## Abstract

An optical scheme for speckle suppression using two or three partially coherent beams in a projection system is proposed. Diffractive optical elements (DOE) placed in the intermediate image plane create several beams carrying the image to a screen. Transparent plates of different thicknesses are placed in the Fourier plane of the projective lens and used for beam decorrelation. The coherence matrix algorithm for speckle suppression is used to calculate the speckle contrast ratio. It is shown that for a small decorrelation length and using the same maximum thickness of the transparent plates, two partially coherent beams would provide better suppression than three beams with different diffraction orders. However, for a large decorrelation length, the three beam setup provides better speckle suppression for all three colors examined with a suppression coefficient close to theoretical limits. Verification of speckle suppression using three-beam decorrelation is reported.

© 2009 Optical Society of America

## 1. Introduction

The development of a portable pico-projector is currently one of the main goals in the field of projection display. One of the promising technical solutions is a 1D laser projector. At the present time, there are several well developed solutions for a 1D laser projector: the Grating Light Valve (GLV) developed by Silicon Light Machine [1], the Grating Electromechanical System (GEMS) developed by Eastman Kodak [2] and the Samsung Optical Modulator (SOM) developed by Samsung Electro-Mechanics [3, 4].

The main goal in all of these devices is the application of a reflective diffraction structure consisting of ribbon-shaped micro mirrors forming a 1D image pixel array. The structure is able to change the phase shift between neighboring elements by the application of voltages to pixel electrodes, providing a height shift of the reflecting ribbons. The structure forms a Spatial Light Modulator (SLM). For electro-mechanical transformation, GLV and GEMS technology use electrostatic attractive forces while SOM technology uses piezoelectric actuators.

The image of a 1D pixel array in a reflected light beam is projected on the screen by the optical system and scanned by an electrically driven mirror. The Schlieren stop (non-transparent shield having a slit) is situated in the focal plane of the projection optical system (spatial frequency domain) and suppresses undesirable diffraction orders. The optical power in the working diffraction order being passed through the Schlieren stop may be changed according to the phase shift and voltage applied to a certain pixel electrode. Thus, owing to the modulation effect, both in the time and space domains, a gray scale image is created on the screen.

Because of their small etendue and high optical efficiency, laser diodes (LD) have strong advantages as light sources for 1D laser projectors. The laser light sources also provide pure color light beams. All these factors enable the construction of small projector systems with small energy consumption, high quality and high color saturated images. It should be said also that it is very difficult to imagine a 1D mobile projector without LD light sources because of the small etendue and large light intensity required for a small 1D array of pixels.

Until now, however, LD sources have not been used in display mass production because highly coherent laser light scattering on a rough screen creates a granular type of intensity modulation in the human eye (subjective speckles) [5, 6, 7, 8]. It is possible to reduce speckle noise via different methods of speckle averaging: destroying time coherence (wideband sources) [9, 10] or destroying spatial coherence by using angle diversity with decorrelation beams coming from different angles [11, 12, 13, 14] (or by vibrating screen [15]) or polarization diversity [16]. The main speckle measurement parameter in laser projectors is the speckle contrast ratio (CR). It is defined in a homogeneously illuminated screen as the ratio of the standard deviation (*σ _{I}*) to the mean (〈

*I*〉) of the light intensity [17]

measured through the screen image by camera with optical parameters identical to those of the human eye.

The speckle of a perfectly coherent polarized beam on a rough surface has CR=1. From standard deviation properties, it follows that using N non-coherent beams hitting the screen at different angles (angular diversity) in the same place should decrease speckle CR to a level

and, in the case of beams with equal intensity in polarized light, to CR=1/sqrt(N).

