1. Introduction

Portable projection displays are currently one of the main topics of interest in the area of projection technology. Portable projection displays should be of small size and have high optical efficiency. Therefore a significant reduction in size and weight is one of the most important goals in portable display development. Small laser sources can create light beams of large power with small attenuation, which allows for the creation of small optical systems with high optical efficiency in projection displays. These laser sources can also create images with high color saturation (good color gamut), much higher than those created with a filament lamp, and they may provide higher efficiency than light-emitting diodes. Therefore laser displays are one of the main directions in the development of portable projection displays.

One of the promising directions in laser display development is one-dimensional (1D) scanning-laser projection displays, in which a narrow line of the image created by a 1D array of pixels is scanned over the screen to create a two-dimensional (2D) image (see Fig. 1). In order to provide the proper resolution, a scanning beam should have a width on the screen equal to or smaller than the width of one pixel. The operating principle of scanning 1D displays is well-known [1, 2, 3, 4, 5]. For our purposes, it is sufficient to know that the scanning beam width is no greater than the pixel width. Among the present solutions in this area are grating light valves, developed by Silicon Light Machine [1], and grating electromechanical systems, developed by Eastman Kodak [2]. In this paper we primarily focus on the application of the spatial optical modulator (SOM) developed by Samsung Electro-Mechanics [3, 4, 5]. However, our main results are applicable to any laser-scanning device that uses a similar speckle-suppression mechanism.

It is a well-known fact that using laser sources in laser projection systems creates speckle noise in the image on the screen [6]. This can significantly decrease the image quality by creating granulation of the image intensity in the human eye (subjective speckle). Subjective speckle noise statistics are well-known for static laser beams [6]. Marom et al. [7] investigated the statistics of the speckle pattern created by a moving beam of coherent light. However, all the analyses in that work were performed for use in bar code scanners, and the effects of aberration and defocusing of the scanning beam were not considered.

Murphy et al. [8] investigated the effects of optical aberrations on laser speckle in the viewer projection system. In that work, partially developed speckles and an aberrated imaging system, such as a camera objective or the human eye’s crystalline lens, were considered. However, neither the optical projection system that transmits the beam to the screen nor its aberrations were investigated. Knowing the requirements of the projection system parameters, including the aberration level and its effect on speckle suppression, is important for developing projection displays.

In our prior work, we used an autocorrelation function of a coherent scanning beam to investigate the possibility of speckle suppression arising from beam scanning along screen [9]. For laser display applications, the most important requirement is first-order statistics, especially the speckle contrast, which describes the level of speckle noise in an image. In our previous paper [9], we derived a simple formula to calculate the subjective speckle contrast in laser displays. That work considered a coherent light beam scanning over the screen with no other moving parts. For increasing speckle suppression, a special phase diffractive optical element (DOE) was considered to be inserted in the object plane (or first image plane). In Fig. 1, it should be inserted in the input plane ($x^{\prime \prime }$) together with a SOM, as proposed in [9, 10], or in the intermediate image plane (same place in Fig. 1), as proposed in [11].

In our previous work [9], we assumed that the projection display used an ideal, aberration-free optical projection system being used in the display. However, for practical construction of a display, it is important to know the influence of a real (not ideal) projective lens’ optical parameters on the speckle contrast. The most interesting questions are as follows:

It is easy to show by using spatial Fourier domain that the autocorrelation function of a time-coherent beam propagating in free space (in our case, beam after projective lens) does not change with beam propagation distance and does not depend on the phase of spatial Fourier components of the field. To position the screen in an exactly focused image plane, beam movement in a projection display could be presented as a linear translational shift along the screen [9], and therefore the speckle properties on screen for this case should not depend on lens aberrations. It is easy to show that speckle contrast in this case should be calculated through $\mathrm{NA}=\sin ({\theta }_1)$ of the objective lens (where ${\theta }_1$ is the convergence angle of the output beam). However, for a laser projection display with mismatch between image plane and screen (defocusing), the beam movement across the screen could not be presented as a simple linear translational motion; it is a complex combination of shift and rotation. Therefore speckle contrast could not be calculated just through the autocorrelation function of incident on the screen beam. It should be calculated through a more complicated correlation relation. Therefore the question about speckle dependence on defocus and aberration (in the case of defocus) is not trivial for a laser projector using a scanning mirror in the Fourier plane to spread the beam along the screen.

