Abstract

The synthesis method including wave-optics and ray-tracing for the acceleration of the simulation of micro-optical systems has been developed. The effects of the spatial coherence and randomization of microlens array (MLA) parameters have been considered. The method based on coherent states representation for the calculation of the optical efficiency of microlens arrays taking into account the light source polarization has been developed. Numerical simulations of the intensity distributions and spreading angle of a diffracted beam have been carried out.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  4. Z. Wang, A. Wang, S. Wang, X. Ma, and H. Ming, “Resolution-enhanced integral imaging using two micro-lens arrays with different focal lengths for capturing and display,” Opt. Express 23(22), 28970–28977 (2015).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  9. N. Lindlein, “Simulation of micro-optical systems including microlens arrays,” J. Opt. A, Pure Appl. Opt. 4(4), S1–S9 (2002).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  31. S.-I. Chang, J.-B. Yoon, H. Kim, J.-J. Kim, B.-K. Lee, and D. H. Shin, “Microlens array diffuser for a light-emitting diode backlight system,” Opt. Lett. 31(20), 3016–3018 (2006).
    [Crossref] [PubMed]
  32. Y. Jin, A. Hassan, and Y. Jiang, “Freeform microlens array homogenizer for excimer laser beam shaping,” Opt. Express 24(22), 24846–24858 (2016).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2017 (2)

2016 (2)

2015 (2)

2013 (1)

2008 (1)

2007 (1)

2006 (1)

2005 (3)

H. Urey and K. D. Powell, “Microlens-array-based exit-pupil expander for full-color displays,” Appl. Opt. 44(23), 4930–4936 (2005).
[Crossref] [PubMed]

J. Garcia-Sucerquia, J. A. H. Ramírez, and D. V. Prieto, “Reduction of speckle noise in digital holography by using digital image processing,” Optik (Stuttg.) 116(1), 44–48 (2005).
[Crossref]

N. I. Petrov, “Reflection and transmission of strongly focused light beams at a dielectric interface,” J. Mod. Opt. 52(11), 1545–1556 (2005).
[Crossref]

2004 (2)

2003 (2)

2002 (3)

N. Lindlein, “Simulation of micro-optical systems including microlens arrays,” J. Opt. A, Pure Appl. Opt. 4(4), S1–S9 (2002).
[Crossref]

A. Büttner and U. D. Zeitner, “Calculation of the average lenslet shape and aberrations of microlens arrays from their far-field intensity distribution,” Appl. Opt. 41(32), 6841–6848 (2002).
[Crossref] [PubMed]

H. Ottevaere, B. Volckaerts, J. Lamprecht, J. Schwider, A. Hermanne, I. Veretennicoff, and H. Thienpont, “Two-dimensional plastic microlens arrays by deep lithography with protons: fabrication and characterization,” J. Opt. A, Pure Appl. Opt. 4(4), S22–S28 (2002).
[Crossref]

2001 (3)

1999 (1)

C. Kopp, L. Ravel, and P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array and random phase plate,” J. Opt. A, Pure Appl. Opt. 1(3), 398–403 (1999).
[Crossref]

1995 (1)

1986 (2)

Y. Z. Ruan and L. B. Felsen, “Reflection and transmission of beams at a curved interface,” J. Opt. Soc. Am. A 3(4), 566–579 (1986).
[Crossref]

S. G. Krivoshlykov, N. I. Petrov, and I. N. Sissakian, “Density-matrix formalism for partially coherent optical fields propagating in slightly inhomogeneous media,” Opt. Quantum Electron. 18(4), 253–264 (1986).
[Crossref]

1966 (1)

1962 (1)

Akatay, A.

Büttner, A.

Casperson, L. W.

Castañeda, R.

Chakrabarti, M.

Chang, S. I.

Chang, S.-I.

Chen, Q. D.

Choi, Y. S.

Dam-Hansen, C.

Felsen, L. B.

