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Abstract
Due to the wave nature of light, the diffraction pattern generated by an optical device is sensitive to the shift of wavelength. This fact significantly compromises the digital micromirror device (DMD) in applications, such as full-color holographic display and multi-color fluorescence microscopy. The existing dispersion compensation techniques for DMD involve adding diffractive elements, which causes a large amount of waste of optical energy. Here, we propose an energy-efficient dispersion compensation method, based on a dispersive prism, for DMD. This method simulates the diffraction pattern of the optical fields reflected from the DMD with an angular spectrum model. According to the simulation, a prism and a set of optical components are introduced to compensate for the angular dispersion of DMD-modulated optical fields. In the experiment, our method reduced the angular dispersion, between the 532 nm and 660 nm light beams, by a factor of ∼8.5.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Digital Micromirror Device (DMD) has millions of independent modulation units and a high modulation rate [1–3]. As a modulator for optical fields, DMD plays a prominent role in the high-speed applications related to the super-resolution microscopy [4–6], 3D laser printing [7,8], and holographic display [9,10]. However, due to the diffraction introduced by DMD, the form of modulated optical field is sensitive to the shift of wavelength [11,12]. This drawback significantly constrains the overall performance of DMD in the full-color holographic display and multi-color fluorescence microscopy. In particular, when the multi-color incident light is necessary, the dispersion caused by DMD will lead to a large mismatch between the diffraction patterns at different wavelengths that can either distort a display [13] or mislead an image interpretation [14–16].
To apply the DMD in the multi-wavelength applications, compensating for the dispersion is necessary [17,18]. Two common approaches are the dual-DMD compensation [19] and grating compensation [20–23]. The dual-DMD method introduces a second DMD, designs a symmetric optical system, and keeps the micromirrors in the second DMD open. Because of the symmetry of optical system, the diffraction introduced by the second DMD can reverse the dispersion in the diffraction introduced by the first DMD. Similarly, the grating compensation method introduces a grating and use a second diffraction to compensate for the dispersion of DMD. The weakness of these methods is that the process of second diffraction is highly energy inefficient [24–26], because the optical system uses only a particular diffraction order and the energy carried by the others will be wasted. The low output energy can lead to either a low-contrast holographic display or a low signal-to-noise ratio in the fluorescence microscopy.
Here, we present a dispersion compensation method for dual-wavelength DMD modulation. The method capitalizes on a dispersive prism to neutralize the dispersion of DMD. Unlike the grating-based dispersion compensation, this method can achieve a high energy efficiency without the optical energy loss introduced in the second diffraction. To accurately compensate for the DMD dispersion, we constructed an diffraction model for the DMD modulation, enabling a precise calculation for the diffraction patterns at various wavelengths and incident angles. Based on the calculation, we designed a custom dispersive prism and carefully adjusted the incident angles of optical fields for the DMD modulation. A 4f system was set between the dispersive prism and the DMD to improve the overlap between the unmodulated optical fields on the DMD. Following this strategy, we experimentally demonstrated a compensation that effectively reduced the angular dispersion between the 532 nm and 660 nm modulated optical fields by a factor of ∼8.5.
2. Principle
Due to the dispersion of DMD, the exit angle of modulated beam varies with the wavelength so that the formed image shows a chromatic aberration and some regions will be dominated by a particular color [27]. To minimize the dependence of exit angle on the wavelength, our proposed method introduces a prism in the front of the DMD (Fig. 1(a)). The prism generates an angular dispersion deliberately designed to offset the angular dispersion of the DMD.
Fig. 1. Principle of prism-based dispersion compensation. (a) Schematic of prism-based achromatic DMD modulation system. L: lens; Lf: lens for Fourier transform. (b) Patterns generated by the 532 nm (green) and 660 nm (red) beams on the Fourier plane, without dispersion compensation. (c) Profiles along the dashed line in (b). (d) Patterns generated by the 532 nm and 660 nm beams on the Fourier plane after the prism compensates for the angular dispersion. (e) Profiles along the dashed line in (d).