In the present analysis we attempt to provide answers to these questions using the theoretical approach for speckle contrast calculations developed in [9]. It should be noted that, with small modifications, the theory that we elaborate for 1D projection systems can also be used for zero-dimension laser projection systems. In the latter case, a 2D image is created by moving a small spot of light in a zigzag pattern over the screen, creating a pixel-by-pixel, raster-type pattern.

An optical diagram of the display-human eye system is presented in Fig. 1. For our analysis, we assumed that the object plane is located in the $x^{\prime \prime }$ plane in Fig. 1. A SOM equipped with a special DOE for speckle suppression may be inserted in this plane, as proposed in [9]. Instead of a DOE in the SOM plane, it can be placed in the intermediate image plane, as proposed in [11]. The previously shaped and modulated optical beam passes through the projection objective and hits the screen. An exactly focused image is reproduced in plane $x^{\prime \prime \prime }$. In the general case, the screen may be situated in a defocused position at a distance h from plane $x^{\prime \prime \prime }$. Scattered light from the screen enters the human eye or any other viewing system situated at a distance a from the screen. The beam is scanned over the screen in the X direction using a scanning mirror.

2. Speckle Contrast Ratio: General Considerations

Figure 1 shows the optical scheme of the 1D laser projection system, and Fig. 2 provides a detailed description of the scanning process and complex beam motion in this system. We previously obtained a simple expression for the speckle contrast ratio (CR) of a scanning laser display [9]. In that work, we assumed that an ideal projection objective produces a magnified image on the screen plane x of the optical field distribution in the input plane $x^{\prime \prime }$ (see Fig. 1) without defocusing ($h=0$ and $x^{\prime \prime \prime }=x$) and without any distortion in either phase or amplitude. However, in the case of a nonideal optical system, defocusing effects ($h\ne 0$) and aberrations may degrade the complex amplitude distribution across the scanning beam on the screen.

The beam’s scanning motion along the screen is provided by reflection from a rotating mirror (e.g., a galvanometric mirror). The best position for this scanning mirror is the projection objective’s rear focal plane, as presented in Figs. 1, 2. Therefore the scanning beam’s behavior on the screen may be represented as a superposition of two different types of movement: linear translational motion and rotation (see Fig. 2).

In our previous work [9], we ignored the rotational component and took into account only linear translational movement of the beam. Therefore the formula for speckle contrast calculations obtained in [9] should be modified to take into account the rotation component of the real beam’s motion. In order to derive formulas for the real beam movement, we used the one for speckle contrast calculation for the case when incident light on the screen is varying more quickly than can be resolved by the human eye (see Fig. 1).

In the Fresnel approximation (the same as in [9]), the complex amplitude of the electromagnetic field on the eye’s retina can be calculated from the formula

From Eq. (1), it is easy to obtain the formula for the field power accepted by an eye during one scan:

In our previous analysis [9], we used a spatially invariant autocorrelation function $A(x_1-x_2)$, because at that time we ignored the rotational component of the beam motion. Here we use a more general definition of the autocorrelation function in Eq. (4), because with taking into account the rotation component of a beam movement, the beam shape on screen changes during the scan along the screen. The second-order moment of the scanning beam intensity on the screen $〈I^2〉$ can be written as

3. Scanning Beam

We assume that the aberrations in the projection system are sufficiently small to produce only phase distortion of the wavefront—with no change in amplitude—at the exit pupil, i.e.,

It is more convenient to present the relative motion of the beam and screen as shown in Fig. 3. Here we consider a stable beam described by Eq. (12) in a motionless $X,Z$ coordinate system and a conventionally rotating screen and its rotating coordinate system $X_S,Z_S$. In order to find the optical field on the conventionally rotating screen in Fig. 3, we need to express the coordinate x in the stable coordinate system through a rotating screen coordinate $x_S$ and to calculate the new defocusing distance $h_S$. For the simple geometry of Fig. 3, we can write the following relationships:

Usually a scan mirror is situated close to the projection objective in such a manner that the distance to lens $L_0$ is much less than the distance to screen $L(L_0\ll L,L_0\ll Z_2)$ if defocusing h is comparatively small. As a result, we may assume $\mathrm{L}\approx Z_2+h$, i.e., the scan-mirror-to-screen distance is approximately the same as the distance from the objective to the screen. In this case, and for small angles β, we may use the following approximations:

4. Aberrations, Defocusing, and Speckle Contrast

The speckle contrast calculation in the scanning beam requires as input the correlation function of the field of the scanning beam on the screen. We can obtain a formula for the autocorrelation by substituting Eq. (22) into Eq. (4):

Equations (27, 28) are invariant with respect to shift in both arguments, and so they depend only on the difference $x_1-x_2$. Thus the expression for the speckle contrast obtained in [9] can also be used, as well as the more general Eq. (7).

