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Optical schemes for speckle suppression by Barker code diffractive optical elements A. Lapchuk,1,* A. Kryuchyn,1 V. Petrov,1 O. V. Shyhovets,1 G. A. Pashkevich,2 O. V. Bogdan,2 A. Kononov,3 and A. Klymenko4 1
Institute for Information Recording of NAS of Ukraine, Shpak Str. 2, Kiev 03113, Ukraine 2 Kyiv Polytechnic Institute, Research Institute for Applied Electronics, Kiev, Ukraine 3 Specialized Enterprise Holography Ltd., Lenin Str. 64, Kiev 02088, Ukraine 4 National Aviation University, Cosmonaut Komarov Pr., Kiev 03058, Ukraine *Corresponding author: [email protected] Received March 29, 2013; revised July 18, 2013; accepted July 18, 2013; posted July 18, 2013 (Doc. ID 188003); published August 8, 2013
A method for speckle suppression based on Barker code and M-sequence code diffractive optical elements (DOEs) is analyzed. An analytical formula for the dependence of speckle contrast on the wavelength of the laser illumination is derived. It is shown that speckle contrast has a wide maximum around the optimal wavelength that makes it possible to obtain large speckle suppression by using only one DOE for red, green, and blue laser illumination. Optical schemes for implementing this method are analyzed. It is shown that the method can use a simple liquid-crystal panel for phase rotation instead of a moving DOE; however, this approach requires a high frequency of liquid-crystal switching. A simple optical scheme is proposed using a 1D Barker code DOE and a simple 1D liquid-crystal panel, which does not require a high frequency of liquid-crystal switching or high-accuracy DOE movement. © 2013 Optical Society of America OCIS codes: (110.6150) Speckle imaging; (110.1650) Coherence imaging. http://dx.doi.org/10.1364/JOSAA.30.001760
1. INTRODUCTION Optical devices using laser illumination have small size, large optical efficiency, and high color saturation. Therefore, laser illumination is the optimal solution for mobile devices such as laser projectors [1–7]. However, speckle noise [6,7] severely deteriorates the image quality obtained under coherent illumination. The speckle problem places severe restrictions on the application of coherent sources to illumination [7–9]. The speckle contrast C, which determines the depth of light intensity modulation by speckle, is the most important parameter of speckle noise: C σ I ∕hIi;
(1)
where σ I and hIi are the standard deviation and mean value of the light intensity on the screen. Speckle suppression methods are based on speckle pattern averaging. Speckle averaging can be achieved by the wavelength, angle, or polarization diversity of a laser beam [8]. The speckle suppression mechanism based on a moving random diffuser or diffractive optical element (DOE) is one of the most effective methods of speckle suppression [9–16]. In our previous publication, the theory of a speckle suppression method based on Barker code DOEs was developed, and its optical parameters were analyzed [17, 18]. It was shown that this method can decrease speckle contrast to below the sensitivity of the human eye, has small optical losses, and is simple to use. It was also shown that the method can use two 1D Barker code DOEs [see Fig. 1(a)] moving in orthogonal directions [17] or one 2D Barker code DOE [see 1084-7529/13/091760-08$15.00/0
Fig. 1(b)] moving with a specified linear velocity and direction [18]. An approximate model and 3D model of the method based on Fresnel approximation was developed. The approximate model was used in [17] to obtain analytical formulae for the speckle contrast of the method. In the approximate model, the formula for speckle contrast is reduced to a product of two factors, and each factor is the speckle contrast from the method with one moving 1D Barker code DOE. Hence, this approach reduces one 3D model to two 2D models; for this reason, we refer to it as the 2D model of the method. The Barker code length has at most 13 elements, and therefore, by using only this type of DOE, speckle contrast can be reduced to less than 4%. However, it should be noted that M sequences have the same autocorrelation function properties as periodic Barker codes [19,20], and therefore, all results obtained for Barker code DOEs are valid for M-sequence DOEs. Since the M sequences can have 15, 31, or more elements in one period, this method can decrease speckle contrast to less than 3%. There are also other phase code modulation methods with narrow autocorrelation functions and code lengths larger than 13: minimum peak sidelobe (MPS) codes; polyphase Barker codes; nested or compound Barker codes; Frank codes; P1, P2, and Px codes; and Chu codes [19,20]. However, they have sidelobe shapes and levels that differ from those of Barker codes, and an additional investigation is needed to clarify the possibility of applying these phase codes to 2D DOE structures for speckle suppression. It should be noted that some results presented in [17,18] were given without proof. Equation (15) in [17], representing the speckle contrast obtained in the 3D approach, is not convenient for calculation. A detailed analysis of the optical © 2013 Optical Society of America
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Fig. 2. Principal optical scheme of the laser projector.
Hx
Fig. 1. 1D Barker code DOE structure. (a) 1D periodic Barker code DOE (N 7). (b) 2D Barker code DOE based on two 1D Barker code DOE periodic structures (with N 7). The upper and left side of the figure shows a 1D Barker code DOE that the 2D DOE is based on. In this figure, white denotes grooves, and gray denotes lands.
expjφ1 1; expjφ2 expjkhn − 1;
(2)
where h is the DOE relief height, n is the refractive index of the DOE medium, k is the wavenumber of the incident light, φ1 0, φ2 kn − 1h and j is an imaginary unit. We analyze the case of a discrete autocorrelation function A0 x; φ2 [see Fig. 3(b)], which is the autocorrelation function of light passing through a Barker code DOE when it makes step-like movements with steps equal to a one-element width T 0 . It is easy to generalize the obtained results to a continuous autocorrelation function Ax; φ2 [see Fig. 3(a)], when the DOE is moving with constant velocity. The periodic autocorrelation function of the complex light amplitude A0 x; φ2 incident on a screen (located in the image plane) for the case of an ideal optical system is given by the following expression: A0 n − iT; φ2
schemes for the technical implementation of this method was not performed. Herein, we analyze the validity of some approximations used in [17,18] for the case of an optical system with a small numerical aperture. The dependence of the efficiency of speckle suppression of this method on the optical wavelength is derived later. We also transform an expression for the speckle contrast obtained in the 3D approximation in [16] to make it convenient for numerical calculation. Finally, several optical schemes for technical implementation of the method are proposed.