We can check the effect of prism-based dispersion compensation by putting a lens (Lf) behind the DMD and a screen on the focal plane of Lf. Lf transforms the modulated beam to its Fourier space and represent its exit angle as a spot on the screen (Fig. 1(a)). For illustration, we assume that there is a DMD modulation system that must function at 532 nm and 660 nm simultaneously. Without the dispersion compensation brought in by the prism, the angular dispersion between the two beams in a specific diffraction order can be as large as a couple of degrees (Fig. 1(b)).
Figure 1(a) shows how the prism compensates for the angular dispersion between the two beams. First, the prism causes an angular dispersion so that the two beams propagating through the prism diverge. Second, the intentionally diverged beams pass through a 4f system that invert their propagation directions. The prism and DMD are set at the two focal planes of the 4f system. As a result, the originally diverged two beams will converge on the DMD. This setting maximizes the spatial overlap of the two incident beams on the DMD. Third, since the two beams respectively propagate towards the DMD at two incident angles with a predesignated angular difference, their exit angles with respect to the chosen diffraction order are the same. Therefore, their corresponding spots overlap on the screen (Fig. 1(d)).
To realize the idea introduced above, we developed a numerical model for the DMD modulation. We assume that the collimated incident light has a wavelength λ and incident angle $({\theta ,\varphi } )$. The origin of our coordinate system (x,y,z) is at the center of the DMD chip (Fig. S1a) and each micromirror rotates with respect to the y axis (Fig. S1b). Modulated by the DMD, the optical field can be expressed as:
Here, ${E_0}$ is the amplitude of incident light. M and $\phi $ describes the amplitude and phase modulation.On the one hand, the form of M is determined by two factors. The first one is the ON and OFF state of the micromirror. If a micromirror is in the OFF state, M = 0. If a micromirror is in the ON state, M can be expressed as:
On the other hand, the function $\phi $ consists of two components dh and dv. dh is the optical path difference introduced by the shift in the xy plane (Fig. S1a):
To compare the simulated result with what we observed in the experiment, we conducted a Fourier transform for the diffracted light and obtained its spatial spectrum $({f_x},{f_y})$. Subsequently, we mapped the spatial spectrum $({f_x},{f_y})$ to an angular spectrum $({\theta _x},{\theta _y})$ shown in Fig. 1(b) and (d), based on the relation:
3. Simulation
We analyzed the diffraction patterns of 532 nm and 660 nm modulated beams. We set w = 7.25 µm, and ${\boldsymbol \psi }$ = 12° for the ON state. Figure 2(a) shows the diffraction patterns if we set ${\boldsymbol \theta }$ = 24° and ${\boldsymbol \varphi }$ = 0°, and all the micromirrors in the ON state. The 0th diffraction orders at 532 nm and 660 nm were located at (-0.36°, -0.36°) and (1.48°, -1.48°), respectively. Here, we define the 0th diffraction order as the diffraction order with the highest intensity propagated approximately along the DMD normal toward the screen (see Supplement 1 S2).
Fig. 2. Simulation of dispersion compensation. (a) Patterns generated by the 532 nm (green) and 660 nm (red) beams on the Fourier plane, when the incident angles are $\theta $ = 24°. The left and right images show the diffraction patterns when the DMD dimension N is equal to 20 and 50, respectively. (b) Trajectories of 0th diffraction order when the incident angles were changed from 14° to 34°. (c) Plots of intensity changing with the exit angles. (d) Diffraction generated by various DMD patterns, including the stripes, dot array, checkerboard, random binary, and white patterns. To maximize the output energy at 532 nm, we set $\theta $ = 28.5° and 26.7° for the green and red light. The lower left image is a close-up view of the overlap of the 0th diffraction orders at 532 nm and 660 nm.