In most practical cases, the phase factor under the integral on the right-hand side of Eq. (28) may be ignored, in which case the autocorrelation function of the exactly focused system becomes identical to that calculated for an ideal projection system in [9]. Thus taking in account the rotational component of the beam scanning motion provides us the possibility to obtain a more simple and appropriate result.

From Eq. (27), it follows that a screen shift at a distance h from the image plane would result in only a rescaling of the modulus of the autocorrelation function. Rescaling would yield exactly the same results for speckle suppression as simply refocusing our optical system to a new position on the screen. Hence we could say that the speckle contrast in a scanning projection depends only on the angular subtense or numerical aperture ${\mathrm{NA}}_{\text{OUT}}=\sin ({\theta }_{\text{OUT}})$ of the output beam hitting the screen (see Fig. 4).

To estimate the speckle contrast, we can use Eq. (7). By substituting Eq. (28) into Eq. (7), we find

Now we assume that a Barker code DOE is used for speckle suppression as proposed in [9]. Then the autocorrelation function in Eq. (28) has a narrow, sharp central peak and a small sidelobe. At the same time, the weight function $Q(x)$ is a flat function and therefore remains constant in the area of the central peak of the autocorrelation function. As long as the defocusing is usually small and the value of μ is close to unity, taking into account the flat shape of weight function $Q(x)$ in the central area [9] of the autocorrelation function, we may assume that $Q(\mu x)\approx Q(x)$. In this case, Eq. (29) gives us a simple relationship among the defocusing, the speckle CR C, and the design speckle CR $C_0$:

Therefore, the speckle contrast in a defocused scanning laser display should increase slightly with defocusing distance h according to Eq. (30), and it should not vary with the defocusing that is caused when the focal length is changed while the distance from objective lens to screen is kept constant.

It should be noted that, in spite of tedious calculation, the final result is very simple. We tried several methods, however, we did not find an easy way to prove this final result.

5. Optimization of Beam Shape for Speckle Suppression

The speckle contrast in a scanning 1D projection system depends on the properties of the scanning beam. What shape should the scanning beam have in order to best suppress speckle? Equation (29) for the speckle contrast calculation in the image plane can be seen as a positive definite functional over the autocorrelation function $A_0(x)$. Hence the problem of optimizing the shape of the scanning beam to obtain maximum speckle suppression can be reduced to a problem of minimizing that functional. If there are no restrictions on the scanning beam, it is easy to find a solution for the optimal beam shape in order to minimize speckle contrast. We get the absolute minimum of speckle contrast (zero level) using a scanning beam with an autocorrelation function that satisfies the condition

However, this solution for the autocorrelation function has an infinite spatial spectrum width that is impossible to achieve in any real application for the reasons mentioned earlier. Therefore, for a practical solution to this minimization problem, we should calculate the minimum of the functional Eq. (29) with a limitation on the spatial spectrum of the scanning beam. Because the optical system put limits on the spatial Fourier plane, it is better to transform Eq. (29) to the spatial frequency domain. Then we can rewrite Eq. (29) in the form

For simplicity, we assume an exactly focused display ($\mu =1$ and $h=0$) in Eq. (33). The limitation in the spatial frequency domain can be written as $|\mathrm{\Delta }{\alpha }_0|\le k·N{A}_{\text{Out}}$, where ${\mathrm{NA}}_{\text{OUT}}=\sin ({\theta }_{\text{OUT}})$ is the output numerical aperture as defined in Fig. 4. Taking into account the truncation in the spatial frequency domain, we can rewrite Eq. (33) for a frequency-limited system as follows:

We used a Fourier set to approximate the field in spatial Fourier space, after which we used direct calculation to find the polynomial coefficients yielding the minimum speckle contrast. We assumed the optical system to have even symmetry and therefore that $A(x)$ is an even function, implying that its Fourier transform is a real function. We expanded the $a(\alpha )$ in a Fourier series:

There is a simple relationship between the spatial spectra of the autocorrelation function $a(\alpha )$ Eq. (34) and of the electrical field $\tilde{E}(\alpha )$:

In this way, Eq. (37) represents optical field intensity in the Fourier plane (rear focal plane of the projection objective). By using the Ritz–Galerkin algorithm, we found coefficients $c_n$, and after substitution of Eq. (36) into Eq. (37), we can calculate the optimal values for the speckle CR.