For destroying spatial light coherence on screen, a complicated projection system having an intermediate image plane was proposed [12, 13]. In the intermediate image plane, the vibrating DOE are placed as shown in Fig. 1. The DOE splits the light beam into several diffraction orders where each carries image information separately. DOE vibration creates different phase shifts for different diffraction orders. If the vibration amplitude is equal to one or several pitch sizes, the vibration totally decorrelates beams of different diffraction orders in a time interval equal to the period of vibration and provides speckle suppression. Trisnadi in [13] proposed a special DOE shape and movement scheme to obtain the best speckle suppression for the case of 2D DOE. The optimal shape of 1D DOE for scanning display is proposed in [14]. From general considerations, to obtain the best speckle suppression, the DOE should provide diffraction orders with approximately the same beam intensity, and the angular separation of the diffraction orders should be sufficient to provide statistically independent speckle patterns in the eye. This means that the angular beam separation should be larger than the angular resolution of the human eye [18].

It is well known that in the case of a scanning projection system, the scanning effect partially destroys spatial coherence in the direction of scanning motion [14]. Therefore, DOE vibration along the scanning direction does not give additional speckle suppression. For the best speckle suppression in the scanning direction, the DOE should provide a homogeneous filling of the projection lens by light [14].

For beam decorrelation in the vertical direction, the DOE should vibrate. The use of mechanical vibration, especially the very fast and precise vibration needed for scanning projection systems (as in the case of 1D or 0D of the scanning projection system), is very inconvenient to apply because the speed of vibration should be very fast. In this case it is advantageous to use wideband laser sources and a passive grating method of beam decorrelation (like different optical paths for different beams) to get speckle suppression. Here, we will present a numerical algorithm and experimental data applying this method to a mobile projection system.

## 2. Numerical algorithm for speckle contrast calculation

A possible optical scheme for obtaining several beams for image projection is shown in Fig. 2, where DOEs are inserted in the intermediate image plane. DOEs separate the incident beam modulated by the image into several diffractive beams. Transparent plates with different thickness should be inserted in the Fourier plane of the projective lens. The difference in the thicknesses should be sufficiently large to decorrelate the beams. Only regular DOEs (periodical DOEs) should be used to get definite diffraction orders in the Fourier plane to have possibility to manipulate with diffraction orders phases without a distorted image.

To calculate the speckle suppression in this optical scheme, one should know the correlation coefficient between two beams with an optical path difference corresponding to the difference between passing through air and through glass of thickness ** s**. It is not difficult to obtain the formula of a correlation coefficient, assuming that the laser radiation is a wide-sense-stationary random process with a spectrum having a Gaussian distribution of intensity

*I*(

*k*) =

*I*

_{0}exp{-[(

*k*-

*k*

_{0})/(0.5*

*δk*)]

^{2}}, where

*k*and

*k*

_{0}are the wavenumber and central wavenumber of the LD bandwidth respectively, and δk is the bandwidth at the 1/e intensity level. The correlation coefficient of two beams created by different diffraction orders of DOE could be written as the autocorrelation function, considering the optical path difference caused by two retarder thicknesses

*s*. Using the Wiener–Khintchine theorem, it is possible to write the normalized autocorrelation function as the Fourier transform of the spectrum of the laser light intensity

_{i}where c is the speed of light, *τ* is the time shift and A is a normalization coefficient to set *μ _{ij}*(0) = 1 . Taking the mutual retardance in neighboring beams as

*cτ*=

_{ij}*s*(

_{i}*n*-1) -

_{i}*s*(

_{j}*n*-1 ) (n

_{j}_{i}are the refractive indexes of the retarder plates), we can find the mutual correlation coefficient between i-th and j-th beam as:

For the case *n _{i}* =

*n*=

_{j}*n*:

From Eq. (5) it is easy to calculate the glass thickness that provides a correlation coefficient smaller then 1/e (decorrelation glass thickness) for the case where all retarders are made from the same glass:

It is not always possible to achieve full beam decorrelation due to limitations of image depth of focus and lack of space in the Fourier plane between diffraction orders. For speckle suppression design, it is necessary to know what will be the impact on speckle suppression if there are several incident partially coherent beams hitting the screen.