land ; groove
N X i1
N X
E i Ein
N X
EiTE i nT
i1
E 0 HiTE0 H i nT;
(3)
i1
2. SOME THEORETICAL ASPECTS OF THE 2D MODEL OF THE METHOD
where x1 , x2 , are coordinates of the screen along the DOEs’ movement, i floorx1 ∕T; i n floorx2 ∕T, N is the length of one period of the Barker code sequence, E i is the complex field amplitude at the screen, E 0 is the amplitude of the incident field, and HiT is the transmission coefficient of the DOE with step T and period T 0 NT. The sizes of the images of the DOE elements on the screen (width T and period T 0 ) relate to the actual sizes of the DOE elements (width T 0 and period T 00 ) through the magnification coefficient κ of the objective: T κ T 0 and T 0 κT 00 . The autocorrelation
First, we analyze the speckle suppression efficiency of the method by using the 2D model. In this approximation, it is sufficient to analyze the optical system with a 1D Barker code diffractive element moving with constant velocity and shifting by one structure period during the resolution time of the human eye. The 1D Barker code DOE is a two-level structure of a periodic sequence of lands and grooves. Figure 1 shows a 1D Barker code DOE with element width T 0 and period T 00 7T 0 (for more details, refer to [15–18]). The Barker code DOE structure modulates the phase of the transmitted light with a periodic Barker code sequence [15–18]. It is assumed that the DOE is located at the object plane, a screen is located at the image plane, and we have an ideal optical system, as shown in Fig. 2. The transmission coefficient of a Barker code DOE with a depth of relief that does not exactly match the wavelength of laser radiation can be written as follows:
Fig. 3. Autocorrelation functions of the complex amplitude of light transmitted through a periodic 1D Barker code DOE: (a) continuous autocorrelation function Ax and (b) discrete autocorrelation function A0 x.
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functions A0 and A for the case of exactly a half-wave phase shift were analyzed in [17,18] (see Fig. 3), and it was proven that the autocorrelation function A0 is a periodic sequence of rectangular-shaped peaks of height A0 max max A0 x; π N with low flat plateaus between them of height A0 min min A0 min x; π (where the value of A0 min can be 1, −1, or 0 depending on the Barker code length). Equation (3) can be rewritten as follows: A0 n T; φ1 –φ2
N X
Ei Ein
i1
N X
C 2xy
R∞ R∞
Sinc2 2π u Sinc2 2π u u2 du1 du2 R ∞D 1 2 2π D 1 As 0 −∞ Sinc D x dx
−∞ −∞ jAs u2 j
4
R∞
2
2 −∞ As DuQudu ; A2s 0
(10)
C xy is the speckle contrast of the method with one 1D Barker code DOE, As is equal to either A0 or A depending on the DOE moving mode (either discrete or continuous),
expjφi − φin
i1
N X cosφi − φin j sinφi − φin i1
N X
cosφi − φin :
(4)
i1
Substituting Eq. (2) into Eq. (4) gives A0 nT; kn − 1h
N X
fBi Bin 1 − sin2 k − k0 n − 1h∕2
i1
sin2 k − k0 n − 1h∕2g A0 nT; π1 − sin2 k − k0 n − 1h∕2 N sin2 k − k0 n − 1h∕2;
(5)
where Bi is a periodic Barker code sequence. Finally, after completing the summation, we can rewrite Eq. (5) as πλ πλ A0 nT; kn − 1h A0 nT; π 1 − cos2 0 N cos2 0 : 2λ 2λ (6) It is assumed that the phase shift φ2 is exactly equal to π for a laser wavelength of λ0 . From Eq. (6), it follows that the autocorrelation function has the same peak heights, Amax max A0 x; kn − 1h πλ πλ A0 max π 1 − cos2 0 A0 max πcos2 0 2λ 2λ A0 max π N;
(7)
and shape, but a higher plateau level between the peaks Amin min A0 kn − 1h A0
min
A0 max − A0
2 min cos
πλ0 : 2λ
(8)
It should be emphasized that there is a misprint in formula (29) in [17], which expresses the dependence of A on wavelength [it was written as sin2 πλ0 ∕2λ instead of cos2 πλ0 ∕2λ]. In the 2D approximation [17], the formula for speckle contrast can be written as follows: C Cx Cy ; where
(9)
Fig. 4. Dependence of the speckle contrast C xy on laser wavelength for the case of fixed DOE relief height: (a) N 4, (b) N 11, and (c) N 13. The solid lines represent the calculation by the approximate Eq. (12), the dashed lines represent the rigorous integration of Eq. (10) for D 5NT 0 , the dash–dot–dotted lines are plots of Eq. (10) for D 2T 0 , and the dashed–dotted lines are plots of Eq. (10) for D T 0 .
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Qx 1 − Sinc4πx∕8π 2 x2 , and D0 D∕2 is the lateral eye resolution on the screen. It is difficult to obtain an analytical formula for the dependence of the speckle contrast on the wavelength in the general case. Therefore, it is assumed here that D ≫ NT T 0 (the small Barker code period approximation). For the small Barker code period, Eq. (10) can be simplified to
C 2xy 4 ≈4
Z
∞
−∞ ∞ X
A2s DuQudu∕A2s 0 Z QiNT∕D
i−∞
NT∕D 0
A2s Dudu∕A2s 0
Z Z NT∕D D ∞ ≈4 Qudu A2s Dudu∕A2s 0 NT −∞ 0 Z NT A2s vdv∕NTA2s 0:
(11)
0
After substitution of Eq. (6) into Eq. (11), the formula for the speckle contrast factors C x and C y is simplified to
C 2xy kn − 1h
A2min N − 2 2
At0 x0 ; x0s C
k2 2π2
Δt 0
level and is significantly wider for the case of A0 min −1, and it remains almost constant over the entire optical spectrum. A wider spectral range of speckle suppression for the case of A0 min −1 is caused by the two minima of the first main term of the numerator when A0 min < 0. The results obtained from the short-period approximation [Eq. (12)] can be seen to almost coincide with the results obtained from Eq. (10) even for a relatively large DOE period of T 0 D∕2 D0 . Hence, this approximation is correct for a range of DOE periods where a large speckle suppression effect exists [17]. However, Eq. (12) gives only qualitative matching with Eq. (10) for a large DOE period of T 0 ≥ 2D0 . From the data in Fig. 4, it is clear that the minimum of the speckle contrast curve has a steeper short-wavelength slope. Therefore, the DOE relief height should be chosen to have optimal half-wave phase rotation situated closer to the violet spectral region. In this case, the speckle contrast variation does not exceed several tens of percent over the entire optical spectrum, and therefore, a single Barker code DOE can be used for red, green, and blue light. The homogeneity of the autocorrelation function of the complex amplitude of light transmitted through the DOE