Figure 5 shows the field intensity distribution in the Fourier plane after beam shape optimization. A numerical optimization with a six-component Fourier series has been carried out for the case of ${\mathrm{NA}}_{\text{OUT}}/{\mathrm{NA}}_{\text{EYE}}=6.18$, where ${\mathrm{NA}}_{\text{OUT}}$ is the output NA of the projection lens (see Fig. 4), and ${\mathrm{NA}}_{\text{EYE}}$ is the numerical aperture of the human eye. The optimal field distribution is presented in Fig. 5. It has a sharp peak at the edge of the spatial frequency band, which is the main difference from the homogeneous field intensity distribution in Fourier domain field distribution corresponding to a Dirac deltalike autocorrelation function in Eq. (32) for a beam with infinite width in the spatial frequency domain. In spite of that fact, the speckle contrast values calculated for this and the homogeneous field distributions over lens numerical distribution are nearly the same, with a difference of less than 3%. Numerical simulation has shown that, even for a smaller NA ratio (${\mathrm{NA}}_{\text{OUT}}/{\mathrm{NA}}_{\text{EYE}}\sim 1$), where the optimization should provide the most effect, the beam shape optimization could provide only up to an 8.5% decrease in speckle CR relative to the homogeneous spatial spectrum. Therefore we can conclude that in the case of a real optical system with a finite NA, a beam with a homogeneous filling of aperture of projection lens would provide the best speckle suppression. This finding agrees with the well-known fact that speckle contrast in a scanning beam depends mainly on the output NA of the objective lens.

6. Experimental Data

We used the optical setup presented in Fig. 6 for experimental verification of the defocusing effect. A green laser beam ($532\mathrm{nm}$) was shaped into a narrow line by an anamorphic beam-shaping system. The intensity distribution in the beam cross section parallel to the picture plane (before DOE) was nearly Gaussian . The beam width in the focal plane where the Barker code DOE was inserted was approximately $20\mathrm{\mu m}$. The top view of the experimental setup is presented in Fig. 6. A long size of a linelike beam cross section was perpendicular to the picture plane.

The Barker code DOE was performed on the flat glass plate in such a manner that the phase modulation function was distributed across the beam in the direction parallel to the picture plane. A Barker code with a length of $\mathrm{13}\mathrm{bits}$ and a single bit length of $2\mathrm{\mu m}$ was used. Thus the total code length was $26\mathrm{\mu m}$. There were many periods of the Barker code on the glass attached to each other at a distance of approximately $0.5\mathrm{mm.}$ By using a micrometer, it was possible to remove the DOE from the beam and to return it again.

The projection system reconstructs the beam image on the screen after reflection from the scan mirror. The scan mirror provided a regular linear scan of the beam over the screen in the horizontal direction (parallel to the picture plane). Initially the screen was positioned at a distance $L=102\mathrm{cm}$. A CCD camera was situated at a distance $a=1.5\mathrm{m}$ from the screen. The screen was made of white scattering material, and it had a rigid flat substrate to suppress vibration or warpage due to air flow.

The projection system was focused at a distance of $L=102\mathrm{cm}$. During the experiment, this distance was kept constant, while the distance between screen and scan mirror was changed. The projection system focal length was $f=50\mathrm{mm}$. So magnification was approximately $M\approx 20$. The distance from the screen to the camera was also kept constant ($a=1.5\mathrm{m}$). The camera was equipped with an objective with a focal length of $160\mathrm{mm}$ and a quadratic diaphragm of $\mathrm{\Delta }=1.8\mathrm{mm}$. The CCD pixel size was approximately $7\mathrm{\mu m}$. The integration time was approximately $300\mathrm{ms}$, while the scanning mirror frequency was $120\mathrm{Hz}$.