Freeman, M. O.

Garcia-Sucerquia, J.

J. Garcia-Sucerquia, J. A. H. Ramírez, and D. V. Prieto, “Reduction of speckle noise in digital holography by using digital image processing,” Optik (Stuttg.) 116(1), 44–48 (2005).
[Crossref]

Ham, Y. N.

Hassan, A.

Hedili, M. K.

Hermanne, A.

H. Ottevaere, B. Volckaerts, J. Lamprecht, J. Schwider, A. Hermanne, I. Veretennicoff, and H. Thienpont, “Two-dimensional plastic microlens arrays by deep lithography with protons: fabrication and characterization,” J. Opt. A, Pure Appl. Opt. 4(4), S22–S28 (2002).
[Crossref]

Herzig, H. P.

N. Lindlein and H. P. Herzig, “Design and modeling of a miniature system containing micro-optics,” Proc. SPIE 4437, 1–13 (2001).
[Crossref]

Hua, H.

Jiang, Y.

Jin, Y.

Kim, H.

Kim, J. M.

Kim, J.-J.

Kim, S. I.

Kogelnik, H.

Kopp, C.

C. Kopp, L. Ravel, and P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array and random phase plate,” J. Opt. A, Pure Appl. Opt. 1(3), 398–403 (1999).
[Crossref]

Krivoshlykov, S. G.

S. G. Krivoshlykov, N. I. Petrov, and I. N. Sissakian, “Density-matrix formalism for partially coherent optical fields propagating in slightly inhomogeneous media,” Opt. Quantum Electron. 18(4), 253–264 (1986).
[Crossref]

Lamprecht, J.

H. Ottevaere, B. Volckaerts, J. Lamprecht, J. Schwider, A. Hermanne, I. Veretennicoff, and H. Thienpont, “Two-dimensional plastic microlens arrays by deep lithography with protons: fabrication and characterization,” J. Opt. A, Pure Appl. Opt. 4(4), S22–S28 (2002).
[Crossref]

Lee, B.-K.

Li, T.

Lindlein, N.

N. Lindlein, “Simulation of micro-optical systems including microlens arrays,” J. Opt. A, Pure Appl. Opt. 4(4), S1–S9 (2002).
[Crossref]

N. Lindlein and H. P. Herzig, “Design and modeling of a miniature system containing micro-optics,” Proc. SPIE 4437, 1–13 (2001).
[Crossref]

Ma, X.

Meyrueis, P.

C. Kopp, L. Ravel, and P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array and random phase plate,” J. Opt. A, Pure Appl. Opt. 1(3), 398–403 (1999).
[Crossref]

Ming, H.

Murty, M. V. R. K.

Ottevaere, H.

H. Ottevaere, B. Volckaerts, J. Lamprecht, J. Schwider, A. Hermanne, I. Veretennicoff, and H. Thienpont, “Two-dimensional plastic microlens arrays by deep lithography with protons: fabrication and characterization,” J. Opt. A, Pure Appl. Opt. 4(4), S22–S28 (2002).
[Crossref]

Park, C. Y.

Pauwels, J.

Pedersen, H. C.

Pedersen, T. F.

Petrov, N.

Petrov, N. I.

N. I. Petrov, “Reflection and transmission of strongly focused light beams at a dielectric interface,” J. Mod. Opt. 52(11), 1545–1556 (2005).
[Crossref]

N. I. Petrov, “Reflection and transmission of strongly focused vector beams at a dielectric interface,” Opt. Lett. 29(5), 421–423 (2004).
[Crossref] [PubMed]

S. G. Krivoshlykov, N. I. Petrov, and I. N. Sissakian, “Density-matrix formalism for partially coherent optical fields propagating in slightly inhomogeneous media,” Opt. Quantum Electron. 18(4), 253–264 (1986).
[Crossref]

Powell, K. D.

Prieto, D. V.

