Diffractive optical elements for shaping Gaussian ...

Diffractive optical elements for shaping Gaussian Schell-model beams Hwi Kim, Taesu Kim, Kyongsik Choi, Seunghoon Han, Il-min Lee, and Byoungho Lee* School of Electrical Engineering, Seoul National University Kwanak-Gu Shinlim-Dong, Seoul 151-744, Korea ABSTRACT A novel design method of diffractive optical element for shaping Gaussian Schell model beams is investigated. In the design, it is important to solve a deconvolution problem for obtaining diffraction image with edge-sharpness degraded by the convolution effect of the partially coherent Gaussian Schell model beam. In this paper, a simple heuristic approach to the deconvolution problem is addressed. It is shown that an extra rim pattern around the target diffraction image can lighten the unwanted edge-blur of the diffraction image. The algorithm for generating the extra rim pattern is explained. Keywords: Diffractive optical element, Gaussian Schell model beam, partial coherence

1. INTRODUCTION Recently, an efficient design method of diffractive optical element (DOE) for partially coherent Gaussian Schell model (GSM) beams has been substantiated based on the mathematical analysis of the diffraction of GSM beams through DOE.1 The diffraction image of a GSM beam through a DOE can be considered as the convolution image of the inherent degree of coherent function2 of the GSM beam and the diffraction image obtained by illuminating the DOE by a complete coherent Gaussian beam having the same intensity profile as the GSM beam. Practically the DOE design for GSM beams is equivalent to the DOE design for completely coherent Gaussian beams with the target image that is the deconvolved image of the really wanted diffraction image by the kernel of the degree of coherent function of the GSM beam. Thus the conventional DOE design method for completely coherent beams as wellknown iterative Fourier transform algorithm (IFTA) can be used in the DOE design for GSM beams. In the DOE design for GSM beams, a main task is determining a proper target image for the conventional IFTA that is for completely coherent Gaussian beam, to form the wanted diffraction image through the convolution by the kernel of the degree of coherent function. In addition, a necessary constraint exits in determining the target image, which is that the deconvolved target image must be non-negative. Commonly used deconvolution algorithm based on the truncated singular value decomposition (TSVD) does not guarantee the non-negativity condition of the deconvolved image, so auxiliary algorithm to satisfy the non-negativity condition is needed. Thus the problem becomes a constrained optimization problem for the deconvolution with non-negative variables. Particularly in many applications, it is important to obtain diffraction image maintaining edge-sharpness that may be significantly smoothed by the convolution effect of partial coherence of the GSM beam. In this paper, maintaining edgesharpness of the diffraction image is mainly considered. In this paper, instead of directly solving the constrained deconvolution optimization problem, an effective heuristic approach for conserving the edge-sharpness is devised. It is shown that the employment of target image with extra rim pattern for the IFTA is effective to improve edge-sharpness of the obtained diffraction image. This paper is organized as follows. In Sec. 2, diffraction of GSM beams through DOE is addressed. In Sec. 3, the deconvolution problem is defined and the construction of the target image with extra rim pattern is proposed. In Sec. 4, numerical results are presented and discussed. In Sec. 5, final remarks are given.

2. DIFFRACTION OF GAUSSIAN SCHELL MODEL BEAM THROUGH DOE In this section, diffraction of GSM beams through a DOE is addressed. Based on the following described theory, the design strategy of DOEs for GSM beams is derived. Let the incident GSM beam and the optical wave transmitted

* Corresponding author Tel: +82-2-880-7245 Fax: +82-2-873-9953 E-mail: [email protected]

Holography, Diffractive Optics, and Applications II, edited by Yunlong Sheng, Dahsiung Hsu, Chongxiu Yu, Byoungho Lee, Proceedings of SPIE Vol. 5636 (SPIE, Bellingham, WA, 2005) · 0277-786X/05/$15 · doi: 10.1117/12.568683

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through the DOE denoted by U i ( u , v, t ) and U t ( u , v , t ) , respectively. Then the cross-spectral density of the

transmitted optical wave Wt ( u1 , v1 ; u2 , v2 ) is defined as2-5

Wt ( u1 , v1 ; u2 , v2 ) = U t ( u1 , v1 , t )U t ( u2 , v2 , t )

*

.

