Dynamic 3D holographic projection of vectorial images with a multimode fiber

Jinghan Zhuang; Panpan Yu; Panpan Yu; Panpan Yu; Yifan Liu; Yijing Wu; Ziqiang Wang; Yinmei Li; Lei Gong; Lei Gong

doi:10.1364/PRJ.514689

1. INTRODUCTION

Multimode fiber (MMF) serves as one of the most powerful tools to deliver light to hard-to-access places, for example, deep tissues and inaccessible cavities [1]. Moreover, compared to graded-index lenses or fiber bundles, MMFs feature high mechanical flexibility and a small footprint [2]. These advantages make them competitive candidates for single-fiber endoscopic imaging [3–7] and optical manipulation [8–10]. Recently, MMFs have been exploited as miniaturized probes for minimally invasive biomedical imaging into deep tissues [11,12] and miniaturized tweezers for versatile holographic optical manipulation [1,13]. In addition, MMFs allow parallel transmission of a large number of spatial modes, which benefits high-capacity data transmission [14–16] and projection of images [17,18]. However, strong mode coupling inherent in MMFs scrambles light propagation inside and yields random speckled patterns at the outputs [19]. Fortunately, wavefront shaping techniques are able to unscramble optical beams that have been arbitrarily mixed in an MMF by the transmission matrix (TM) method [20,21], the digital optical phase conjugation method [22], or deep learning methods [23–26].

Recent studies reveal that polarization scrambling also occurs in MMFs, causing a well-prepared input polarization state to turn into randomly distributed polarization states. So, light propagating inside an MMF experiences both polarization mixing and mode mixing due to the inherent imperfections of the fiber and environmental perturbations. Besides, the spatial and polarization degrees of freedom (DoFs) are strongly coupled in the propagation process [27]. Notably, the strong coupling between different DoFs opens the possibility of utilizing wavefront shaping for polarization control of the output field [28–30]. In 2018, Xiong et al. adopted this strategy to achieve complete control of polarization states by only manipulating the spatial wavefront of a laser beam into the fiber [27]. Later, vector focusing through an MMF was achieved [31,32]. Recently, this strategy was further exploited for dynamic polarization holography [33]. Polarization holography generally depends on polarization-sensitive materials to reconstruct vectorial images, but such kinds of polarization holograms lack the reconfigurability. A scattering material combined with a spatial light modulator has enabled dynamic vectorial holographic projections. However, the existing techniques demonstrated only projections with focal spots, and the relatively low transmission through a scattering medium limits the efficiency [30,33]. In contrast, an MMF-based projector can offer relatively high efficiency and high mechanical flexibility [14]. More importantly, it can deliver optical patterns to hard-to-access places for structured illumination or manipulation. Although delivering 2D images or even colorful images has been demonstrated with a fiber-based projector [23,25,34], 3D holographic projection of complex vectorial images remains a challenging task to date. This is because dynamic 3D polarization holography has not been realized with the MMF, which hinders polarization-dependent applications using a single fiber.

In this work, we propose a holographic approach for MMF-based projection of 3D complex images with desired polarization states. This approach allows us to modulate the 3D distributions of intensity and polarization of the output field simultaneously by shaping the wavefront of the incident light. As such, switching phase-only holograms to shape the incident light achieves dynamic 3D projection of vectorial images at the distal end of the fiber. In the experiment, we demonstrate complete polarization control of the optical foci projected on multiple planes through a 1-m MMF, and an averaged degree of polarization over 94% is achieved. To project 3D vectorial images with continuous intensity distributions, we further exploit a modified Gerchberg–Saxton (GS) algorithm based on the measured TMs of the fiber to optimize the calculation of holograms. In this way, multi-plane projections of polarization-multiplexed grayscale images are presented, and an averaged Pearson correlation coefficient over 0.91 with respect to the ground truth is achieved for the complex images. Furthermore, by employing the switching ability of a digital micromirror device (DMD), we also demonstrate dynamic 3D projection through the MMF. Our work is expected to benefit a variety of fiber-optic polarization-related applications, such as 3D displays, optical endoscopy, optical manipulation, and multiplexed data transmission.