J. Garcia-Sucerquia, J. A. H. Ramírez, and D. V. Prieto, “Reduction of speckle noise in digital holography by using digital image processing,” Optik (Stuttg.) 116(1), 44–48 (2005).
[Crossref]

Ramírez, J. A. H.

J. Garcia-Sucerquia, J. A. H. Ramírez, and D. V. Prieto, “Reduction of speckle noise in digital holography by using digital image processing,” Optik (Stuttg.) 116(1), 44–48 (2005).
[Crossref]

Ravel, L.

C. Kopp, L. Ravel, and P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array and random phase plate,” J. Opt. A, Pure Appl. Opt. 1(3), 398–403 (1999).
[Crossref]

Ruan, Y. Z.

Sales, T. R. M.

T. R. M. Sales, “Structured microlens arrays for beam shaping,” Opt. Eng. 42(11), 3084–3085 (2003).
[Crossref]

Schwider, J.

H. Ottevaere, B. Volckaerts, J. Lamprecht, J. Schwider, A. Hermanne, I. Veretennicoff, and H. Thienpont, “Two-dimensional plastic microlens arrays by deep lithography with protons: fabrication and characterization,” J. Opt. A, Pure Appl. Opt. 4(4), S22–S28 (2002).
[Crossref]

Shin, D. H.

Sissakian, I. N.

S. G. Krivoshlykov, N. I. Petrov, and I. N. Sissakian, “Density-matrix formalism for partially coherent optical fields propagating in slightly inhomogeneous media,” Opt. Quantum Electron. 18(4), 253–264 (1986).
[Crossref]

Spencer, G. H.

Stubager, J.

Sun, H. B.

Thienpont, H.

H. Ottevaere, B. Volckaerts, J. Lamprecht, J. Schwider, A. Hermanne, I. Veretennicoff, and H. Thienpont, “Two-dimensional plastic microlens arrays by deep lithography with protons: fabrication and characterization,” J. Opt. A, Pure Appl. Opt. 4(4), S22–S28 (2002).
[Crossref]

Tian, Z. N.

Tovar, A. A.

Urey, H.

Veretennicoff, I.

H. Ottevaere, B. Volckaerts, J. Lamprecht, J. Schwider, A. Hermanne, I. Veretennicoff, and H. Thienpont, “Two-dimensional plastic microlens arrays by deep lithography with protons: fabrication and characterization,” J. Opt. A, Pure Appl. Opt. 4(4), S22–S28 (2002).
[Crossref]

Verschaffelt, G.

Volckaerts, B.

H. Ottevaere, B. Volckaerts, J. Lamprecht, J. Schwider, A. Hermanne, I. Veretennicoff, and H. Thienpont, “Two-dimensional plastic microlens arrays by deep lithography with protons: fabrication and characterization,” J. Opt. A, Pure Appl. Opt. 4(4), S22–S28 (2002).
[Crossref]

Wang, A.

Wang, S.

Wang, X.

Wang, Z.

Xu, J. J.

Yao, W. G.

Yoon, J. B.

Yoon, J.-B.

Yu, Y. H.

Zeitner, U. D.

Appl. Opt. (4)

J. Mod. Opt. (1)

N. I. Petrov, “Reflection and transmission of strongly focused light beams at a dielectric interface,” J. Mod. Opt. 52(11), 1545–1556 (2005).
[Crossref]

J. Opt. A, Pure Appl. Opt. (3)

H. Ottevaere, B. Volckaerts, J. Lamprecht, J. Schwider, A. Hermanne, I. Veretennicoff, and H. Thienpont, “Two-dimensional plastic microlens arrays by deep lithography with protons: fabrication and characterization,” J. Opt. A, Pure Appl. Opt. 4(4), S22–S28 (2002).
[Crossref]

C. Kopp, L. Ravel, and P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array and random phase plate,” J. Opt. A, Pure Appl. Opt. 1(3), 398–403 (1999).
[Crossref]