(1)

If the transmittance3 of the DOE is denoted by DOE ( u , v ) , then Wt ( u1 , v1 ; u2 , v2 ) is represented more explicitly as

Wt ( u1 , v1 ; u2 , v2 ) = DOE ( u1 , v1 ) DOE * ( u2 , v2 ) U i ( u1 , v1 , t ) U i ( u2 , v2 , t )

*

(2)

= DOE ( u1 , v1 ) DOE ( u2 , v2 ) Wi ( u1 , v1; u2 , v2 ) . *

It is assumed that the optical wave propagates through the Fourier transforming optical system (the optical Fourier transformer).3 The kernel of the optical Fourier transformer takes the form

K ( x, y , u , v ) =

  e j 4π f / λ 2π exp  − j ( xu + yv )  , λf jλ f  

(3)

where ( x, y ) and ( u , v ) are the spatial coordinates of the output plane and the input plane of the optical Fourier

λ and f are operating wavelength and focal length of the optical Fourier transformer, respectively. Then the intensity distribution I ( x, y ) in the output plane holds the relation 2, 4 transformer, respectively, and

I ( x, y ) = ∫ ∫ ∫ ∫ K ( x, y; u1 , v1 ) K * ( x, y; u2 , v2 ) DOE ( u1 , v1 ) DOE * ( u2 , v2 ) Wi ( u1 , v1 , u2 , v2 ) du1dv1du2 dv2 . (4) Substituting the kernel (3) into the integral (4), we can obtain the following expression of the intensity distribution

I ( x, y ) =

1

(λ f )

2

∫ ∫ ∫ ∫ DOE



( u1 , v1 ) DOE* ( u2 , v2 )Wi ( u1 , v1 , u2 , v2 ) exp  j 

 2π ( x ( u2 − u1 ) + y ( v2 − v1 ) )  du1dv1du2 dv2 , λf 

(5) which is the Fresenl diffraction integral for GSM beams. It is assumed that the incident optical wave is an isotropic GSM beam, so the cross-spectral density of the incident optical wave Wi ( u1 , v1 ; u2 , v2 ) is given by2, 5

Wi ( u1 , v1 , u2 , v2 ) = where

σ s and σ g

  u 2 + v2   u 2 + v2  I 0 exp  − 1 2 1  exp  − 2 2 2  exp  −  σs  σs    

( u1 − u2 ) + ( v1 − v2 ) 2

σ

2 g

2

   

(6)

are transverse spot width and transverse coherence width, respectively. Substituting Eq. (6) into Eq.

(5) leads to the explicit expression of the intensity distribution as

I ( x, y) =

I0

(λ f )

2

 ∫∫∫∫ DOE 



2 2  ∆u2 +∆v2   2π  u +v   u2 +v12  DOE* ( u2,v2 ) exp− 2 2 2 exp− exp− j ( x( u1 −u2 ) + y( v1 −v2 )) dudvdu 1 1 2dv2. 2  2   f σs  σ σ λ  s  g    

( u1,v1) exp− 1 

(7) 432

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To obtain a concise expression, we further manipulate Eq. (7) by means of consecutively changing variables. With changing variables as

∆u = u1 − u2 , ∆v = v1 − v2 , u =

1 1 ( u1 + u2 ) , v = ( v1 + v2 ) , 2 2

(8)

Eq. (7) is manipulated as

I ( x, y ) = =

1

(λ f )

2

1

(λ f )

 ∫ ∫ ∫ ∫T u 

  ∫ ∫ ∫ ∫ T  u  



1



(λ f )

=∫∫

2

2

+

+

 ∆u 2 + ∆v 2    ∆u ∆v  *  ∆u ∆v  2π ,v + ,v − ( x∆u + y∆v )  dudvd ∆ud ∆v  exp  − j T  u −  exp  − 2  2 2   2 2  σ λ f   g  

 ∆u 2 + ∆v 2    2π ∆u ∆v  *  ∆u ∆v   ,v + ,v − ( x∆u + y∆v )  d ∆ud ∆v  exp  − j T  u −  dudv  exp  − 2  2 2   2 2  σ λ f    g   





 

* ∫ ∫ T ( u , v ) T ( u − ∆u , v − ∆v ) dudv  exp  −

∆u 2 + ∆v 2 

σ

  exp  − j   

2 g

 2π ( x∆u + y∆v )  d ∆ud ∆v λf 

(9)

where T ( u , v ) takes the form 

T ( u , v ) = I 0 DOE ( u , v ) exp  − 

By scaling variables as

x′ =

variables ( x′, y ′ ) as

u 2 + v2 

σ s2

. 