R1
Amin Amax − Amin x2 dx A20 max N 2 A0 min A0 min 2 2 πλ0 2 πλ0 min 0 1 − 1 − AA00 max 3 − 2 1 − N sin2 πλ 1 − sin sin 2λ A0 max 2λ A0 max 2λ 3 0
N
Figure 4 shows the dependence of C xy on the wavelength calculated by the approximate Eq. (12) and by performing the numerical integration in Eq. (10) for Barker code DOEs of different code lengths. The three samples of the Barker code DOE were intentionally chosen in such a way as to have a different plateau level A0 min between the peaks: A0 min 0 for N 4, A0 min −1 for N 11 (also for Barker codes with N 3 and 7 and for any M-sequence code), and A min 1 for N 13 (valid also for N 5). For all three cases, the speckle contrast has a wide minimum around the optimal wavelength. However, the width of the minimum depends on the plateau
Z
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:
(12)
was used [that is, A0 x; x0 ≈ A0 x − x0 ] in [17] to reduce the 2D autocorrelation function of a 2D Barker code grating to a product of two independent 1D autocorrelation functions of two 1D Barker code gratings. Hence, the 2D approximation is valid when the autocorrelation function of the complex amplitude of the light on the screen is a homogeneous function. However, is this approximation also valid for an objective lens with a finite numerical aperture, which truncates part of the diffracted light? The autocorrelation function on the screen At0 (thin lens model and Fresnel approximation) for this case can be written as follows:
ik x2 x02 SinckNAx lx0 ∕sdx ExHx − vt exp −∞ l s 2 dt R∞ 2 02 x − vt exp − ik xs xs 0 ∕sdx SinckNAx E x H lx s s s s s −∞ l s 2 R∞
02 k2 I ik x x02 x exp − 2 s s 2π2 2 Z∞Z∞ ik x − x2s x0 x0 × A0 x − xs exp Sinc kNA xs l s Sinc kNA x l dxdxs ; 2 l s s −∞ −∞
C
(13)
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where Ex is the electric field at the object plane (see Fig. 2), C is constant, H is the complex amplitude distribution of the optical beam on the back side of the Barker code DOE, Δt is the resolution time of the human eye, k is the wavenumber, x and xs are the coordinates in the object plane, x0 and x0s are the coordinates in the image plane, and l and s are, respectively, the distances from the object and the image plane to the objective. After a simple transformation, Eq. (12) can be rewritten as follows:
where ys1 y1 ∕MN; ys2 y2 ∕MN Substituting Eq. (17) into Eq. (16), we obtained
Z
jAt0 x0 ; x0s j C 1
where xs x2 − x1 u x1 − ys1 ; v x1 − ys2
∞ −∞
Z
∞
−∞
A0 uSinckNAv
lx0 lx0 × Sinc kNA u v − s dvdu
; s s
(14)
Z
C 1
x0 − x0s A0 uSinc kNA u l du
s −∞
Fxs ; u; v
(15)
and, hence, it is a homogeneous function. Therefore, even for an optical system with a small numerical aperture that truncates part of the diffracted light, the separation of the 2D autocorrelation function into a product of two independent 1D autocorrelations is accurate.
3. SOME THEORETICAL ASPECTS OF THE 3D MODEL A formula for the speckle contrast of the method (in the Fresnel approximation) is derived in [17]:
Sinc2
(19)
Z Sdu
∞
At0 x0 − x0s ;
Sinc2
2π 2π x1 Sinc2 MNx1 − u dx1 D D
D 1 MN − MN − 1cos4πun − 2Sinc4πun ; 16π 2 u2n (20)
un u∕D. For MN ≫ 1, further simplification of Eqs. (18)– (20) is possible by using the fact that the last two sinc functions in Eq. (19) have narrow maximum peaks at an argument of zero with a very fast decrease of the function when the argument moves away from the origin. The first two sinc functions in Eq. (16) have a significantly weaker dependence on x1 (by a factor of MN), and therefore, with a small error, they can be moved outside the integral. In this approximation, Eq. (19) can be rewritten as follows:
vRR u RR jA2d x1 ; x2 ; y1 ; y2 j2 Sinc2 2π x1 Sinc2 2π y1 Sinc2 2π x2 Sinc2 2π y2 dx1 dx2 dy2 dy2 u D D D D RR 2 ; Ct 2 2π Sinc2 2π D x1 Sinc D y1 Ax1 ; x1 ; y1 ; y1 dx1 dy1
where x1 , y1 , x2 , and y2 are screen coordinates, and A2d is the autocorrelation function of the light incident on the screen. Equation (16) is not convenient for calculation because it does not use relative screen coordinates. It can be modified to a more convenient expression as follows. The autocorrelation function of the complex amplitude of the light that passed through the Barker code DOE for the case of an ideal optical system can be simplified to A2d x1 ; x2 − x1 ; y1 ; y2 − y1 ZT N∕T HNMx − x1 − ys2 0
× H NMx − x1 − ys2 A0 x − x1 − x2 dx A2d x1 − x2 ; x1 − ys1 ; x1 − ys2 ;
(17)
(18)
2π 2π x1 Sinc2 x1 − xs D D 2π 2π MNx1 − u Sinc2 MNx1 − v dx1 ; × Sinc2 D D Z
where v xs lx0s ∕s and u x − xs , and C 1 is constant (C 1 ≠ C). It was assumed in Eq. (14) that the object size satisfies the condition h∕2 < DL l∕s (see Fig. 2), and that assumption allows the omission of the exponent term in Eq. (13). After a simple transformation, Eq. (14) can be rewritten as jAt0 x0 ; x0s j
vRRR u u jA2d xs ; u; vj2 Fxs ; u; vdxs dudv 2 R t ; C SduA2d 0; u; udu
(16)
2π v u 2π v u Sinc2 − xs D 2 D 2 0 1 1 − Sinc MN4πv−u C D D B B C: (21) × A 2πMNv−u 2 2MN @
Fxs ; u; v ≈ Sinc2
D
One of the most important parameters in this method is the sensitivity of the speckle suppression effect to the accuracy of the DOE movement, which specifies a constraint on the accuracy of the DOE mechanical movement. In the case when the DOE moves accurately along the slow axis and with some error along the fast axis (the DOE shift during the time resolution of the human eye is not exactly equal to MNT), we can rewrite the formula for the
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autocorrelation function of the complex amplitude of the light as A2d x1 ;x2 ;y1 ;y2 Z 1−1∕MN T HxNM y1 −NMx1 H xNM y2 −NMx1 T 0 A0 x2 −x1 modx∕Tdx Zε 1N HxNM y1 −NMx1 H xNM y2 −NMx1 A0 MT 0 ×x2 −x1 modx∕Tdx:
(22)
From Eq. (22), it is clear (since jεj < T∕2) that for large MM ≥ N, the effect of speckle suppression has weak sensitivity to small errors in DOE movement.
