ({j,k} )$ and ${\phi _\textrm{g}}^\ast ({j,k} )$ are the conjugate phase values of ${T_\textrm{r}}({j,k} )$ and ${T_\textrm{g}}({j,k} )$, $\textrm{Arg}({\cdot} )$ computes the argument principal value of a complex number. To estimate the normalized focusing intensity for the red component, we discuss the phase distribution of $\textrm{exp}({{{\tilde{\phi }}_{FS}}({j,k} )- {\phi_\textrm{r}}^\ast ({j,k} )} )$. As illustrated in Fig. 5, $\textrm{exp}({i{\phi_\textrm{r}}^\ast ({j,k} )} )$ is denoted as a red arrow with a constant length of 1. Depending on the values of $A = \sqrt {{\alpha _\textrm{g}}} /\sqrt {{\alpha _\textrm{r}}} $, $\textrm{exp}({i{\phi_\textrm{r}}^\ast ({j,k} )} )+ \left( {\sqrt {{\alpha_\textrm{g}}} /\sqrt {{\alpha_\textrm{r}}} } \right)\textrm{exp}({i{\phi_\textrm{g}}^\ast ({j,k} )} )$ is evenly distributed on concentric circles with different radius, centering at a fixed point of $\textrm{exp}({i{\phi_\textrm{r}}^\ast ({j,k} )} )= 1$. As a result, its probability density function (PDF) in terms of $({r,\theta } )$ can be written as

(21)$${P_{r,\theta }}({r,\theta } )= \frac{{\delta ({r - A} )}}{{2\pi }},as\left\{ {\; \begin{array}{c} {r = \sqrt {{{({Re - 1} )}^2} + I{m^2}} }\\ {\theta = {\textrm {Arg}}({({Re - 1} )+ iIm} )} \end{array}} \right.$$

where $Re$ and $Im$ are the real and imaginary parts of the data points on circles in Cartesian coordinates. Following the mathematical rules of coordinate transformation, the PDF can be rewritten in terms of $Re$ and $Im$

(22)$$\begin{array}{c} {{P_{R,I}}({Re,Im} )= {P_{r,\theta }}\left( {\sqrt {{{({Re - 1} )}^2} + I{m^2}} ,\textrm{Arg}({({Re - 1} )+ iIm} )} \right) \times {J_{({r,\theta } )|({Re,Im} )}}}\\ { = \delta \left( {\sqrt {{{({Re - 1} )}^2} + I{m^2}} - A} \right)/2\pi \sqrt {{{({Re - 1} )}^2} + I{m^2}} } \end{array}$$

where ${J_{({r,\theta } )|({Re,Im} )}}$ is the Jacobian of this transformation. Using a similar approach, the PDF can be further transformed to the polar coordinates $({R,\varphi } )$ as

(23)$${P_{R,\varphi }}({R,\varphi } )= R\delta \left( {\sqrt {{{({R\textrm{cos}\varphi - 1} )}^2} + {{({R\textrm{sin}\varphi } )}^2}} - A} \right)/2\pi \sqrt {{{({R\textrm{cos}\varphi - 1} )}^2} + {{({R\textrm{sin}\varphi } )}^2}} $$

 

Fig. 5. Illustration of field synthesis with phase-only modulation. The subscript α and β represent different situations of ${A_\alpha } < 1$ and ${A_\beta } > 1$.

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The phase distribution of $\textrm{exp}({{{\tilde{\phi }}_{FS}}({j,k} )- {\phi_\textrm{r}}^\ast ({j,k} )} )$ can be estimated by integrating out the dependence of R in Eq. (23). Based on the residue theorem, the result is

(24)$$\begin{array}{c} {{P_\varphi }(\varphi )= \left\{ {\begin{array}{c} {\frac{{\textrm{cos}\varphi }}{{\pi \sqrt {{A^2} - \textrm{si}{\textrm{n}^2}\varphi } }},A \le 1,\varphi \in [{ - \textrm{arcsin}(A ),\textrm{arcsin}(A )} ]}\\ {\frac{{\textrm{cos}\varphi + \sqrt {{A^2} - \textrm{si}{\textrm{n}^2}\varphi } }}{{2\pi \sqrt {{A^2} - \textrm{si}{\textrm{n}^2}\varphi } }},\; A > 1,\varphi \in [{ - \pi ,\pi } ]} \end{array}} \right.} \end{array}$$

It is worth mentioning that, the values of A determine the locations of the poles in Dirac function, leading to different integral results. Following the strategy developed in Ref. [7], the integral of phase errors leads to the deviated factor from the ideal case. Thus, the reduction in the optical field ${\eta _E}$ is given by

(25)$${\eta _E} = \mathop \smallint \nolimits_{{\varphi _{\textrm{min}}}}^{{\varphi _{\textrm{max}}}} {e^{i\varphi }}{P_\varphi }(\varphi )d\varphi = \left\{ {\begin{array}{c} {\frac{2}{\pi }E\left( {A,\frac{\pi }{2}} \right)\; \; \; \; \; \; \; \; \; \; ,A \le 1}\\ {\frac{2}{\pi }\frac{{{A^2}E\left( {\frac{1}{A},\frac{\pi }{2}} \right) - ({{A^2} - 1} )K\left( {\frac{1}{A},\frac{\pi }{2}} \right)}}{A},A > 1} \end{array}} \right.$$

Here, ${\varphi _{\textrm{max}}}$ and ${\varphi _{\textrm{min}}}$ are the respective upper and lower limit of the integral denoted in Eq. (24). $E({x,\pi /2} )$ is the complete elliptic integral of the second kind with parameter $0 \le x \le 1$ (in the interval from $0$ to $\pi /2$) while $K({x,\pi /2} )$ is the complete elliptic integral of the first kind with parameter $0 \le x \le 1$ (in the interval from $0$ to $\pi /2$) [34]. Then, the normalized focusing intensity for the red component becomes

(26)$${\langle \tilde{I}_\textrm{r}}{(j)\rangle_{\textrm{nor}}}={\eta _E}^2$$

The normalized focusing intensity for the green component can be obtained by simply replacing A with $1/A$. When ${\alpha _\textrm{r}} = {\alpha _\textrm{g}} = 0.5$, we get $A = 1$ and the normalized intensity of this special case becomes ${|{2/\mathrm{\pi }} |^2} \approx 40.528\%$.

Funding

National Key Research and Development Program of China (2018YFB1802300); Fundamental Research Funds for the Central Universities (DUT18RC(3)047).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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