Figure 2(b) and (c) show how the position and intensity of 0th diffraction order will change, if we gradually increase ${\boldsymbol \theta }$ from 14° to 34°. Since the lasers available at 532 nm often had a lower output energy than those working at 660 nm, we tried to maximize the output energy at 532 nm. Following this scenario, we set the incident angle at 532 nm to ${\boldsymbol \theta }$ = 28.5° (Fig. 2(c)). To match the exit angle at 532 nm and minimize the angular dispersion, we set the incident angle at 660 nm to ${\boldsymbol \theta }$ = 26.7° (Fig. 2(c)). In practice, a DMD was often loaded with complicated patterns for modulation. We examined how the location of 0th diffraction order would change with the DMD patterns, including stripes, dot array, checkerboard, and random patterns. Once the incident angles were set, the center of 0th diffraction order would not be shifted, no matter what pattern that we loaded (Fig. 2(d)). Therefore, our analysis confirmed that compensating for the angular dispersion by carefully presetting the incident angles could meet the requirements for achromatic DMD modulation in theory.
With this finding, we proceeded to design the geometry of the dispersive prism as well as the incident angle with respect to the prism. We chose H-ZF52 as the prism material whose Abbe number was 23.78, corresponding to a refractive index of 1.855037 at 532 nm and 1.836496 at 660 nm. More detail about this design can be found in Supplement 1 S3.
4. Experiment
Figure 3(a) illustrates the experimental setup for the achromatic DMD modulation. In this system, a dual-wavelength laser (532 nm and 660 nm wavelengths, customized, Lasos Inc.) is employed as the light source for 532 nm and 660 nm collimated beams. The lenses L1 and L2 expanded the beams by a factor of 58.1 to cover the entire DMD chip (JUOPT-DLP9000X, JUOPT Technology Co., Ltd.). The mirror M1 between L1 and L2 reflected the beams so that both of them entered the custom prism (H-ZF52, vertex angle ∼53.8°) for dispersion compensation at an incident angle of ~55.9°. Because of the dispersion of the prism, the two beams which were originally coincident were diverged with an angular dispersion of 2.1°.
Fig. 3. Optical system for achromatic DMD modulation. (a) Schematic of the optical system. L, lens; M, mirror. (b) Photograph of the system.
Made up of L3 and L4, the 4f system reversed the divergence so that they would cover the same area of DMD chip. To make the system compact, we added a series of mirrors M2, M3, M4, and M5 which confined the light in a small space. After the achromatic DMD modulation, the diffraction patterns of the two beams would be captured by a camera (MV-SUA501GC-T, MindVision, Inc). The volume of the entire setup was 418 mm × 350 mm × 205 mm (Fig. 3(b)).
To evaluate the proposed method, we first studied its performance in the projection display. We loaded the DMD with “ABCD” characters and checked the chromatic aberration in the camera. To ensure that the camera could capture the diffraction patterns at both wavelengths even without a dispersion compensation, we put a 4f system (L5, f = 300 mm and L6, f = 100 mm) downstream (Fig. 4(a) and (b)). By moving the location of the camera along the optical axis, we monitored the divergence of modulated patterns at two wavelengths (Fig. 4(c) and (d)). We found that, without an appropriate compensation, the “ABCD” pattern only well aligned at the focal plane of L6. Moving away from that plane, the patterns at two wavelengths diverged quickly (Fig. 4(c)). In contrast, with our dispersion compensation, the “ABCD” patterns at two wavelengths well overlapped along the optical axis within the range that we monitored. Furthermore, we quantitatively evaluated the dispersion compensation by projecting a dot with the DMD at each wavelength and measuring the distance between the two dots (Fig. 4(e)). As we had expected in the theory, we observed an divergence of 5.9°. This divergence could be significantly reduced, through our prism-based compensation, to ∼0.7°. Thus, our method reduced the angular dispersion by a factor of ∼8.5. These evidences indicate that the proposed method can compensate for the angular dispersion induced by DMD.