4. OPTICAL SCHEMES The analysis has shown that the speckle suppression effect of the method depends on the location of the DOE and the screen in the optical scheme. To understand that fact, it is sufficient to analyze the case when a screen is located at the Fourier plane of the objective plane. From the discussion above and from the results obtained in [17,18], it is clear that the DOE located at the object or at intermediate image planes provides a good speckle suppression effect. Hence, optimal speckle suppression is achieved when the DOE is located at a conjugate to the screen plane, and therefore there are three possible planes for the DOE location: in an illumination system using an additional lens to create the DOE image in the object plane of the projector [Fig. 5(a)], in the optical modulator plane [Fig. 5(b)], and in the intermediate image plane [Fig. 5(c)]. In addition, in all cases, the optical system should not truncate the light propagating within the first maximum of the diffraction envelope function of the Barker code DOE in order to have close to the maximum speckle suppression effect. Two different 2D DOE structures can be used for speckle suppression, as shown in Figs. 6 and 7: (1) a rectangular 2D Barker code DOE structure (Fig. 6) and (2) a logarithmic spiral 3D Barker code DOE structure (Fig. 7). The squares in Figs. 6 and 7 represent one period of the 2D Barker code DOE structure. The logarithmic spiral DOE should have a constant angle φ between the spiral and the radius that satisfies the condition tanφ MN. Using a spiral DOE has advantages in terms of the accuracy and simplicity of DOE movement. However, this type of DOE requires more volume and therefore is less appropriate for mobile applications. There are other optical schemes for this method of speckle suppression. For example, it is possible to use one 1D Barker code DOE and one liquid-crystal 1D Barker code DOE structure. The liquid-crystal 1D Barker code DOE is a liquid-crystal panel with transparent electrodes, placed close to each other, stretched along one direction (see Fig. 8). The modulation of the complex amplitude of the light by a periodic Barker code sequence is obtained here by applying a voltage pulse sequence, with an appropriate amplitude (to obtain a phase shift equal to half the wavelength), to the electrodes, as shown in Fig. 8. The phase shift of the light transmitted through the liquid-crystal panel should not change its polarization. The most convenient place for the liquid-crystal DOE is the optical modulator plane. A typical 1D Barker code structure should fulfill the requirement of fast motion, and the liquid-crystal
Fig. 5. Optical schemes for the method of speckle suppression by using Barker code DOEs.
structure should fulfill the requirement of slow switching with a rate F f · N, where f is the frame frequency and N is the Barker code length. For f 50 Hz and N 13, we have F 650 Hz, which is within the current liquid-crystal switching frame frequency. A feedback loop can be used to synchronize the liquid-crystal switching with the DOE movement. Therefore, high-accuracy DOE movement is not needed in this case. A microelectromechanical spatial light modulator such as the grating light valve [1] can be used in the method instead of the 1D Barker code DOE in this optical scheme. The phase distribution and its switching in correspondence with the moving Barker code DOE in this case can be obtained by changing the height of moving mirrors. However, the length of the mirror strips required for display and laser projectors should be at least of several millimeters, and therefore it is difficult to achieve a required switching rate for such long strips. One 2D liquid-crystal DOE structure can also be used for speckle suppression. The liquid-crystal panel should have upper transparent electrodes of rectangular shape with width T and bottom transparent electrodes that are stretched in orthogonal directions. The modulation of the complex amplitude by a 2D Barker code sequence is obtained by applying to
Fig. 6. Rectangular 2D Barker code DOE structure.
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Fig. 7. Logarithmic spiral Barker code DOE structure: tan φ MN.
Fig. 10. Photo of 2D Barker code DOE structure and its diffraction pattern: (a) DOE photo, (b) diffraction pattern, and (c) intensity distribution along a horizontal central line of the diffraction pattern for N 13, T 0 6 μm, and T 00 78 μm. Fig. 8. DOE.
1D liquid-crystal panel for the method using a 1D Barker code
the optimal speckle suppression effect of the method. However, in this case, the frequency of liquid-crystal panel switching should be at least F f · N 2 and hence well above 1 kHz. By using electron beam lithography and photolithography, we produced a Barker code DOE of length N 13. The DOE has an element width of T 6 μm and a period T 0 of 78 μm. The depth of relief is chosen to give a half-wave phase shift for a laser beam wavelength of 532 nm. A photo of the Barker code DOE is shown in Fig. 10(a). Figure 10(b) shows the diffraction pattern created by a green laser beam of wavelength λ 532 nm diffracted by the DOE. The intensity distribution in the diffraction pattern corresponds to the DOE envelope function, indicating that the DOE was accurately fabricated. The mechanism for moving the DOE is under construction. Experimental data on the speckle contrast decrease obtained with this method will be presented in our next publication.
5. CONCLUSION
Fig. 9. 2D liquid-crystal panel for the Barker code DOE speckle suppression method without mechanical vibration.
the bottom and top electrodes a voltage that replicates the shape of the periodic Barker code sequence (see Fig. 9). The amplitude of the applied voltage should provide a halfwavelength shift of the phase of the light and should not rotate the polarization of the light. It is evident that this structure will realize a 2D DOE Barker code structure. By switching the voltage of every electrode at an appropriate frequency (f x ∕f y N), we can achieve proper phase movement to obtain
An analytical formula for the dependence of the speckle suppression method on wavelength is derived. It is shown that a single Barker code DOE can be used for red, green, and blue laser illumination. It was shown that the speckle suppression efficiency is almost constant over the entire optical spectrum for the case of using a DOE based on a Barker code length of N 3, 7, or 11 or any M-sequence code length where these cases result in the autocorrelation function sidelobes having negative values. Our analysis has shown that there are several different effective optical schemes for applying our method. It was also proven that the method can use a simple liquidcrystal panel without vibration to obtain large speckle suppression. A simple optical scheme to implement this method was also proposed using a 1D Barker code DOE and a simple 1D liquid-crystal panel, which does not require fast liquidcrystal switching or high-accuracy DOE movement.
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J. I. Trisnadi, C. B. Carlisle, and V. Monteverde, “Overview and applications of grating light valve based optical write engines for high-speed digital imaging,” Proc. SPIE 5348, 52–64 (2004). M. W. Kowarz, J. C. Brazas, and J. G. Phalen, “Conformal grating electromechanical system (GEMS) for high-speed digital light modulation,” in Technical Digest of IEEE 15th International MEMS Conference (IEEE, 2002), pp. 568–573. R. Sprague, M. Champion, M. Brown, D. Brown, M. Freeman, and M. Niesten, “Mobile projectors using scanned beam displays,” in Mobile Displays, Technology and Applications, K. Bhowmik, Z. Li, and P. J. Bos, eds. (Wiley, 2008). W. O. Davis, R. Sprague, and J. Miller, “MEMS-based picoprojector display,” in Proceedings of 2008 IEEE/LEOS International Conference on Optical MEMs and Nanophotonics (IEEE, 2008), pp. 31–32. 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). S. K. Yun, J. H. Song, S. D. An, I. J. Yeo, Y. J. Cho, 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, Ch. 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. Ph. Jung, S. J. Kim, K. U. Lee, J. H. Park, J. H. Yang, G. Y. Byun, S. H. Byun, Y. J. Cho, Ch. G. Kim, J. H. Kim, S. K. Kim, S. I. Lee, W. H. Lee, K. Y. Oh, S. K. Oh, W. Ch. Shin, B. K. Song, and E. M. Bourim, “A novel diffractive micro-optical modulator for mobile display applications,” Proc. SPIE 6887, 688702 (2008).