Below we will compare speckle contrast for two cases (see Fig. 3): a two beam scheme and a three beam scheme. In the first case, we assume that we have two beams with equal intensities for all colors partially decorrelated by a total glass thickness of s_{2}. This regime is easily adapted through +1 and −1 diffraction orders directly from active DOE such as SOM, GLV imaging devices. In the second case, any passive DOE in the intermediate image plane would provide an unequal intensity distribution between diffraction orders for different beams (red, green, blue). Our numerical simulation has shown that given a rectangular shaped DOE with equal groove and land width (see Fig. 4) and a groove depth having equal intensities for 0 and ±1 diffraction orders for a green laser, the minimal difference in intensity distribution for all three colors is achieved with a shallow grating relief (h ~ (n−1)*λ*/2). Figure 5 shows the dependence of diffraction order intensity on wavelength for rectangular shape DOE with the optimal depth and land/groove ratio equal to 1. In the numerical simulation, we will use red, green and blue lasers with wavelengths of 640, 532 and 440 nm, respectively. We have diffraction order intensity distributions I_{R±1}1= 0.5I_{R0} for red and I_{B±1}= 2.9 I_{B0} for blue.

We will compare these two different optical schemes based on the assumption that they have the same total thickness of retarders s_{2} (since the maximum thickness is correlated with the focal length of the projective lens). For two partially correlated beams, the formula for speckle CR calculation is well known and can be written as follows [18]:

However, for three partially coherent beams, there is a not simple formula for speckle CR calculation. Therefore we will use a more complicated algorithm based on the coherence matrix worked out by Goodman (see [17]) to get formulae for speckle suppression for the three beams optical scheme. This algorithm uses a linear transformation of three beam fields to obtain a representation of the field amplitude on the screen from the interferences of three uncorrelated beams with appropriate intensities. For the case of uncorrelated beams, the speckle CR could be calculated using Eq. (2). The intensities of these uncorrelated beams are the eigenvalues of the three beam coherence matrix. In our case, we could write the coherence matrix as:

To find the matrix eigenvalues, we should solve the equation:

After a simple mathematical transformation, Eq. (9) can be written as follows:

$$+{I}_{0}{I}_{1}^{2}{\mid {\mu}_{1,-1}\mid}^{2}+{I}_{0}{I}_{1}^{2}{\mid {\mu}_{0,1}\mid}^{2}+{I}_{0}{I}_{1}^{2}{\mid {\mu}_{0,-1}\mid}^{2}-{I}_{1}{I}_{0}{I}_{1}{\mu}_{0,-1}{\mu}_{0,1}*\mu {*}_{1,1}-{I}_{0}{I}_{1}{I}_{1}{\mu}_{1,-1}{\mu}_{\mathrm{0,1}}{\mu}_{\phantom{\rule[-0ex]{.3em}{0ex}}0,-1}^{*}\text{}=0.$$

We will assume that the second retarder has twice the thickness of the first one (see Fig. 3(b)) to obtain the same decorrelation coefficient between diffraction orders -1 and 0 and between diffraction orders -1 and 1, *μ*
_{0,1} =*μ*
_{1,-1} and from Eq. (5) follows for this case that *μ*
_{0,-1} = (*μ*
_{0,1})^{4} . Equation (10) is solved numerically, and we use the simple formula

to calculate speckle CR, where *λ*
_{1}, *λ*
_{2}, and *λ*
_{3} are the roots of Eq. (10).