(10)

2π 2π x , y′ = y , we can see that the intensity distribution is represented by the λf λf

λ

f λf  x′, y′  2 2π  π 

I% ( x′, y ′) = I  =

 ∫∫ 

1

(λz)

2

∫ ∫T

( u , v ) T * ( u − ∆u, v − ∆v )

(11)   ∆u 2 + ∆v 2 dudv  exp  −  σ g2  

  exp  

( − j ( x′∆u + y′∆v )) d ∆ud ∆v.

Equation. (11) can be viewed as two-dimensional Fourier transform of the multiplication of two functions A ( ∆u , ∆v )

and B ( ∆u , ∆v ) as

I% ( x′, y ′) = ∫ ∫ A ( ∆u , ∆v ) B ( ∆u , ∆v ) exp ( − j ( x′∆u + y′∆v ) ) d ∆ud ∆v

(12)

where A ( ∆u , ∆v ) and B ( ∆u , ∆v ) indicate, respectively,

A ( ∆u, ∆v ) =

1

(λ z )

2

∫∫

T ( u , v ) T * ( u − ∆u, v − ∆v ) dudv

(13)

and

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

B ( ∆u, ∆v ) = exp  −

∆u 2 + ∆v 2 

.  

σ g2

 

(14)

With the aid of the convolution theorem7, I% ( x′, y′ ) can be the convolution form as

I% ( x′, y′ ) = ∫ ∫ A ( x′ − x′′, y′ − y′′ ) B ( x′′, y′′ ) dx′′dy′′ = A ( x′, y′ ) ⊗ B ( x′, y′ )

(15)

where A ( x′, y ′ ) and B ( x′, y ′ ) indicate, respectively,

A ( x′, y′ ) = ∫ ∫ A ( ∆u , ∆v ) exp ( − j ( x′∆u + y′∆v ) ) d ∆ud ∆v ,

(16)

and

B ( x′, y′ ) = ∫ ∫ B ( ∆u , ∆v ) exp ( − j ( x′∆u + y′∆v ) ) d ∆ud ∆v .

(17)

A ( x′, y′ ) of Eq. (16) can be further manipulated to give I%coh ( x′, y′ ) as 

1

 

(λ z )

A ( x′, y′ ) = ∫ ∫  = = = =

( u , v ) ∫∫ T * (u − ∆u, v − ∆v ) exp ( − j ( x′∆u + y′∆v ) ) d ∆ud ∆vdudv

2

∫∫ T

( u , v ) exp ( − j ( x′u + y′v ) ) ∫∫ T * ( ∆u′, ∆v′ ) exp ( j ( x′∆u′ + y ′∆v′ ) ) d ∆u′d ∆v′dudv

2

∫∫ T

( u , v ) exp ( − j ( x′u + y′v ) ) dudv  ∫∫ T ( ∆u′, ∆v′ ) exp ( − j ( x′∆u′ + y ′∆v′) ) d ∆u′d ∆v′ 

∫∫ T

(u , v ) exp ( − j ( x′u + y ′v ) ) dudv

1

(λ z )

 

∫∫ T

1

(λ z )

* ′ ′ ∫ ∫ T ( u , v ) T ( u − ∆u , v − ∆v ) dudv  exp ( − j ( x ∆u + y ∆v ) ) d ∆ud ∆v

2

1

(λ z )



2

(18) *

1

2

(λ z ) = I%coh ( x′, y′ ) . 2

B ( x′, y′ ) of Eq. (17) also gives I%raw ( x′, y′ ) as 

B ( x′, y′ ) = ∫ ∫ exp  −

∆u 2 + ∆v 2 

σ g2

 

 σg = πσ g2 exp  −  4 

2

( x′

2

+ y ′2 )

 exp  

  = I%raw 

( − j ( x′∆u + y′∆v )) d ∆ud ∆v

( x′, y′ ) .

Then I% ( x′, y′ ) of Eq. (15) takes the form

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(19)

I% ( x′, y′ ) = πσ g2 ∫ ∫ I%coh ( x′′, y′′ ) I%coh ( x′ − x′′, y′ − y′′ ) dx′′dy′′ . Finally, by scaling variables as

x′ =

(20)

2π 2π x and y′ = y , the formula of the diffraction of GSM beam through DOE λf λf

is obtained as

I ( x, y ) = ∫ ∫ I coh ( x′′, y′′ ) I raw ( x − x′′, x − y′′ ) dx′′dy′′ =

4π 3σ g2

(λ f )

2



′′ ′′ ∫ ∫ I coh ( x , y ) exp  −  

π 2σ g2 λ2 f 2

(( x − x′′) + ( y − y′′) 2

2

)

(21)

 dx′′dy′′.   