2. PRINCIPLE

Figure 1 illustrates the concept of our holographic approach for MMF-based 3D projection. The MMF features cylindrical symmetry, whose orthogonal eigenmodes are radially symmetric. These modes preserve their spatial distributions upon propagation, but their interference yields a random speckle pattern at the output of the fiber. When a plane wave with a well-prepared state of polarization propagates through an MMF, multiple scattering intermixes all the DoFs of the incident light. As a result, the output field exhibits 3D speckle optical fields, where weak optical foci and various polarization states are randomly distributed at different axial planes, as shown in Fig. 1(a). Physically, this scattering process inside the MMF can be described with a TM that correlates the input and scattered fields. By harnessing the coupling between the spatial and polarization DoFs and calibrating the polarization-related TMs at each axial plane [Fig. 1(b)], simultaneous active control of the output intensity and polarization in 3D space can be achieved by modulating the incident wavefront. As such, the 2D phase-holograms [Fig. 1(c)] are created corresponding to the desired intensity distributions and polarization states on each targeted plane, and the 3D hologram [Fig. 1(d)] can be obtained via the complex superposition of 2D holograms. By using a DMD as the reconfigurable hologram, dynamic 3D projection of vectorial images through the MMF can be implemented by switching the corresponding holograms in sequence, as illustrated in Fig. 1(e).

Fig. 1. Concept of dynamic 3D projection of vectorial images with an MMF. (a) Strong coupling between the spatial and polarization degrees of freedom of light propagating through an MMF, yielding speckle fields with random polarization states in a 3D space. (b) Polarization-related TMs at targeted planes allow simultaneous active control of the output intensity and polarization by shaping the wavefront of light. (c) Phase-only holograms rendered from the 2D optical wavefront. (d), (e) 3D hologram for projecting multiple vectorial images through an MMF. Dynamic holographic vectorial projection can be implemented by switching the 3D holograms in sequence.

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Considering the vectorial nature of light waves, the relationship between the input and output fields of an MMF can be fully described with a vector transmission matrix (VTM) $t$ , which reads [35]

\begin{matrix} (\begin{matrix} E_{o}^{V} \\ E_{o}^{H} \end{matrix}) = (\begin{matrix} t^{V X} & t^{V Y} \\ t^{H X} & t^{H Y} \end{matrix}) (\begin{matrix} E_{in}^{X} \\ E_{in}^{Y} \end{matrix}) \end{matrix},

where the superscripts

X

,

Y

,

V

, and

H

represent two orthogonal polarization states of the input and output ends, respectively. For each axial plane of

z = z_{k}

, the VTM

t_{z_{k}}

correlates the output fields at such a plane with the input field. Thus, we can create a hologram in the form of

H_{k} (ξ, η) = t_{z_{k}}^{†} U_{k} (x, y)

to reconstruct a vectorial image

U_{k} (x, y) = [U_{k}^{V} (x, y); U_{k}^{H} (x, y)]

on the

z_{k}

-plane, where the symbol

†

denotes the complex conjugate transpose. However, it requires modulating the complex amplitudes of incident fields at both orthogonal polarizations and is technically demanding. The coupling between them allows us to modulate only one polarization component. In this case, the hologram can be constructed in the form of

H_{k} (ξ, η) = {[t_{z_{k}}^{V X}]}^{†} U_{k}^{V} (x, y) + {[t_{z_{k}}^{H X}]}^{†} U_{k}^{H} (x, y),

which modulates the

X

-polarization component of the input field, and only two polarization-related TMs are required.