N. Lindlein, “Simulation of micro-optical systems including microlens arrays,” J. Opt. A, Pure Appl. Opt. 4(4), S1–S9 (2002).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

T. R. M. Sales, “Structured microlens arrays for beam shaping,” Opt. Eng. 42(11), 3084–3085 (2003).
[Crossref]

Opt. Express (8)

N. Petrov, “Focusing of beams into subwavelength area in an inhomogeneous medium,” Opt. Express 9(12), 658–673 (2001).
[Crossref] [PubMed]

S. I. Chang and J. B. Yoon, “Shape-controlled, high fill-factor microlens arrays fabricated by a 3D diffuser lithography and plastic replication method,” Opt. Express 12(25), 6366–6371 (2004).
[Crossref] [PubMed]

A. Akatay and H. Urey, “Design and optimization of microlens array based high resolution beam steering system,” Opt. Express 15(8), 4523–4529 (2007).
[Crossref] [PubMed]

M. K. Hedili, M. O. Freeman, and H. Urey, “Transmission characteristics of a bidirectional transparent screen based on reflective microlenses,” Opt. Express 21(21), 24636–24646 (2013).
[Crossref] [PubMed]

Z. Wang, A. Wang, S. Wang, X. Ma, and H. Ming, “Resolution-enhanced integral imaging using two micro-lens arrays with different focal lengths for capturing and display,” Opt. Express 23(22), 28970–28977 (2015).
[Crossref] [PubMed]

M. Chakrabarti, C. Dam-Hansen, J. Stubager, T. F. Pedersen, and H. C. Pedersen, “Replication of optical microlens array using photoresist coated molds,” Opt. Express 24(9), 9528–9540 (2016).
[Crossref] [PubMed]

Y. Jin, A. Hassan, and Y. Jiang, “Freeform microlens array homogenizer for excimer laser beam shaping,” Opt. Express 24(22), 24846–24858 (2016).
[Crossref] [PubMed]

J. Pauwels and G. Verschaffelt, “Speckle reduction in laser projection using microlens-array screens,” Opt. Express 25(4), 3180–3195 (2017).
[Crossref] [PubMed]

Opt. Lett. (4)

Opt. Quantum Electron. (1)

S. G. Krivoshlykov, N. I. Petrov, and I. N. Sissakian, “Density-matrix formalism for partially coherent optical fields propagating in slightly inhomogeneous media,” Opt. Quantum Electron. 18(4), 253–264 (1986).
[Crossref]

Optik (Stuttg.) (1)

J. Garcia-Sucerquia, J. A. H. Ramírez, and D. V. Prieto, “Reduction of speckle noise in digital holography by using digital image processing,” Optik (Stuttg.) 116(1), 44–48 (2005).
[Crossref]

Proc. SPIE (1)

N. Lindlein and H. P. Herzig, “Design and modeling of a miniature system containing micro-optics,” Proc. SPIE 4437, 1–13 (2001).
[Crossref]

Other (4)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

N. I. Petrov, J.-J. Kim, H.-S. Jeong, and D. H. Shin, “Diffraction of partially-coherent light beams by micro-lens arrays,” in Proceedings of Frontiers in Optics2006/Laser Science Conference, Rochester, USA, 2006, Paper FTuM2.
[Crossref]

D. Marcuse, Light Transmission Optics (New York, 1972).

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Figures (13)

Fig. 1
Fig. 1

Optical scheme of the problem.

Fig. 2
Fig. 2

The region of integartion.

Fig. 3
Fig. 3

Array with convex profile of microlenses: x, y, and z in µm.

Fig. 4
Fig. 4

Integration area for hexagonal lens base.

Fig. 5
Fig. 5

Intensity distributions at the distance z = 5 mm for different coherent radii r0: (a) r0 = 3 μm; (b) r0 = 10 μm; (c) r0 = 30 μm; (d) r0 = 300 μm.