We can see that I coh ( x, y ) is the diffraction image obtained by illuminating the DOE by a complete coherent Gaussian beam. The diffraction image of a GSM beam through a DOE is the convolution image of the inherent degree of coherent

function I raw ( x, y ) of the GSM beam and the diffraction image I coh ( x, y ) obtained by illuminating the DOE by a complete coherent Gaussian beam.

3. A HEURISTIC APPROACH FOR THE DECONVOLUTION PROBLEM As seen in Eq. (21), the diffraction image I is solely determined by the coherent target image I coh . Practically the DOE design for GSM beams is equivalent to the DOE design shaping I coh with completely coherent Gaussian beam having the same intensity profile as the GSM beam. The DOE to generate I coh can be easily produced by the conventional IFTA scheme. The design procedure consists of two stages. The first is extracting the target image I coh in Eq. (21) to generate the wanted diffraction image I with edge as sharp as possible. In general, an exact target image I coh may not exist for the wanted diffraction image. The second is the conventional IFTA process to obtain the phase profile of DOE forming the coherent target image I coh . In this section, the deconvolution problem of the first stage is addressed. Mathematically the deconvolution of I coh in Eq. (21) is the Fredholm integral equation of the first kind. As well known, the Fredholm integral equation of the first kind with the compact kernel as I raw may be ill-posed.7 Hence for the deconvolution, regularized inversion techniques as TSVD or Tikhonov regularization theory are required. Numerically the deconvolution is equivalent to the inversion of the Toeplitz matrix of the kernel. In many cases, the Toeplitz matrix of the kernel is sparse, efficient calculation is feasible. However, as previously mentioned, an inevitable non-negativity constraint of I coh must be considered in the deconvolution problem that we want to deal with. That is because I coh indicate the intensity distribution not the amplitude. Commonly used deconvolution algorithm based on the inversion of the Toeplitz matrix of the kernel does not guarantee the non-negativity condition of the deconvolved image, thus auxiliary algorithm to satisfy the non-negativity condition is needed. Anyway the extraction of I coh is not a pure deconvolution problem. Hence it is plausible that the problem is considered as a constrained optimization problem with non-negative variables. In this paper, we consider a main goal as obtaining the diffraction image maintaining edge-sharpness that may be significantly smoothed by the convolution effect of partial coherence of the GSM beam. For this, instead of directly dealing with an optimization technique for the constrained optimization problem, we take a simple heuristic approach for conserving the edge-sharpness. We address that the employment of target images with extra rim patterns for the IFTA is

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effective to improve edge-sharpness of the obtained diffraction image. With only simple addition of the rim pattern to the coherent target image I coh , the loss of edge-sharpness of the diffraction image can be prevented. Figure 1 presents the main idea of this paper by displaying several convolution curves I coh ⊗ I raw obtained by the convolution of the kernel I raw and several target images I coh with extra rim pattern. The coherent target image for the IFTA, I coh and the degree of coherence function I raw are indicated by dotted line and dash-dotted line, respectively, in Fig. 1. The convolution curve I coh ⊗ I raw is drawn by solid line. In Fig. 1(a), I coh is set to simple square profile. However the obtained convolution curve shows significantly smoothed pattern. The edge is indistinguishable and the top profile of the convolution curve near the center is not flat. On the other hand, in the cases of I coh with extra rim pattern, the convolution curves are somewhat different from that of Fig. 1(a).

1.2

1.2 1

I coh

I coh ⊗ I raw

0.8

0.8

I raw

0.6

0.6

0.4

0.4

0.2

0.2

0 -1

-0.5

0

x

0.5

α =2

1

1

0 -1

-0.5

(a)

0.5

1

(b)

1.2

1.2

α =4

1

0.8

0.6

0.6

0.4

0.4

0.2

0.2

-0.5

0

x

0.5

α =7

1

0.8

0 -1

0

x

1

(c)

0 -1

-0.5

0

x

0.5

1

(d)

Fig. 1. The convolution curves of I raw and I coh . (a) the case of taking simple rectangular profile as I coh and the case of using the profile with extra rim pattern as I coh with (b)

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α = 2 , (c) α = 4 , and (d) α = 7

The parameter α shown in Fig. 1 is defined as the ratio of the magnitudes of the center rectangular and the extra rim pattern. Figures 1(b), (c) and (d) show the cases of α equals to 2, 4, and 7, respectively. In Fig. 1(c), the top profile appears to be flat compared to any other cases in Fig. 1. As α increases, the portion near the extra rim pattern is stressed as seen in Fig. 1(d). It is desirable to choose the value of α to make the top profile as flat as possible. The extra rim generation algorithm is very simple. At first, the edge pattern of the target image I coh is extracted, which is denoted