The camera had a digital output connected to a PC through a buffer device. During the experiment, the mean value $〈I〉$ and standard deviation σ of the brightness was calculated over the same small central area on the screen. Afterward the speckle CR was obtained as the ratio $C=\sigma /〈I〉$. These measurements and calculations were repeated for a number of different distances L in two regimes: the presence or absence of a Barker code DOE.

For choosing Barker code single mesh size b, we used the next considerations. Being projected on the screen, it should be larger than screen microroughness characteristic feature size ${\delta }_S$, and at the same time, it should be less possible than the scanning beam width, which we suppose is matched to the camera resolution element on screen $D/2\sim 0.4\mathrm{mm}$. So we can use the next relationship: $D/2>b·M\gg {\delta }_S$, where $M=20$ is the projection system magnification.

To check our theory, Eq. (30) was used. To calculate the expected contrast at the exactly focused distance $L=102\mathrm{cm}$, instead of Eq. (31), we used the simple equation from [9]:

As described in [9], $N_0$ can also be defined as the approximate number of the Barker code bit length N. Thus in the regime without the Barker code DOE, we can make the approximation that $N_0=N=1$, giving $C_0=0.47$ from Eq. (39). Finally the formula in the regime without Barker code DOE could be written as

Figure 7 shows the experimentally measured and numerically calculated dependence of speckle CR on a distance L. Graphs 1 and 3 correspond to the measured (curve 1) and calculated (curve 3) CRs, respectively, for the regime without the Barker code DOE. Graphs 2 and 4 correspond to the measured (curve 2) and calculated (curve 4) CRs, respectively, for the regime with the Barker code DOE. From Fig. 7, one can see that the experimental curves align with the theoretical prediction of Eq. (30). The magnification of the display projection system changes with distance L as $M=L/Z_1=(Z_2+h)/Z_1$, independent of whether the image is focused. At the same time, the autocorrelation function is rescaled in the same proportion [Eq. (27)], and the speckle CR changes according to Eq. (30). The speckle suppression mechanism functions as if the projection system is exactly refocused for each new distance L each time it is changed. Thus defocusing at any degree has no effect on the speckle suppression mechanism.

We also tried introducing defocusing by a slight shift of the Barker code DOE along the optical axis and by keeping all distances constant at $L=102\mathrm{cm}$ and $a=150\mathrm{cm}$. Using such a procedure, we controlled the defocusing degree in the initial exactly focused plane. Simultaneously, we checked the speckle CR in the same area on the screen. The CR remained the same over a wide range of defocusing shifts.

We made no special measurements of the speckle contrast in aberrated and nonaberrated systems under the same conditions. But during our other experiments with “good” and “bad” objectives and lenses, we never noticed any large differences in the speckle suppression effect.

7. Conclusion

By applying a Fresnel scalar approach and a thin lens model to the objective lens in order to calculate the speckle CR, we deduced the following:

 

Fig. 1 Conventional scheme of beam propagation in scanning projection displays. The linear pixel array is perpendicular to the picture plane.

Download Full Size | PDF

 

Fig. 2 Beam scanning mechanism.

Download Full Size | PDF

 

Fig. 3 Conceptual beam scanning scheme. The convention is to fix the position of the beam and to rotate the screen.

Download Full Size | PDF

 

Fig. 4 Depiction of the scanning beam and definition of NA: ${\mathrm{NA}}_{\text{IN}}=\sin {\theta }_{\text{IN}}\approx d_1/Z_1$ and ${\mathrm{NA}}_{\text{OUT}}=\sin {\theta }_{\text{OUT}}\approx d_1/(Z_2+h)$.

Download Full Size | PDF

 

Fig. 5 Optimization for the best speckle suppression electric field intensity in the spatial frequency domain with a twelfth-order polynomial for the case of ${\mathrm{NA}}_{\text{OUT}}=0.0055$ and ${\mathrm{NA}}_{\text{EYE}}=8.9\times {10}^{-4}$. The calculated speckle contrast for the optimized electric field distribution is $C_o=0.22$ and for a homogeneous intensity distribution in spatial frequency domain is $C_H=0.227$, 100% $(C_H-C_o)/C_o=2.7\%$

Download Full Size | PDF

 

Fig. 6 Optical diagram of the experiment.