Fig. 6
Fig. 6

Intensity distributions at different distances z for RL = 10 μm and Rsc = 3 μm: (a) z = 1 mm; (b) z = 5 mm.

Fig. 7
Fig. 7

Intensity distributions (a, b) and radiation patterns (c, d) for LD (a, c) and LED (b, d) sources: z = 5mm.

Fig. 8
Fig. 8

Intensity distributions of diffracted light for coherent (a, b) and low-coherent (c, d) sources. (a, c) – regular MLA, (b, d) – random MLA.

Fig. 9
Fig. 9

Geometrical configuration and coordinate system for a boundary.

Fig. 10
Fig. 10

Reflectance r (red curves) and transmittance t (blue curves) versus depth h at lower/higher index and higher/lower index interfaces for different values of incident wavefront curvature radiuses: left - TE polarized beam (a, c), right – TM polarized beam (b, d); curves 1, Rf = 1500 μm; curves 2, plane wavefront.

Fig. 11
Fig. 11

Optical efficiency and spreading angle as function of aspect ratio.

Fig. 12
Fig. 12

User friendly interface: input data (a) and graphical outputs (b).

Fig. 13
Fig. 13

Measured intensity distributions of the diffracted radiation by microlens array at different distances with LD source: (a) z = 45cm; (b) z = 145cm; (c) z = 290cm; (d) LED source.