′ is constructed by subtracting Ω0 from I coh ( I coh ′ = I coh − Ω 0 ). The edge pattern by Ω0 . Then the first image I coh ′ is extracted. The second image I coh ′′ is obtained by I coh ′′ = I coh ′ − Ω1 . Next, the edge pattern Ω 2 of the Ω1 of I coh

′′′ + Ω1 is taken as the modified ′′ is extracted. Then I coh ′′′ = I coh ′′ − Ω 2 is obtained. Finally I coh = I coh second image I coh target image for the IFTA. p

4. NUMERICAL RESULTS In this section, an example of the proposed method is presented. The size of the image plane is set to 2cm × 2cm . The transverse coherence width σ g of the GSM beam in the image plane and the operating wavelength are taken as 0.04cm and 633nm , respectively. Figure 2 shows the simulation results. In Figs. 2(a) and (b), the coherent target image and the phase profile of DOE obtained by the conventional IFTA with the simple target image with no rim pattern are indicated, respectively. Figs. 2(c) and (d) shows the coherent target image and the phase profile of DOE with the proposed extra rim pattern, respectively. In this case, the ratio of the magnitudes of the internal image and the rim pattern is tuned to 4.5. In Figs. 2(e) and (f), the resulting partially coherent diffraction images for the respective cases of the conventional and proposed target images are presented. As seen in Fig. 3, the flatness of the top profile is improved by the proposed method compared to the profile of the conventional case. Figure 3 shows the comparison of the cross-section profile of the diffraction images produced by the simple DOE (denoted by dotted line) and the proposed DOE (denoted by solid line). It is explicitly shown that the extra rim pattern shown in Fig. 2(c) can improve the flatness of the diffraction image. To obtain a more refined diffraction image, an effective optimization technique of constrained optimization problem with non-negative variables should be adopted. The proposed method of extra rim image is somewhat intuitive, but the simulation shows the practical effectiveness of the proposed method. The proposed technique of using extra rim pattern is of use in the development of the advanced optimization technique.

5. CONCLUSION We propose a simple design method of DOE for the beam shaping of partially coherent GSM beams. By investigating the convolution integral of the diffraction, it is shown that the extra rim pattern of the target image is proved to make flatter intensity distribution of the diffraction image. The proposed technique of using extra rim pattern is expected to be useful for the development of the more advanced optimal DOE design for GSM beams.

ACKNOWLEDGMENT This work was supported by the Ministry of Science and Technology through the National Research Laboratory Program.

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6 Icoh (a.u.)

4 3 x ( m)

Phase (rad)

5

2 1

y ( m) (a)

(b)

6 Icoh (a.u.)

4 3 x ( m)

Phase (rad)

5

2 1

y ( m) (c)

(d)

I (a.u.)

I (a.u.)

x (m)

x ( m)

y (m)

y (m )

(e)

(f)

Fig. 2. (a) the diffraction image and (b) the phase profile of DOE with the simple target image, and (c) the diffraction image and (d) the phase profile of DOE with the proposed target image with extra rim pattern (e) the resulting diffraction image of the simple DOE and (f) the resulting diffraction image of the proposed DOE with extra rim image.

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6

x 10

-4

I (a.u.)

5 4 3 2 1 0 -0.02

-0.01

0

x (m)

0.01

0.02

Fig. 3. Comparison of the cross-section profile of the diffraction images produced by the simple DOE (denoted by dotted line) and the proposed DOE (denoted by solid line)

REFERENCES 1. D. Schafer, “Design concept for diffractive elements shaping partially coherent laser beams,” Journal of Optical Society of America A, 18, pp. 2915-, 1979. 2. M. Born and E. Wolf, Principles of Optics, 7 th Ed., Cambridge university press, New York, 1999. 3. J. Goodman, Introduction to Fourier Optics, 2nd Ed., McGraw-Hill, New York, 1988. 4. J. Goodman, Statistical Optics, Wiley, New York, 1985. 5. L. Mandel and E. Wolf, Optical coherence and Quantum Optics, Cambridge university press, New York, 1995. 6. A. V. Oppenheim, A. S. Willsky, Signals and Systems, 2 nd Ed., Prentice-Hall, New Jersey, 1997. 7. A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Illposed Problems, Kluwer Academic, Boston, Mass., 1995.

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