Figure 6 shows speckle suppression for the green beam versus total retarder plate thickness *s _{2}* for the different green laser bandwidths. From the dependence of the correlation coefficient on the decorrelation length and bandwidth, one can see that the change in bandwidth or wavelength shift should result in a rescaling of the dependence of the speckle suppression curve on the decorrelation length. Increases in bandwidth automatically squeeze the same amount in the speckle CR curve dependence on the decorrelation length. One can see from Fig. 6 that the two diffraction order regime provides larger speckle suppression for a relatively small decorrelation length that does not provide large decorrelation (

*μ*

_{0,1}=

*μ*

_{1,-1}> 0.621;

*μ*

_{0,-1}> 0.15). The improvement in speckle suppression for the two beams is obtained because the small thickness s

_{2}results in a small decorrelation (large correlation coefficient) between 0 and 1 and between 1 and -1 and a relatively small correlation coefficient between 0 and -1 for the three beam optical scheme. However, for the case of only two diffraction orders we have a relatively small correlation coefficient between the two beams, with the same value as that between diffraction orders 0 and -1 in the three beam optical scheme. Therefore, it should provide a speckle suppression coefficient close to 1/sqrt(2). Hence if we could not obtain a significant decorrelation length between all three beams in the three beam optical regime, the optical scheme with two diffraction orders would provide smaller speckle CR.

Figure 7 shows dependence of correlation coefficient and speckle CR on the retarder plate thickness. There are small differences in correlation coefficient curves from the green laser case since a red laser larger decorrelation length for the same bandwidth causes the dependence on decorrelation length to be stretched along the plate thickness axis. In spite of a large difference in diffraction order intensity (I_{1}=0.5*I_{0}), the red laser speckle CR has a very similar dependence on the transparent plate thickness s_{2}, as one can see in Fig. 7. However even for the case of full decorrelation length, the optical system for red light has a smaller speckle suppression than that for green light, since the former does not have an equal intensity in the 0 and ±1 diffraction orders.

The blue laser has quite a different intensity distribution between 0 and ±1 diffraction orders compared to the red and green beams. However, the dependence of speckle CR on decorrelation length is almost the same (after rescaling to account for the change in decorrelation length) as in the green and red beam cases, as clearly shown in Fig. 8. It should be noted also that the final speckle suppression for totally decorrelated beams for blue and red lasers is smaller than that for green lasers because of the unequal intensity of the 0 and ±1 diffraction orders.

## 3. Experimental data and discussion

We apply the method described above for speckle suppression in the SOM pico-projection system. The SOM pico-projector creates a VGA image (640×480 pixels) of area 200×150 mm. The SOM-based mobile projector had a pixel width of 10 um (one pixel one ribbon scheme). We use the SOM with small pixel width because we need a small projector thickness. To obtain a large decorrelation length between beams of different diffraction orders, the projector should have long separation between diffraction orders and hence a large focal length. However, the mobile projector should have as small a projector volume as possible. The total volume of the SOM-based projector used for speckle evaluation is 13 cc (the projection system of the projector is shown on Fig. 2). To obtain a small size, we used a projection lens with a focal length F=12.5 mm. Usually, the pixel size on the screen should be equal to or smaller than the resolution of the eye. To achieve the angle decorrelation effect for different diffraction orders, the DOE period should be equal to or smaller than one pixel length. We use at most * s_{2}*=2.5 mm thick glass (for beam decorrelation) to avoid image distortion caused by diffraction orders overlapping on the glass-air interface and on the side surfaces of the glass plates (see Fig. 9). To achieve the maximum optical path difference between different diffraction orders, we use glass plates with a large refractive index, n=1.88 (exactly the same

*n*as was used for speckle CR calculation presented on Fig. 6–8). We use one red laser, one green laser and one high power blue laser in one projector.

It is well known that when combining several different speckle suppression methods, the results are multiplicative. Speckle decrease in a real projector has several independent speckle suppression phenomena which all provide some decrease in speckle CR and which could not be rejected in experiments. Therefore, to find the speckle suppression caused only by the speckle averaging due to using three partially correlated beams, we used the speckle suppression coefficient which is the ratio of speckle CR after and before the application of the three partially correlated beam method. In our case, this method is independent from all other speckle suppression methods using the SOM (large NA with scanning and polarization rotation), and therefore we expect that the speckle suppression coefficient should be the same as the speckle CR calculated above.