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

s(x,y)= c x (x x c ) 2 1+ 1(1+ κ x ) (x x c ) 2 c x 2 + c y (y y c ) 2 1+ 1(1+ κ y ) (y y c ) 2 c y 2 + p A xp (x x c ) p + A yp (y y c ) p ,
u(x,y,0)= A 0 exp[ ( x 2 w x 2 + y 2 w y 2 ) ]exp[ i π λ ( x 2 R x + y 2 R y ) ],
E(x,y,z)= E 0 iλz j D j exp( ξ 2 + η 2 a 0 2 )exp[ iφ ] dξdη ,
Γ( r, r ,z )= E * (r,z)E(r',z) ,
Γ( r 1 , r 2 )= I 0 exp{ r 1 2 + r 2 2 a 0 2 ( r 1 r 2 ) 2 r 0 2 iπ λ R f ( r 2 2 r 1 2 ) } ,
I(r,z)= | E(r,z) | 2 = c ε 0 2 ( k 2πz ) 2 Γ 0 ( r 1 ' , r 2 ') expi[ Φ( r 1 ')Φ( r 2 ') ] G * ( r 1 ',r,z)G( r 2 ',r,z) d 2 r 1 ' d 2 r 2 '
G( x',y',x,y,z )= 1 iλ zexp(ikR) R 2 ,
G( x',y',x,y,z )= k 2πiz exp{ ikz+ ik 2z [ ( xx' ) 2 + ( yy' ) 2 ] } .
I(x,y,z)= I 0 (λz) 2 d y 1 ' d y 2 ' F y ( y 1 ' , y 2 ' ) G ˜ (y, y 1 ' , y 2 ' ,z) D mn d x 1 ' d x 2 ' F x ( x 1 ' , x 2 ' ) G ˜ (x, x 1 ' , x 2 ' ,z) ,
F x ( x 1 ' , x 2 ' )=exp{ x 1 '2 + x 2 '2 a 0x 2 ( x 1 ' x 2 ' ) 2 r 0x 2 ik 2 R fx ( x 1 '2 x 2 '2 ) }exp{ ikΔn[ s m ( x 1 ' ) s n ( x 2 ' ) ] } F y ( y 1 ' , y 2 ' )=exp{ y 1 '2 + y 2 '2 a 0y 2 ( y 1 ' y 2 ' ) 2 r 0y 2 ik 2 R fy ( y 1 '2 y 2 '2 ) }, G ˜ (x, x 1 ' , x 2 ' ,z)=exp{ ikz+ ik 2z [ ( x x 1 ' ) 2 ( x x 2 ' ) 2 ] }
I(x,y,z)= I 0 (λz) 2 A y mn D mn d x 1 ' d x 2 ' F x ( x 1 ' , x 2 ' )exp{ ik 2z [ ( x x 1 ' ) 2 ( x x 2 ' ) 2 ] } ,
A y = π a y a ˜ y exp{ ( a yr 1/ r 0y 2 ) y 2 2 a y a ˜ y k 2 z 2 }, a y = 1 a 0y 2 + 1 r 0y 2 ik 2 R fy + ik 2z , a ˜ y = 1 a 0y 2 + 1 r 0y 2 + ik 2 R fy ik 2z 1 r 0y 4 a y , a yr = 1 a 0y 2 + 1 r 0y 2 , a x = 1 a 0x 2 + 1 r 0x 2 + ik 2 R fx ik 2z , a ˜ x = 1 a 0x 2 + 1 r 0x 2 ik 2 R fx + ik 2z 1 r 0y 4 a x , a xr = 1 a 0x 2 + 1 r 0x 2 .
I(x,y,z)= I 0 (λz) 2 A y mn D mn dr'dR' F x (r',R')exp{ ik 2z ( 2R'r'2xr' ) } .
F x (r',R')=exp{ 2R ' 2 a 0x 2 r ' 2 2 a 0x 2 r ' 2 r 0x 2 ikR'r' R fx + ikΔn 2 R sc [ 2R'r'2R'( x 0m x 0n )r'( x 0m + x 0n )+( x 0m 2 x on 2 ) ] }.
I(x,y,z)= I 0 (λz) 2 A y π a m,n a m b m dR'f(R') ,
f(R')=exp{ b 2 4a 2R ' 2 a 0x 2 }cos[ kΔn R sc ( x 0m 2 x 0n 2 ) kΔn R sc R'( x 0m x 0n ) ], a= 1 r 0 2 + 1 2 a 0x 2 , b= ik R fx R' ikΔn R sc R'+ ik z (R'x)+ ikΔn 2 R sc ( x 0m + x 0n ) .
I(x,y,z)= I 0 (λz) 2 A y mn D mn dr'dR' F x (r',R')exp{ ik 2z ( 2R'r'2xr' ) } .
I(x,y,z)= I 0 ( λz ) 2 D pl d y 1 ' d y 2 ' F y ( y 1 ' , y 2 ' ) G ˜ (y, y 1 ' , y 2 ' ,z) D mn d x 1 ' d x 2 ' F x ( x 1 ' , x 2 ' ) G ˜ (x, x 1 ' , x 2 ' ,z) ,