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7. K. V. Chellappan, E. Erden, and H. Urey, “Laser-based displays: a review,” Appl. Opt. 49, F79–F98 (2010). 8. J. C. Dainty, Laser Speckle and Related Phenomena (SpringerVerlag, 1975). 9. J. W. Goodman, Speckle Phenomena in Optics. Theory and Applications (Roberts, 2006). 10. P. Janssens and K. Malfait, “Future prospects of high-end laser projectors,” Proc. SPIE 7232, 72320Y (2009). 11. L. Wang, T. Tschudi, T. Halldorsson, and P. R. Pétursson, “Speckle reduction in laser projection systems by diffractive optical elements,” Appl. Opt. 37, 1770–1775 (1998). 12. S. Kubota and J. W. Goodman, “Very efficient speckle contrast reduction realized by moving diffuser device,” Appl. Opt. 49, 4385–4391 (2010). 13. J. I. Trisnadi, “Hadamard speckle contrast reduction,” Opt. Lett. 29, 11–13 (2004). 14. W. Gao, Z. Tong, V. Kartashov, M. N. Akram, and X. Chen, “Replacing two-dimensional binary phase matrix by a pair of one-dimensional dynamic phase matrices for laser speckle reduction,” J. Disp. Technol. 8, 291–295 (2012). 15. 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). 16. V. Yurlov, A. Lapchuk, S.-K. Yun, J.-H. Song, I. Yeo, H. Yang, and S.-D. An, “Speckle suppression in scanning laser displays: aberration and defocusing of the projection system,” Appl. Opt. 48, 80–90 (2009). 17. A. Lapchuk, A. Kryuchyn, V. Petrov, V. Yurlov, and V. Klymenko, “Full speckle suppression in laser projectors using two Barker code-type optical diffractive elements,” J. Opt. Soc. Am. A 30, 22–31 (2013). 18. A. Lapchuk, A. Kryuchyn, V. Petrov, and V. Klymenko, “Optimal speckle suppression in laser projectors using a single twodimensional Barker code diffractive optical element,” J. Opt. Soc. Am. A 30, 227–232 (2013). 19. N. Levanon and E. Mozeson, Radar Signals (Wiley, 2004). 20. M. N. Akram, V. Kartashov, and Z. Tong, “Speckle reduction in line-scan laser projectors using binary phase codes,” Opt. Lett. 35, 444–445 (2010).
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Optical schemes for speckle suppression by Barker code diffractive optical elements A. Lapchuk,1,* A. Kryuchyn,1 V. Petrov,1 O. V. Shyhovets,1 G. A. Pashkevich,2 O. V. Bogdan,2 A. Kononov,3 and A. Klymenko4 1
Institute for Information Recording of NAS of Ukraine, Shpak Str. 2, Kiev 03113, Ukraine 2 Kyiv Polytechnic Institute, Research Institute for Applied Electronics, Kiev, Ukraine 3 Specialized Enterprise Holography Ltd., Lenin Str. 64, Kiev 02088, Ukraine 4 National Aviation University, Cosmonaut Komarov Pr., Kiev 03058, Ukraine *Corresponding author: [email protected] Received March 29, 2013; revised July 18, 2013; accepted July 18, 2013; posted July 18, 2013 (Doc. ID 188003); published August 8, 2013
A method for speckle suppression based on Barker code and M-sequence code diffractive optical elements (DOEs) is analyzed. An analytical formula for the dependence of speckle contrast on the wavelength of the laser illumination is derived. It is shown that speckle contrast has a wide maximum around the optimal wavelength that makes it possible to obtain large speckle suppression by using only one DOE for red, green, and blue laser illumination. Optical schemes for implementing this method are analyzed. It is shown that the method can use a simple liquid-crystal panel for phase rotation instead of a moving DOE; however, this approach requires a high frequency of liquid-crystal switching. A simple optical scheme is proposed using a 1D Barker code DOE and a simple 1D liquid-crystal panel, which does not require a high frequency of liquid-crystal switching or high-accuracy DOE movement. © 2013 Optical Society of America OCIS codes: (110.6150) Speckle imaging; (110.1650) Coherence imaging. http://dx.doi.org/10.1364/JOSAA.30.001760
1. INTRODUCTION Optical devices using laser illumination have small size, large optical efficiency, and high color saturation. Therefore, laser illumination is the optimal solution for mobile devices such as laser projectors [1–7]. However, speckle noise [6,7] severely deteriorates the image quality obtained under coherent illumination. The speckle problem places severe restrictions on the application of coherent sources to illumination [7–9]. The speckle contrast C, which determines the depth of light intensity modulation by speckle, is the most important parameter of speckle noise: C σ I ∕hIi;
(1)
where σ I and hIi are the standard deviation and mean value of the light intensity on the screen. Speckle suppression methods are based on speckle pattern averaging. Speckle averaging can be achieved by the wavelength, angle, or polarization diversity of a laser beam [8]. The speckle suppression mechanism based on a moving random diffuser or diffractive optical element (DOE) is one of the most effective methods of speckle suppression [9–16]. In our previous publication, the theory of a speckle suppression method based on Barker code DOEs was developed, and its optical parameters were analyzed [17, 18]. It was shown that this method can decrease speckle contrast to below the sensitivity of the human eye, has small optical losses, and is simple to use. It was also shown that the method can use two 1D Barker code DOEs [see Fig. 1(a)] moving in orthogonal directions [17] or one 2D Barker code DOE [see 1084-7529/13/091760-08$15.00/0
Fig. 1(b)] moving with a specified linear velocity and direction [18]. An approximate model and 3D model of the method based on Fresnel approximation was developed. The approximate model was used in [17] to obtain analytical formulae for the speckle contrast of the method. In the approximate model, the formula for speckle contrast is reduced to a product of two factors, and each factor is the speckle contrast from the method with one moving 1D Barker code DOE. Hence, this approach reduces one 3D model to two 2D models; for this reason, we refer to it as the 2D model of the method. The Barker code length has at most 13 elements, and therefore, by using only this type of DOE, speckle contrast can be reduced to less than 4%. However, it should be noted that M sequences have the same autocorrelation function properties as periodic Barker codes [19,20], and therefore, all results obtained for Barker code DOEs are valid for M-sequence DOEs. Since the M sequences can have 15, 31, or more elements in one period, this method can decrease speckle contrast to less than 3%. There are also other phase code modulation methods with narrow autocorrelation functions and code lengths larger than 13: minimum peak sidelobe (MPS) codes; polyphase Barker codes; nested or compound Barker codes; Frank codes; P1, P2, and Px codes; and Chu codes [19,20]. However, they have sidelobe shapes and levels that differ from those of Barker codes, and an additional investigation is needed to clarify the possibility of applying these phase codes to 2D DOE structures for speckle suppression. It should be noted that some results presented in [17,18] were given without proof. Equation (15) in [17], representing the speckle contrast obtained in the 3D approach, is not convenient for calculation. A detailed analysis of the optical © 2013 Optical Society of America
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Fig. 2. Principal optical scheme of the laser projector.