Fig. 4. Achromatic DMD projection display. (a) Schematic of a display with chromatic aberration. (b) Schematic of a display with dispersion compensation. (c) Images captured by the camera at different distances, if no compensation was applied. (d) Images captured by the camera at different distances, after the dispersion compensation was applied. The number below each image denotes the displacement with respect to the focal plane of L6 along the optical axis, in the unit of millimeter. (e) Measurement of the angular dispersion between 532 nm and 660 nm beams with and without dispersion compensation.
Next, we validated the performance of prism-based dispersion compensation in the structured illumination. Without the dispersion compensation, the two beams at 532 nm and 660 nm had their own oblique incident angles (Fig. 5(a)). In a microscopic imaging system, the rear aperture of objective, with a limited size, could block their propagation and contributed to dark regions in the stripe patterns (Fig. 5(b)). In contrast, if the dispersion could be appropriately compensated for (Fig. 5(c)), before entering the objective, the completeness of stripe patterns at both wavelengths could be preserved (Fig. 5(d)). These results shows the potential application of our method.
Fig. 5. Dispersion compensation for structured illumination. (a) Schematic showing the propagation of green and red lights in the optical system without the dispersion compensation. (b) Images of stripe patterns without the compensation. (c) Schematic showing the propagation of green and red lights in the optical system after the dispersion compensation. (d) Images of stripe patterns after the compensation.
In the end, we analyzed the energy partition in various diffraction orders. We uploaded a stripe pattern to the DMD and then observed multiple diffraction spots in the Fourier space (Fig. 6). Three diffraction spots around the center were most prominent in the field of view. We measured the power of these three diffraction spots at the two wavelengths. Their total powers were 93.7µw and 130.65µw at 532 nm and 660 nm, respectively. Their corresponding energy efficiency was above 10% at both wavelengths.
Fig. 6. Analysis of output energy efficiency. (a) Diffraction pattern in the Fourier space at 660 nm. (b) Diffraction pattern in the Fourier space at 532 nm.
5. Conclusion and discussion
We have proposed and demonstrated a prism-based dispersion compensation method that can considerably improve the optical energy efficiency of achromatic DMD modulation. Our DMD diffraction model can accurately compute the angular dispersion and intensity distribution generated by the DMD modulation. The computed result allows us to design the geometry of prism that generates a pre-incident dispersion neutralizing the angular dispersion of DMD. Due to the absence of energy waste in an additional diffraction, our method can obtain an optical energy efficiency of 11.13% at 532 nm and 10.05% at 660 nm.
In theory, our method can be extended to subdue an angular dispersion involving more than two wavelengths, given enough degrees of freedom in the design of dispersion-compensated optical component in the front of DMD. Considering the real capability of fabricating optical components as well as the nonlinear effects in optical materials, however, the accuracy of prism-based dispersion compensation will be degraded over a broad spectral range. Further studies should include quantifying the performance of dispersion compensation over a broader spectral range, as well as more sophisticated design methods for the dispersion compensation components [28].
Our cost-effective dispersion compensation method offers a solution to address the significant angular dispersion in the DMD modulation over a broadband (Supplement 1 S4), encountered in various applications, such as super-resolution microscopy, laser printing, and full-color holographic displays. For instance, the proposed method can help structured illumination microscopy mitigate the dispersion in the structured illumination patterns so that the image acquired at each wavelength will be well aligned and prevent unexpected misinterpretation [29].
Funding
National Natural Science Foundation of China (62205189, 62375171); Shanghai Pujiang Program (22PJ1407500); Shanghai Jiao Tong University 2030 Initiative (WH510363001-10); Oceanic Interdisciplinary Program of Shanghai Jiao Tong University (SL2022ZD205); Science Foundation of Donghai Laboratory (DH-2022KF01001).
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.
Supplemental document
See Supplement 1 for supporting content.
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