The reconstructed output field can then be calculated as

\begin{matrix} [\begin{matrix} E_{o}^{V} (x, y) \\ E_{o}^{H} (x, y) \end{matrix}] = [\begin{matrix} t_{z_{k}}^{V X} {(t_{z_{k}}^{V X})}^{†} U_{k}^{V} (x, y) + t_{z_{k}}^{V X} {(t_{z_{k}}^{H X})}^{†} U_{k}^{H} (x, y) \\ t_{z_{k}}^{H X} {(t_{z_{k}}^{H X})}^{†} U_{k}^{V} (x, y) + t_{z_{k}}^{H X} {(t_{z_{k}}^{V X})}^{†} U_{k}^{H} (x, y) \end{matrix}] \end{matrix} .

Because of the uncorrelation between the polarization-related TMs from multiple scattering, the values of the cross-terms equal the inner product of nearly orthogonal random vectors and thus converge to zero. In addition, the energy is almost distributed uniformly in all polarization states, leading to

t_{z_{k}}^{V X} {[t_{z_{k}}^{V X}]}^{†} = t_{z_{k}}^{H X} {[t_{z_{k}}^{H X}]}^{†} = α

. As a result, the output field can be rewritten as

\begin{matrix} [\begin{matrix} E_{out}^{V} (x, y) \\ E_{out}^{H} (x, y) \end{matrix}] = [\begin{matrix} α U_{k}^{V} (x, y) \\ α U_{k}^{H} (x, y) \end{matrix}] \end{matrix} .

It shows that such a scalar hologram represented by Eq. (2) reconstructs the desired vectorial image at the target plane.

Based on the polarization-related TMs at different planes, the construction of a single 3D hologram for $K$ -plane vectorial projection is straightforward via a superposition operation:

H (ξ, η) = \sum_{k = 1}^{K} H_{k} (ξ, η) = \sum_{k = 1}^{K} {{[t_{z_{k}}^{VX}]}^{†} U_{k}^{V} (x, y) + {[t_{z_{k}}^{HX}]}^{†} U_{k}^{H} (x, y)} .

It is noted that, when a plane wave propagates through a scattering medium, multiple scattering decorrelates all the speckle fields at different depths and different polarizations. This property leads to minimized coherent crosstalk between the vectorial images projected at different planes. The overall effect of the other images on the target plane is adding a very weak speckle background. Thus, the scalar 3D hologram is able to project the vectorial images at each depth with a small loss in image contrast. In such a vectorial projection, a polarizer can be used to separately reveal the images encoded on different polarizations from a single hologram.

3. EXPERIMENTAL RESULTS

To implement 3D vectorial image projection through an MMF, the polarization-related TMs at each projected plane should be calibrated first. We adopt single-beam interferometry based on parallel wavefront optimization [36] so that the same setup can be used for image projection purposes. For calibration of the polarization-related TMs at different planes, the calibration process is the same, but a polarizer is required to select the corresponding polarization components. We utilize a high-speed DMD with $1024 \times 768 pixels$ and a pixel pitch of 13.7 μm (V-7001, Vialux) that is capable of switching at 22.27 kHz. The experimental setup is schematically illustrated in Fig. 2. A He–Ne laser with a wavelength of $λ = 633 nm$ (HNL210LB, Thorlabs) is used as the coherent light source. The laser beam is expanded and steered to fully illuminate the surface of the DMD with an incident angle of 24°. A half waveplate is placed before the DMD and rotated to adjust the incident polarization. The DMD with the help of a $4 - f$ system ( $4 \times$ demagnification) and a pinhole is capable of modulating the wavefront of incident light. The modulated laser beam is then coupled into a 1-m MMF (SUH600, XIRI) via an objective lens ( $10 \times$ , NA = 0.25; Olympus). The MMF features an NA of 0.22 and a core diameter of $d = 600 μm$ . It supports approximately $2.15 \times 10^{5}$ spatial modes in each polarization, given by ${(π d NA / λ)}^{2} / 2$ [37]. At the distal end of the fiber, another objective lens ( $10 \times$ , NA = 0.25; Olympus) and a tube lens ( $f = 180 mm$ ) image the output field, whose intensity distribution is recorded by a CMOS camera (PL-D752MU, PixeLINK). A polarizer is inserted before the camera to reveal the polarization-multiplexed images. To observe the images on different axial planes, the collected objective lens is mounted on a translation stage to cover a large axial range.