Hx
Fig. 1. 1D Barker code DOE structure. (a) 1D periodic Barker code DOE (N 7). (b) 2D Barker code DOE based on two 1D Barker code DOE periodic structures (with N 7). The upper and left side of the figure shows a 1D Barker code DOE that the 2D DOE is based on. In this figure, white denotes grooves, and gray denotes lands.
expjφ1 1; expjφ2 expjkhn − 1;
(2)
where h is the DOE relief height, n is the refractive index of the DOE medium, k is the wavenumber of the incident light, φ1 0, φ2 kn − 1h and j is an imaginary unit. We analyze the case of a discrete autocorrelation function A0 x; φ2 [see Fig. 3(b)], which is the autocorrelation function of light passing through a Barker code DOE when it makes step-like movements with steps equal to a one-element width T 0 . It is easy to generalize the obtained results to a continuous autocorrelation function Ax; φ2 [see Fig. 3(a)], when the DOE is moving with constant velocity. The periodic autocorrelation function of the complex light amplitude A0 x; φ2 incident on a screen (located in the image plane) for the case of an ideal optical system is given by the following expression: A0 n − iT; φ2
schemes for the technical implementation of this method was not performed. Herein, we analyze the validity of some approximations used in [17,18] for the case of an optical system with a small numerical aperture. The dependence of the efficiency of speckle suppression of this method on the optical wavelength is derived later. We also transform an expression for the speckle contrast obtained in the 3D approximation in [16] to make it convenient for numerical calculation. Finally, several optical schemes for technical implementation of the method are proposed.
land ; groove
N X i1
N X
E i Ein
N X
EiTE i nT
i1
E 0 HiTE0 H i nT;
(3)
i1
2. SOME THEORETICAL ASPECTS OF THE 2D MODEL OF THE METHOD
where x1 , x2 , are coordinates of the screen along the DOEs’ movement, i floorx1 ∕T; i n floorx2 ∕T, N is the length of one period of the Barker code sequence, E i is the complex field amplitude at the screen, E 0 is the amplitude of the incident field, and HiT is the transmission coefficient of the DOE with step T and period T 0 NT. The sizes of the images of the DOE elements on the screen (width T and period T 0 ) relate to the actual sizes of the DOE elements (width T 0 and period T 00 ) through the magnification coefficient κ of the objective: T κ T 0 and T 0 κT 00 . The autocorrelation
First, we analyze the speckle suppression efficiency of the method by using the 2D model. In this approximation, it is sufficient to analyze the optical system with a 1D Barker code diffractive element moving with constant velocity and shifting by one structure period during the resolution time of the human eye. The 1D Barker code DOE is a two-level structure of a periodic sequence of lands and grooves. Figure 1 shows a 1D Barker code DOE with element width T 0 and period T 00 7T 0 (for more details, refer to [15–18]). The Barker code DOE structure modulates the phase of the transmitted light with a periodic Barker code sequence [15–18]. It is assumed that the DOE is located at the object plane, a screen is located at the image plane, and we have an ideal optical system, as shown in Fig. 2. The transmission coefficient of a Barker code DOE with a depth of relief that does not exactly match the wavelength of laser radiation can be written as follows:
Fig. 3. Autocorrelation functions of the complex amplitude of light transmitted through a periodic 1D Barker code DOE: (a) continuous autocorrelation function Ax and (b) discrete autocorrelation function A0 x.
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functions A0 and A for the case of exactly a half-wave phase shift were analyzed in [17,18] (see Fig. 3), and it was proven that the autocorrelation function A0 is a periodic sequence of rectangular-shaped peaks of height A0 max max A0 x; π N with low flat plateaus between them of height A0 min min A0 min x; π (where the value of A0 min can be 1, −1, or 0 depending on the Barker code length). Equation (3) can be rewritten as follows: A0 n T; φ1 –φ2
N X
Ei Ein
i1
N X
C 2xy
R∞ R∞
Sinc2 2π u Sinc2 2π u u2 du1 du2 R ∞D 1 2 2π D 1 As 0 −∞ Sinc D x dx
−∞ −∞ jAs u2 j
4
R∞
2
2 −∞ As DuQudu ; A2s 0
(10)
C xy is the speckle contrast of the method with one 1D Barker code DOE, As is equal to either A0 or A depending on the DOE moving mode (either discrete or continuous),
expjφi − φin
i1
N X cosφi − φin j sinφi − φin i1
N X
cosφi − φin :
(4)
i1
Substituting Eq. (2) into Eq. (4) gives A0 nT; kn − 1h
N X
fBi Bin 1 − sin2 k − k0 n − 1h∕2
i1
sin2 k − k0 n − 1h∕2g A0 nT; π1 − sin2 k − k0 n − 1h∕2 N sin2 k − k0 n − 1h∕2;
(5)
where Bi is a periodic Barker code sequence. Finally, after completing the summation, we can rewrite Eq. (5) as πλ πλ A0 nT; kn − 1h A0 nT; π 1 − cos2 0 N cos2 0 : 2λ 2λ (6) It is assumed that the phase shift φ2 is exactly equal to π for a laser wavelength of λ0 . From Eq. (6), it follows that the autocorrelation function has the same peak heights, Amax max A0 x; kn − 1h πλ πλ A0 max π 1 − cos2 0 A0 max πcos2 0 2λ 2λ A0 max π N;
(7)
and shape, but a higher plateau level between the peaks Amin min A0 kn − 1h A0
min
A0 max − A0
2 min cos
πλ0 : 2λ
(8)
It should be emphasized that there is a misprint in formula (29) in [17], which expresses the dependence of A on wavelength [it was written as sin2 πλ0 ∕2λ instead of cos2 πλ0 ∕2λ]. In the 2D approximation [17], the formula for speckle contrast can be written as follows: C Cx Cy ; where
(9)
Fig. 4. Dependence of the speckle contrast C xy on laser wavelength for the case of fixed DOE relief height: (a) N 4, (b) N 11, and (c) N 13. The solid lines represent the calculation by the approximate Eq. (12), the dashed lines represent the rigorous integration of Eq. (10) for D 5NT 0 , the dash–dot–dotted lines are plots of Eq. (10) for D 2T 0 , and the dashed–dotted lines are plots of Eq. (10) for D T 0 .