Download Full Size | PDF

 

Fig. 7 Experimental data compared with calculations from Eq. (31). The exact image plane is at a distance of $102\mathrm{cm}$. Curves 1 and 2 correspond to the experimental CRs, while curves 3 and 4 correspond to the calculated ratios. Curves 1 and 3 correspond to the regime of scanning without the Barker code DOE, while curves 2 and 4 correspond to the regime of scanning with the Barker code DOE.

Download Full Size | PDF

1. D. M. Bloom, “Grating light valve: revolutionizing display technology,” Proc. SPIE 3013, 165–171 (1997). [CrossRef]  

2. M. W. Kowarz and J. G. Phalen, “High-contrast display system with scanned conformal grating device,” U.S.patent 6,678,085 (13 January 2004).

3. S. Yun, “Open hole-based diffractive light modulator,” U.S.patent 7,206,118 (17 April 2007).

4. S. K. Yun, J. H. Song, I. J. Yeo, Y. J. Choi, V. I. Yurlov, S. D. An, H. W. Park, H. S. Yang, Y. G. Lee, K. B. Han, I. Shyshkin, A. S. Lapchuk, K. Y. Oh, S. W. Ryu, J. W. Jang, C. S. Park, C. G. Kim, S. K. Kim, E. J. Kim, K. S. Woo, J. S. Yang, E. J. Kim, J. H. Kim, S. H. Byun, S. W. Lee, O. K. Lim, J. P. Cheong, Y. N. Hwang, G. Y. Byun, J. H. Kyoung, S. K. Yoon, J. K. Lee, T. W. Lee, S. K. Hong, Y. S. Hong, D. H. Park, J. C. Kang, W. C. Shin, S. I. Lee, S. K. Oh, B. K. Song, H. Y. Kim, C. M. Koh, Y. H. Ryu, H. K. Lee, and Y. K. Raek, “Spatial optical modulator (SOM): high density diffractive laser projection display,” Proc. SPIE 6487, 648710 (2007). [CrossRef]  

5. S. Yun, J. H. Song, S. D. An, I. J. Yeo, Y. J. Choi, Y. G. Lee, H. W. Park, K. B. Han, H. S. Yang, V. Yurlov, I. Shyshkin, A. Lapchuk, H. Y. Kim, J. W. Jang, J. H. Kyoung, J. S. Yang, S. K. Yoon, C. S. Park, J. P. Cheong, Y. N. Hwang, K. S. Woo, S. W. Ryu, S. W. Lee, C. M. Koh, Y. K. Baek, D. H. Bae, H. K. Lee, J. H. Lee, Y. H. Ryu, H. Y. Hwang, C. M. Yang, O. K. Lim, D. H. Park, S. H. An, J. H. Bae, S. M. Cho, B. S. Go, S. K. Hong, H. P. Jung, S. J. Kim, K. U. Lee, J. H. Park, J. H. Yang, G. Y. Byun, S. H. Byun, Y. J. Cho, C. G. Kim, J. H. Kim, S. K. Kim, S. I. Lee, W. H. Lee, K. Y. Oh, S. K. Oh, W. C. Shin, B. K. Song, and E. M. Bourim, “A novel diffractive micro-optical modulator for mobile display applications,” Proc. SPIE 6887, 688702 (2008). [CrossRef]  

6. J. W. Goodman, Speckle Phenomena in Optics (Roberts and Co., 2006).

7. E. Marom, S. Kresic-Juric, and L. Bergstein, “Analysis of speckle noise in bar-code scanning systems,” J. Opt. Soc. Am. A 18, 888–901 (2001). [CrossRef]  

8. P. K. Murphy, J. P. Allebach, and N. C. Gallagher, “Effect of optical aberrations on laser speckle,” J. Opt. Soc. Am. A 3, 215–222 (1986). [CrossRef]  

9. V. Yurlov, A. Lapchuk, S-K. Yun, J-H. Song, and H-S. Yang, “Speckle suppression in scanning laser display,” Appl. Opt. 47, 179–187 (2008). [CrossRef]   [PubMed]  

10. A. Lapchuk and J-H. Song, “Optical modulator and optical modulator module for reducing laser speckle,” U.S.patentapplication 20080055698 (6 March 2008).

11. J. I. Trisnadi, “Method and apparatus for reducing laser speckle,” U.S.patent 6,323,984 (27 November 2001).

12. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1968).

13. W. Ritz, “Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik,” J. Reine Angew. Math. 135, 1–61 (1908).