F x ( x 1 ' , x 2 ' )=exp{ x 1 '2 + x 2 '2 a 0x 2 ( x 1 ' x 2 ' ) 2 r 0x 2 ik 2 R fx ( x 1 '2 x 2 '2 ) }exp{ ikΔn[ s m ( x 1 ' ) s n ( x 2 ' ) ] }, F y ( y 1 ' , y 2 ' )=exp{ y 1 '2 + y 2 '2 a 0y 2 ( y 1 ' y 2 ' ) 2 r 0y 2 ik 2 R fy ( y 1 '2 y 2 '2 ) }exp{ ikΔn[ s p ( y 1 ' ) s l ( y 2 ' ) ] }.
I(x,y,z)= I 0 (λz) 2 D pl dρ'dS' F y (ρ',S')exp{ ik 2z ( 2S'ρ'2yρ' ) }U ,
U= D mn dr'dR' F x (r',R') exp{ ik 2z ( 2R'r'2xr' ) }, F x (r',R')=exp{ 2R ' 2 a 0x 2 r ' 2 2 a 0x 2 r ' 2 r 0x 2 ikR'r' R fx + ikΔn 2 R sc [ 2R'r'2R'( x 0m x 0n )r'( x 0m + x 0n )+( x 0m 2 x on 2 ) ] }
F y (ρ',S')=exp{ 2S ' 2 a 0y 2 ρ ' 2 2 a 0y 2 ρ ' 2 r 0y 2 ikS'ρ' R fy + ikΔn 2 R sc [ 2S'ρ'2S'( y 0p y 0l )ρ'( y 0p + y 0l )+( y 0p 2 y ol 2 ) ] }.
I(x,y,z)= I 0 (λz) 2 mn D mn dS'dR'f(S',R',x,y,z) ,
f(S',R',x,y,z)= π a a ˜ exp{ b i 2 4a 2R ' 2 a 0x 2 b ˜ i 2 4 a ˜ 2S ' 2 a 0y 2 }cos[ kΔn R sc Φ ], Φ= x 0m 2 x 0n 2 2 R'( x 0m x 0n )+ y 0m 2 y 0n 2 2 S'( y 0m y 0n ), b i = k R fx R'+ k z (R'x) kΔn 2 ( 2R' R sc x 0m + x 0n R sc ), b ˜ i = k R fy S'+ k z (S'y) kΔn 2 ( 2S' R sc y 0m + y 0n R sc ), a= 1 r 0x 2 + 1 2 a 0x 2 , a ˜ = 1 r 0y 2 + 1 2 a 0y 2 .
E(x)= E 0 m=N N exp[ ( xmw w ) 2 ] m=N N exp( m 2 ) ,
E( x,0 )= π 1 d 2 αx|αf(α)
a |α=α|α,
|α= Ψ α (x)= ( 2 π w 0 2 ) 1 4 exp( x 2 w 0 2 + 2x w 0 α α 2 2 | α | 2 2 )
| f(α) | 2 = 2 w 0 / a 0 1+ w 0 2 / a 0 2 exp{ | α | 2 + | α | 2 1 w 0 2 / a 0 2 1+ w 0 2 / a 0 2 cos2ϑ }.
Γ( x 1 , x 2 )= E( x 1 )E( x 2 ) = π 2 d 2 α d 2 β Ψ α * ( x 1 ) Ψ β ( x 2 ) f( α * )f(β) ,
f(α*)f(β) = d x 1 d x 2 Ψ α * ( x 1 ) Ψ β ( x 2 )Γ( x 1 , x 2 )exp( 1 2 | α | 2 + 1 2 | β | 2 ) .
Γ( x 1 , x 2 ,z)= E( x 1 ,z)E( x 2 ,z) = π 2 d 2 α d 2 β Ψ α * ( x 1 ,z) Ψ β ( x 2 ,z) f( α * )f(β) ,
r= P r P i = d 2 α | f(α) | 2 R( θ 1 i ) d 2 α | f(α) | 2 ,t= P t P i = d 2 α | f(α) | 2 T( θ 1 i ) d 2 α | f(α) | 2 ,
R E = ( n 1 cos θ 1 i n 2 2 n 1 2 sin 2 θ 1 i ) 2 ( n 1 cos θ 1 i + n 2 2 n 1 2 sin 2 θ 1 i ) 2 , T E = 4 n 1 cos θ 1 i n 2 2 n 1 2 sin 2 θ 1 i ( n 1 cos θ 1 i + n 2 2 n 1 2 sin 2 θ 1 i ) 2 , R H = ( n 2 cos θ 1 i n 1 n 2 n 2 2 n 1 2 sin 2 θ 1 i ) 2 ( n 1 cos θ 1 i + n 1 n 2 n 2 2 n 1 2 sin 2 θ 1 i ) 2 , T H = 4 n 1 cos θ 1 i n 2 2 n 1 2 sin 2 θ 1 i ( n 2 cos θ 1 i + n 1 n 2 n 2 2 n 1 2 sin 2 θ 1 i ) 2 ,

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