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Qx 1 − Sinc4πx∕8π 2 x2 , and D0 D∕2 is the lateral eye resolution on the screen. It is difficult to obtain an analytical formula for the dependence of the speckle contrast on the wavelength in the general case. Therefore, it is assumed here that D ≫ NT T 0 (the small Barker code period approximation). For the small Barker code period, Eq. (10) can be simplified to
C 2xy 4 ≈4
Z
∞
−∞ ∞ X
A2s DuQudu∕A2s 0 Z QiNT∕D
i−∞
NT∕D 0
A2s Dudu∕A2s 0
Z Z NT∕D D ∞ ≈4 Qudu A2s Dudu∕A2s 0 NT −∞ 0 Z NT A2s vdv∕NTA2s 0:
(11)
0
After substitution of Eq. (6) into Eq. (11), the formula for the speckle contrast factors C x and C y is simplified to
C 2xy kn − 1h
A2min N − 2 2
At0 x0 ; x0s C
k2 2π2
Δt 0
level and is significantly wider for the case of A0 min −1, and it remains almost constant over the entire optical spectrum. A wider spectral range of speckle suppression for the case of A0 min −1 is caused by the two minima of the first main term of the numerator when A0 min < 0. The results obtained from the short-period approximation [Eq. (12)] can be seen to almost coincide with the results obtained from Eq. (10) even for a relatively large DOE period of T 0 D∕2 D0 . Hence, this approximation is correct for a range of DOE periods where a large speckle suppression effect exists [17]. However, Eq. (12) gives only qualitative matching with Eq. (10) for a large DOE period of T 0 ≥ 2D0 . From the data in Fig. 4, it is clear that the minimum of the speckle contrast curve has a steeper short-wavelength slope. Therefore, the DOE relief height should be chosen to have optimal half-wave phase rotation situated closer to the violet spectral region. In this case, the speckle contrast variation does not exceed several tens of percent over the entire optical spectrum, and therefore, a single Barker code DOE can be used for red, green, and blue light. The homogeneity of the autocorrelation function of the complex amplitude of light transmitted through the DOE
R1
Amin Amax − Amin x2 dx A20 max N 2 A0 min A0 min 2 2 πλ0 2 πλ0 min 0 1 − 1 − AA00 max 3 − 2 1 − N sin2 πλ 1 − sin sin 2λ A0 max 2λ A0 max 2λ 3 0
N
Figure 4 shows the dependence of C xy on the wavelength calculated by the approximate Eq. (12) and by performing the numerical integration in Eq. (10) for Barker code DOEs of different code lengths. The three samples of the Barker code DOE were intentionally chosen in such a way as to have a different plateau level A0 min between the peaks: A0 min 0 for N 4, A0 min −1 for N 11 (also for Barker codes with N 3 and 7 and for any M-sequence code), and A min 1 for N 13 (valid also for N 5). For all three cases, the speckle contrast has a wide minimum around the optimal wavelength. However, the width of the minimum depends on the plateau
Z
1763
:
(12)
was used [that is, A0 x; x0 ≈ A0 x − x0 ] in [17] to reduce the 2D autocorrelation function of a 2D Barker code grating to a product of two independent 1D autocorrelation functions of two 1D Barker code gratings. Hence, the 2D approximation is valid when the autocorrelation function of the complex amplitude of the light on the screen is a homogeneous function. However, is this approximation also valid for an objective lens with a finite numerical aperture, which truncates part of the diffracted light? The autocorrelation function on the screen At0 (thin lens model and Fresnel approximation) for this case can be written as follows:
ik x2 x02 SinckNAx lx0 ∕sdx ExHx − vt exp −∞ l s 2 dt R∞ 2 02 x − vt exp − ik xs xs 0 ∕sdx SinckNAx E x H lx s s s s s −∞ l s 2 R∞
02 k2 I ik x x02 x exp − 2 s s 2π2 2 Z∞Z∞ ik x − x2s x0 x0 × A0 x − xs exp Sinc kNA xs l s Sinc kNA x l dxdxs ; 2 l s s −∞ −∞
C
(13)
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where Ex is the electric field at the object plane (see Fig. 2), C is constant, H is the complex amplitude distribution of the optical beam on the back side of the Barker code DOE, Δt is the resolution time of the human eye, k is the wavenumber, x and xs are the coordinates in the object plane, x0 and x0s are the coordinates in the image plane, and l and s are, respectively, the distances from the object and the image plane to the objective. After a simple transformation, Eq. (12) can be rewritten as follows:
where ys1 y1 ∕MN; ys2 y2 ∕MN Substituting Eq. (17) into Eq. (16), we obtained
Z
jAt0 x0 ; x0s j C 1
where xs x2 − x1 u x1 − ys1 ; v x1 − ys2
∞ −∞
Z
∞
−∞
A0 uSinckNAv
lx0 lx0 × Sinc kNA u v − s dvdu
; s s
(14)
Z
C 1
x0 − x0s A0 uSinc kNA u l du
s −∞
Fxs ; u; v
(15)
and, hence, it is a homogeneous function. Therefore, even for an optical system with a small numerical aperture that truncates part of the diffracted light, the separation of the 2D autocorrelation function into a product of two independent 1D autocorrelations is accurate.
3. SOME THEORETICAL ASPECTS OF THE 3D MODEL A formula for the speckle contrast of the method (in the Fresnel approximation) is derived in [17]:
Sinc2
(19)
Z Sdu
∞
At0 x0 − x0s ;
Sinc2
2π 2π x1 Sinc2 MNx1 − u dx1 D D
D 1 MN − MN − 1cos4πun − 2Sinc4πun ; 16π 2 u2n (20)
un u∕D. For MN ≫ 1, further simplification of Eqs. (18)– (20) is possible by using the fact that the last two sinc functions in Eq. (19) have narrow maximum peaks at an argument of zero with a very fast decrease of the function when the argument moves away from the origin. The first two sinc functions in Eq. (16) have a significantly weaker dependence on x1 (by a factor of MN), and therefore, with a small error, they can be moved outside the integral. In this approximation, Eq. (19) can be rewritten as follows:
vRR u RR jA2d x1 ; x2 ; y1 ; y2 j2 Sinc2 2π x1 Sinc2 2π y1 Sinc2 2π x2 Sinc2 2π y2 dx1 dx2 dy2 dy2 u D D D D RR 2 ; Ct 2 2π Sinc2 2π D x1 Sinc D y1 Ax1 ; x1 ; y1 ; y1 dx1 dy1
where x1 , y1 , x2 , and y2 are screen coordinates, and A2d is the autocorrelation function of the light incident on the screen. Equation (16) is not convenient for calculation because it does not use relative screen coordinates. It can be modified to a more convenient expression as follows. The autocorrelation function of the complex amplitude of the light that passed through the Barker code DOE for the case of an ideal optical system can be simplified to A2d x1 ; x2 − x1 ; y1 ; y2 − y1 ZT N∕T HNMx − x1 − ys2 0
× H NMx − x1 − ys2 A0 x − x1 − x2 dx A2d x1 − x2 ; x1 − ys1 ; x1 − ys2 ;
(17)
(18)
2π 2π x1 Sinc2 x1 − xs D D 2π 2π MNx1 − u Sinc2 MNx1 − v dx1 ; × Sinc2 D D Z
where v xs lx0s ∕s and u x − xs , and C 1 is constant (C 1 ≠ C). It was assumed in Eq. (14) that the object size satisfies the condition h∕2 < DL l∕s (see Fig. 2), and that assumption allows the omission of the exponent term in Eq. (13). After a simple transformation, Eq. (14) can be rewritten as jAt0 x0 ; x0s j
vRRR u u jA2d xs ; u; vj2 Fxs ; u; vdxs dudv 2 R t ; C SduA2d 0; u; udu
(16)
2π v u 2π v u Sinc2 − xs D 2 D 2 0 1 1 − Sinc MN4πv−u C D D B B C: (21) × A 2πMNv−u 2 2MN @
Fxs ; u; v ≈ Sinc2
D
One of the most important parameters in this method is the sensitivity of the speckle suppression effect to the accuracy of the DOE movement, which specifies a constraint on the accuracy of the DOE mechanical movement. In the case when the DOE moves accurately along the slow axis and with some error along the fast axis (the DOE shift during the time resolution of the human eye is not exactly equal to MNT), we can rewrite the formula for the
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autocorrelation function of the complex amplitude of the light as A2d x1 ;x2 ;y1 ;y2 Z 1−1∕MN T HxNM y1 −NMx1 H xNM y2 −NMx1 T 0 A0 x2 −x1 modx∕Tdx Zε 1N HxNM y1 −NMx1 H xNM y2 −NMx1 A0 MT 0 ×x2 −x1 modx∕Tdx:
(22)
From Eq. (22), it is clear (since jεj < T∕2) that for large MM ≥ N, the effect of speckle suppression has weak sensitivity to small errors in DOE movement.