The measurement of the speckle CR decrease for the green laser using three partially correlated beams (see Fig. 3(b)) shows a different speckle suppression level for different temperatures and for different laser samples. The smallest bandwidth is ~0.05 nm and the widest bandwidth is ~1.4 nm, as shown in Fig. 10. For a bandwidth of 0.05 nm, the speckle suppression ratio (1–CR/CR_{0}) is 15%, and for a bandwidth of 1.4 nm, the speckle suppression ratio is 34%. From comparison of experimental and theoretical data, one could see a definite correlation between measured and calculated speckle suppression data.

For the red laser, the speckle CR decreases no more than 20%, corresponding to a red laser bandwidth of <0.08 nm. Our experimental data and technical documentation has shown that the red laser has a bandwidth of less than 0.1 nm, well matched to the speckle CR we expect from the theory worked out above. To increase the red laser bandwidth, we apply several MHz high frequency modulation (similarly as in [19, 20]). After applying high frequency modulation, we observe a 36% decrease in speckle CR when using three partially correlated beams where the bandwidth is now 1.0 nm. This speckle CR decrease corresponds to 0.14 nm spectrum bandwidth according to the model based on the Gaussian approximation of the laser spectrum shape developed above. It also is close to theoretical limit for speckle CR for 3 uncorrelated red beams (see Fig. 3) since *δλ*=0.14 nm provides almost total decorrelation of three laser beams. Theory for *δλ*=1.0 nm gives 38% decrease in speckle CR (see Tab. 1).

The blue light laser has quite a different intensity distribution between the 0 and ±1 diffraction orders from the red and green beams. It should be noted that the final speckle suppression for totally decorrelated beams for blue and red lasers is a little smaller than that for green lasers because of unequal intensity in the 0 and ±1 diffraction orders. The blue laser that we used in SOM has a sufficiently large bandwidth (0.8 nm) that there is no problem for three beam decorrelation and for speckle suppression at the theoretical limit.

Table 1 summarizes the theoretically calculated and experimentally measured speckle suppression for each RGB laser bandwidth. From comparison of experimental and theoretical data, one can see definite correlations between measured and calculated speckle suppression. The data in Table 1 also show that the theoretically calculated CR gives a slightly larger speckle suppression effect than that obtained in experiment. There could be several reason for the absence of complete correspondence between theory and experiment: 1) the spectrum does not have a Gaussian distribution; 2) some small correlation remains between the different methods of speckle suppression; 3) some errors in speckle measurement. Our experimental data are not sufficient to pinpoint the cause of the difference between theory and experiment. In spite of these small differences, the strong correlation between theory and experimental results shows the validity of the theory and the efficiency of the proposed method of speckle suppression.

In the SOM pico-projector, the speckle CR is measured when all methods of speckle suppression have been applied. First, we utilize the scanning effect with barker code DOE [8] to increase the numerical aperture in the scanning direction. Second, a polarization rotator plate is electrically driven for polarization diversity. Third, the newly developed three-beam decorrelation method reported in this paper is applied. The total speckle CR is ~5% in white color as shown in Fig. 11.

## 4. Conclusion

We developed a mathematical algorithm for speckle CR calculation for an optical projector using several partial coherence beams for speckle suppression. It is shown that the optical scheme with two beams could provide better speckle CR reduction than one with three beams when retarders do not provide sufficient decorrelation for all beams. However, if the optical path difference is sufficiently large to obtain decorrelation for all the separated beams, the optical scheme with many beams is better. The pico-projector using 0 and ±1 diffraction orders for image projection with a simple rectangular shape DOE provided speckle suppression close to the theoretical limit (1/sqrt(3)) for all three colors beams (red, blue and green). Using all the speckle suppression methods developed for SOM Laser display, the speckle CR level has now decreased to ~5%. The development of the SOM pico-projector is now more closely approaching a speckle-less laser display.

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