>Spatial light modulator based on spatial mode projection
Spatial light modulators (SLMs) are devices used to manipulate the phase of incident wavefronts, which can modulate the phase or amplitude of light beams and are widely used in optical capture, dynamic diffractive optical elements, and display technologies. The most common is liquid crystal (LC) SLM, which uses liquid crystal on silicon (LCoS) technology to modulate the phase of the incident wavefront without modulating the amplitude. LC SLMs work on the principle of electronically controlled birefringence (ECB). Electrically and optically anisotropic LC molecules have a voltage-dependent refractive index, which is an effect of the ECB. This in turn allows a controlled amount of phase shift to be introduced at the incident wavefront. In the case of electrically addressed SLMs, the voltage is interpreted by the computer as a gray level in the image. Most SLMs use high-resolution 8-bit images with 256 different gray levels to control the phase.
We propose a new method based on spatial pattern projection. The key to this method is to provide a specific mode on the incident beam and measure the power produced using a single-mode fiber. By applying this method to the SLM calibration, we can detect the phase that the SLM transmits at a certain gray value.
As a mode filter, single-mode fiber only accepts Gaussian mode. The power P as a result of this projection is described by
E f 分别指的是入射场和接收单模光纤的场。 E 0 and E f refer to the incident field and the field to receive the single-mode fiber, respectively. These fields are
w 0 和 w f 是束腰， U ( x,y ) 是相位前沿。 Where w 0 and w f are beam waists, and U ( x, y ) is the phase front.
U ( x, y ) = exp ( iΦ ( x, y )) (4)
Φ 是相位，并直接对应于上传到设备的灰度图案。 Where Φ is the phase and directly corresponds to the grayscale pattern uploaded to the device. To measure the phase response, we uploaded a simple pattern in which half of the planes maintained a constant phase and the other changed from black to white. φ1 设置为零的恒定灰度值，而 φ2 从0-225变化。 We set φ1 to a constant gray value of zero, while φ2 varies from 0-225. Δφ=φ2-φ1 ，校准模式由公式给出 By letting Δφ = φ2-φ1 , the calibration mode is given by the formula
Equation (5) generates 256 specific calibration modes, each mode being associated with a unique gray value. Finally, by integrating and simplifying (1), we express the power of the spatial mode projection as
Even a slight optical misalignment can seriously affect the coupling efficiency of single-mode fibers. w 0 不能超过单模光纤的束腰 The beam waist w 0 cannot exceed the beam waist of the single-mode fiber ，式(7)表明束腰失配导致耦合效率下降，但不考虑角度失配。 w f , Equation (7) shows that the coupling efficiency is reduced due to the beam waist mismatch, but the angle mismatch is not considered. Equation (6) shows that the power can be expressed as a simple function of the phase difference, that is, a function controlled by the gray difference.
The specific SLM model used experimentally is the Holoeye PLUTO-2 reflective LCOS SLM. It has a resolution of 1920 × 1080 pixels and a pixel pitch of 8 μm. The LC uses an 8-bit grayscale image for addressing with a refresh rate of 60 Hz. The calibration settings are shown in Figure 1 (a). As a light source, a tunable He-Ne laser can switch between several discontinuous wavelengths. This paper uses 633 nm, 612 nm, and 604 nm. The lenses L1 and L2 are retractable structures that collimate and expand the incident beam to cover more of the surface of the SLM. SLM's phase shift mechanism is sensitive to polarization, so we use a linear polarizer (LP) to ensure that our beam is p-polarized. The SLM is set at a small angle to separate different diffraction orders. Mirrors M1 and M2 are used to guide the zero-order diffraction from the SLM into the single-mode fiber. Although some unmodulated light is reflected by the protective glass surface covering the LC layer, this light is minimal for most LC SLMs. The lenses L3 and L4 are also retractable, they reduce the beam waist for fiber coupling. This structure also ensures that the electric field at the fiber entrance is the same as the field at the SLM plane. Single mode fiber (SMF) filters the beam in Gaussian mode. Finally, the intensity was measured with a photodetector. According to formula (5), the left side of the mode uploaded to SLM remains unchanged, while the right side gray value changes from 0-255. This mode is shown in Figure 1 (b). The power p of the displayed gray value is measured by taking the average of several intensity readings. The normalized original result is shown in Figure 2 (a). The ECB mechanism of SLM varies with the wavelength of the incident beam, so we must create a calibration curve for each necessary wavelength.
Figure 1 (a) Experimental setup for calculating the phase response of SLM.