4. OPTICAL SCHEMES The analysis has shown that the speckle suppression effect of the method depends on the location of the DOE and the screen in the optical scheme. To understand that fact, it is sufficient to analyze the case when a screen is located at the Fourier plane of the objective plane. From the discussion above and from the results obtained in [17,18], it is clear that the DOE located at the object or at intermediate image planes provides a good speckle suppression effect. Hence, optimal speckle suppression is achieved when the DOE is located at a conjugate to the screen plane, and therefore there are three possible planes for the DOE location: in an illumination system using an additional lens to create the DOE image in the object plane of the projector [Fig. 5(a)], in the optical modulator plane [Fig. 5(b)], and in the intermediate image plane [Fig. 5(c)]. In addition, in all cases, the optical system should not truncate the light propagating within the first maximum of the diffraction envelope function of the Barker code DOE in order to have close to the maximum speckle suppression effect. Two different 2D DOE structures can be used for speckle suppression, as shown in Figs. 6 and 7: (1) a rectangular 2D Barker code DOE structure (Fig. 6) and (2) a logarithmic spiral 3D Barker code DOE structure (Fig. 7). The squares in Figs. 6 and 7 represent one period of the 2D Barker code DOE structure. The logarithmic spiral DOE should have a constant angle φ between the spiral and the radius that satisfies the condition tanφ MN. Using a spiral DOE has advantages in terms of the accuracy and simplicity of DOE movement. However, this type of DOE requires more volume and therefore is less appropriate for mobile applications. There are other optical schemes for this method of speckle suppression. For example, it is possible to use one 1D Barker code DOE and one liquid-crystal 1D Barker code DOE structure. The liquid-crystal 1D Barker code DOE is a liquid-crystal panel with transparent electrodes, placed close to each other, stretched along one direction (see Fig. 8). The modulation of the complex amplitude of the light by a periodic Barker code sequence is obtained here by applying a voltage pulse sequence, with an appropriate amplitude (to obtain a phase shift equal to half the wavelength), to the electrodes, as shown in Fig. 8. The phase shift of the light transmitted through the liquid-crystal panel should not change its polarization. The most convenient place for the liquid-crystal DOE is the optical modulator plane. A typical 1D Barker code structure should fulfill the requirement of fast motion, and the liquid-crystal
Fig. 5. Optical schemes for the method of speckle suppression by using Barker code DOEs.
structure should fulfill the requirement of slow switching with a rate F f · N, where f is the frame frequency and N is the Barker code length. For f 50 Hz and N 13, we have F 650 Hz, which is within the current liquid-crystal switching frame frequency. A feedback loop can be used to synchronize the liquid-crystal switching with the DOE movement. Therefore, high-accuracy DOE movement is not needed in this case. A microelectromechanical spatial light modulator such as the grating light valve [1] can be used in the method instead of the 1D Barker code DOE in this optical scheme. The phase distribution and its switching in correspondence with the moving Barker code DOE in this case can be obtained by changing the height of moving mirrors. However, the length of the mirror strips required for display and laser projectors should be at least of several millimeters, and therefore it is difficult to achieve a required switching rate for such long strips. One 2D liquid-crystal DOE structure can also be used for speckle suppression. The liquid-crystal panel should have upper transparent electrodes of rectangular shape with width T and bottom transparent electrodes that are stretched in orthogonal directions. The modulation of the complex amplitude by a 2D Barker code sequence is obtained by applying to
Fig. 6. Rectangular 2D Barker code DOE structure.
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Fig. 7. Logarithmic spiral Barker code DOE structure: tan φ MN.
Fig. 10. Photo of 2D Barker code DOE structure and its diffraction pattern: (a) DOE photo, (b) diffraction pattern, and (c) intensity distribution along a horizontal central line of the diffraction pattern for N 13, T 0 6 μm, and T 00 78 μm. Fig. 8. DOE.
1D liquid-crystal panel for the method using a 1D Barker code
the optimal speckle suppression effect of the method. However, in this case, the frequency of liquid-crystal panel switching should be at least F f · N 2 and hence well above 1 kHz. By using electron beam lithography and photolithography, we produced a Barker code DOE of length N 13. The DOE has an element width of T 6 μm and a period T 0 of 78 μm. The depth of relief is chosen to give a half-wave phase shift for a laser beam wavelength of 532 nm. A photo of the Barker code DOE is shown in Fig. 10(a). Figure 10(b) shows the diffraction pattern created by a green laser beam of wavelength λ 532 nm diffracted by the DOE. The intensity distribution in the diffraction pattern corresponds to the DOE envelope function, indicating that the DOE was accurately fabricated. The mechanism for moving the DOE is under construction. Experimental data on the speckle contrast decrease obtained with this method will be presented in our next publication.
5. CONCLUSION
Fig. 9. 2D liquid-crystal panel for the Barker code DOE speckle suppression method without mechanical vibration.
the bottom and top electrodes a voltage that replicates the shape of the periodic Barker code sequence (see Fig. 9). The amplitude of the applied voltage should provide a halfwavelength shift of the phase of the light and should not rotate the polarization of the light. It is evident that this structure will realize a 2D DOE Barker code structure. By switching the voltage of every electrode at an appropriate frequency (f x ∕f y N), we can achieve proper phase movement to obtain
An analytical formula for the dependence of the speckle suppression method on wavelength is derived. It is shown that a single Barker code DOE can be used for red, green, and blue laser illumination. It was shown that the speckle suppression efficiency is almost constant over the entire optical spectrum for the case of using a DOE based on a Barker code length of N 3, 7, or 11 or any M-sequence code length where these cases result in the autocorrelation function sidelobes having negative values. Our analysis has shown that there are several different effective optical schemes for applying our method. It was also proven that the method can use a simple liquidcrystal panel without vibration to obtain large speckle suppression. A simple optical scheme to implement this method was also proposed using a 1D Barker code DOE and a simple 1D liquid-crystal panel, which does not require fast liquidcrystal switching or high-accuracy DOE movement.
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