Fig. 2. Experimental setup. $λ / 2 - WP$ , half waveplate; L, lens; Obj, objective lens; MMF, multimode fiber; TL, tube lens; P, polarizer.

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Actually, the decorrelation between optical fields on the two planes leads to the elimination of any coherent trace of the images on each other, which is guaranteed by the uncorrelated relation between the TMs. To validate this hypothesis, we performed cross-correlation analyses between the measured TMs [Fig. 3(a)], $t_{z_{1}}^{H H}$ and $t_{z_{2}}^{V H}$ in terms of two adjacent planes with a depth interval of 0.5 mm. Figure 3(b) presents the amplitude of the autocorrelation $| C_{t_{z_{1}}^{H H}, t_{z_{1}}^{H H}} |$ of $t_{z_{1}}^{H H}$ and the normalized cross-correlation $| C_{t_{z_{1}}^{H H}, t_{z_{2}}^{V H}} |$ in terms of the amplitude autocorrelation. In the cross-correlation, no obvious peak appears, and the amplitude is much smaller than that of the autocorrelation, which confirms the low correlation between the TMs at different planes. In addition, we characterize the ability of complete polarization control at different planes using our method. For this purpose, we projected two focal spots with orthogonal polarizations on two planes, as shown in Fig. 3(c). To quantify the polarization purity, a polarizer is used to check the degree of polarization. Figure 3(d) plots the normalized intensities of the two foci as a function of the polarizer angle, and an average degree of polarization over 94% is achieved for the two projected focal spots.

Fig. 3. Characterization of the MMF-based 3D projector. (a) Measured polarization-related TMs in terms of two adjacent planes, $t_{z_{1}}^{H H}$ and $t_{z_{2}}^{V H}$ . $N = 64 \times 64$ segments on the DMD and $M = 192 \times 192$ pixels on the camera are taken as the input and output modes, respectively. (b) Autocorrelation $| C_{t_{z_{1}}^{H H}, t_{z_{1}}^{H H}} |$ of $t_{z_{1}}^{H H}$ and cross-correlation $| C_{t_{z_{1}}^{H H}, t_{z_{2}}^{V H}} |$ of the measured TMs. (c) Complete polarization control of focal spots projected at different planes. Yellow arrows label the polarization states. (d) Normalized intensity of the two focal spots measured behind a rotating polarizer. Scale bar, 25 μm.

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The polarization control ability of our method is applied to perform 3D vectorial projections through an MMF. To project images with continuous intensity distributions, a modified GS algorithm based on the measured TMs is developed to improve the fidelity of the complex images [38]. In this algorithm, the iterative optimization process is the same as the traditional GS algorithm, but the measured TMs instead of the Fourier transformation serve as an operator to calculate the hologram via Eq. (5). For each iteration, the intensity of the desired images is set as the constraint, and the iteration quits until the given fidelity is achieved. In this manner, we created a 3D hologram to project grayscale images with different polarization states at four planes. The target images are presented in Fig. 4(a), where yellow arrows represent the polarization states. Figures 4(b)–4(e) show the corresponding images recorded at the targeted planes behind a polarizer with varying orientations, for example, 0°, 45°, 90°, and 135°. Apart from the linear polarization, our method is also able to control the circular polarization of light through the MMFs. We demonstrate a hologram that projected two grayscale images in linear polarization and circular polarization states, and Fig. 5 presents the result of such a projection.

Fig. 4. Experimental demonstration of polarization-multiplexed multiplane projection of grayscale images through an MMF. $N = 96 \times 96$ segments on the DMD and $M = 256 \times 256$ pixels on the camera are taken as the input and output modes, respectively. (a) Four desired grayscale images to be projected on different planes with various polarizations represented by the yellow arrows. (b)–(e) Images recorded at the targeted planes behind a polarizer with varying orientations (red arrows). Scale bar, 15 μm.