The sine expression in equation (6) indicates that the maximum value should have equal strength, and the minimum value should also be the same. However, Figure 2 (a) does not look like a perfect sine curve, mainly because the beam is difficult to align with the single-mode fiber. In fact, due to the high sensitivity of single-mode fibers to changes in position and angular direction, perfect alignment is almost impossible. Therefore, a data processing technique is used, which is called mobile normalization, that is, both peaks are scaled to 1 and the minimum is scaled to 0. The processed data is shown in Figure 2 (b). Unprocessed data is called raw data, and data with mobile normalization is called processed data.
The gray value and phase shift curve measured in Figure 2 (c) does not seem to change significantly with the incident wavelength. Instead, we observe in Figure 2 (d) that the results for each wavelength begin to deviate from the gray value of approximately 120. This directly corresponds to the second peak of the data we processed with mobile normalization. We have also observed that longer wavelengths of 633 nm produce smaller maximum phase shifts than shorter wavelengths of 604 nm and 612 nm. This may be because 604nm and 612nm are closer to each other than their 633nm. If we repeat the calibration procedure with shorter wavelengths than these, we expect to see higher maximum phase shifts.
Figure 2 Calibration curve using spatial mode projection measurement. (A) Normalized and unprocessed data measured using photodetectors, the same data as (b), but processed using normalization techniques. (C) Expanded phase shift of unprocessed data and (d) Data processed using mobile normalization techniques.
Use Figures 2 (a) and 2 (b) to create lookup tables (LUTs) to calibrate the images to be uploaded to SLM. LUTs map each gray value in the image to a new gray value.
Considering that beam shaping is one of the most prominent applications of LC phase SLM, we chose to evaluate the quality of our calibration method by generating a higher order Laguerre-Gauss (LG) mode. The electric fields of these modes are defined as
） 是束腰， Where w ( z ) is the waist, R（z） 是波束的曲率半径，k是波数， φ（z） 是相移。 Represents the generalized Laguerre polynomial, R (z) is the radius of curvature of the beam, k is the wave number, and φ (z) is the phase shift. l 指数是指波束的拓扑电荷，而p指数是指径向指数。 Physically, the l index refers to the topological charge of the beam, and the p index refers to the radial index.
Computer-generated holograms (CGHs) are images generated by the superposition of the electric field of a beam of light we want with Gauss. By making a Gaussian beam incident on an SLM with CGH, we can convert the beam to LG mode. CGH is created by
The device used to propagate and capture these LG beams is shown in Figure 3 (a).
Figure 3 (a) Experimental setup for creating a Laguerre-Gaussian (LG) mode from a computer-generated hologram (CGH). We also show here the effect of calibration on the CGH used, where (b) is the original hologram and (c) is the calibrated hologram. An intensive lateral line scan is overlaid on each CGH, with a line cut through the center representing the area where the line scan was performed.
To show the width of the LG mode, we selected three specific beams for comparison. The first is simple The pattern is defined by equation (8), as shown in Figure 4; the second is a commonly used petal pattern, as shown in Figure 5. The third is less common Mode, as shown in Figure 6. We also provide simulations of these same beams for comparison. An uncalibrated beam is a beam without an LUT applied. An incorrectly calibrated beam is a beam of an LUT generated using unprocessed data in Figure 2 (c). A beam that is successfully calibrated refers to a beam that has been processed using a moving normalization technique.
Figure 4 Use Comparison of beam performance. (A) Simulated beam. (B), (e), (h) Beams from an uncalibrated SLM. (C), (f), (i) Calibrate the hologram beam using our original calibration curve. (D), (g), (j) use calibrated SLM and our processed LUT. The first, second and third lines correspond to the incident wavelengths of 604nm, 612nm and 633nm, respectively.
Figure 5 Use Performance comparison of modes. (A) Simulated beam. (B), (e), (h) Beams from uncalibrated SLM. (C), (f), (i) Calibrate the hologram beam using our original calibration curve. (D), (g), (j) use calibrated SLM and our processed LUT. The first, second and third lines correspond to the incident wavelengths of 604nm, 612nm and 633nm, respectively.
Figure 6 Use Beam performance comparison. (A) Simulated beam. (B), (e), (h) Beams from an uncalibrated SLM. (C), (f), (i) Calibrate the hologram beam using our original calibration curve. (D), (g), (j) use calibrated SLM and our processed LUT. The first, second and third lines correspond to the incident wavelengths of 604nm, 612nm and 633nm, respectively.
This paper proposes a new SLM calibration method based on spatial mode projection. Apply data processing techniques to our data to generate accurate lookup tables from our calibration curves. We then used a computer-generated hologram to propagate the laguerre-Gauss beam to evaluate the success of our method. The results show that qualitative analysis of propagation modes using SLM is a feasible performance evaluation method. Overall, we show that spatial mode projection provides significant performance improvements in the phase modulation capabilities of SLMs.
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