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Fig. 5. Simultaneous control of linear and circular polarization states through an MMF. (a) Two grayscale images to be projected with linear and circular polarization states that are represented by the yellow arrows. (b)–(e) Images recorded at the targeted planes behind a polarizer with varying orientations (red arrows). Scale bar, 15 μm.

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Furthermore, we demonstrate dynamic polarization-multiplexed projection of grayscale images that were encoded in four different polarization states by exploiting the fast refreshing rate of the DMD. For this purpose, a series of 3D holograms corresponding to various letter images were projected sequentially. Figure 6 shows the reconstructed vectorial images from a sequence. In this four-plane vectorial projection, an average Pearson correlation coefficient over 0.91 with respect to the ground truth is achieved. The dynamic process is presented in Visualization 1 as well as the polarization-discernible images.

Fig. 6. Dynamic multiplane projection of 3D vectorial images through the MMF. (a)–(c) Image sequences projected on four axial planes with various polarization states (represented by yellow arrows) that were reconstructed from a sequential projection of dynamic 3D holograms. The dynamic process and their polarization-discernible images are available in Visualization 1. Scale bar, 15 μm.

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4. DISCUSSION

Our current 3D holograms demonstrated projections on only four axial planes, and the complexity of reconstructed 3D images is limited by the pixel resolution of the holograms. Holograms with a higher pixel resolution allow projections with finer details and increased contrast at more image planes. It is noted that the polarization control is not directly related to the pixel resolution of the hologram because the orthogonality between the polarization states is eventually guaranteed by the depolarization property of the fiber. In the current experiments, $96 \times 96$ independent modes on the DMD were used in the hologram, and we already got a high degree of polarization [Fig. 2(e)]. Actually, liquid crystal spatial light modulators (LC-SLMs) could also be used for shaping the light through the MMF, but DMDs offer an advantage over the SLMs in the refreshing rate, which will benefit a quick VTM calibration.

In summary, we have proposed and implemented a holographic approach that enables the MMF to be a 3D holographic projector with the ability of complete polarization control on multiple planes. This approach achieves simultaneous modulation of the intensity and polarization of the output field through the MMF by shaping the wavefront of the incident light. As such, phase-only holograms are capable of 3D holographic projection of vectorial images at the distal end of the fiber. In particular, a modified GS algorithm based on the measured TMs is developed to optimize hologram generation so that vectorial images with continuous intensity distributions can be projected beyond multiple scattering. Experimentally, we built an MMF-based 3D projector with a DMD and demonstrated complete polarization control and polarization-multiplexed projection of grayscale images on multiple planes. The polarization state at each projection plane could be defined at will, and a high polarization purity was achieved. In the meantime, projecting vectorial images with high fidelity was verified by an averaged Pearson correlation coefficient over 0.92 in the experiment. Furthermore, we demonstrated dynamic 3D vectorial projections through an MMF by exploiting the DMD as a reconfigurable hologram. Our fiber-optic holographic approach with dynamic projection and polarization multiplexing functionalities opens perspectives for various fields, such as 3D display, endoscopic imaging, and holographic optical trapping.

Funding

National Natural Science Foundation of China (11974333, 12204456); Natural Science Foundation of Anhui Province (2208085J24); Open Project of Advanced Laser Technology Laboratory of Anhui Province (AHL2022KF02).

Acknowledgment

We acknowledge support from the Center for Micro and Nanoscale Research and Fabrication, University of Science and Technology of China.

Disclosures

The authors declare no conflicts of interest.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Dynamic 3D holographic projection of vectorial images with a multimode fiber

Abstract

1. INTRODUCTION

2. PRINCIPLE

3. EXPERIMENTAL RESULTS

4. DISCUSSION

Funding

Acknowledgment

Disclosures

Data Availability

REFERENCES

Supplementary Material (1)

Data Availability

Cited By

Figures (6)

Equations (5)

Photonics Research