Abstract

With the development of microfabrication technology, it has become possible to modulate the optical wavefront with high accuracy by a device such as a liquid crystal spatial light modulator (SLM). In this study, we conducted a theoretical analysis and experimental study on the generation of Laguerre Gaussian beam of vortex light wave using phase modulation SLM. Numerical simulation and experimental results on the generation of Laguerre Gaussian beam by both binary and phase only modulation are discussed based on the angular spectrum method and the Fresnel transformation method in hologram (CGH) diffraction. The experimental results show that Fresnel diffraction calculation based on Fourier transform method can simulate the light diffraction of SLM very well. This method is very suitable for the action simulation of digital optical wavefront control devices such as SLM. It can be imagined that the simulation method in this study can also play an important role in the design and development of digital optical diffraction elements.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. G. J. Zhang, M. Zhang, and Y. Zhao, “Phase modulation characteristics of spatial light modulator and the system for its calibration,” J. Electr. Eng. 6, 193–205 (2018).
  2. L. Shi, J. Li, and T. Tao, “Micro-particles’ rotation by Laguerre-Gaussian beams produced by computer- generated holograms,” Laser Infrared 42(11), 1226–1229 (2012).
  3. F. Li, C. Q. Gao, Y. D. Liu, and M. W. Tao, “Experimental study of the generation of Laguerre-Gaussian beam using a computer-generated amplitude grating [J],” Wuli Xuebao 57(2), 860–866 (2008).
  4. A. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
    [Crossref]
  5. A. Bekshaev, S. Sviridova, A. Popov, A. Rimashevsky, and A. Tyurin, “Optical vortex generation by volume holographic elements with embedded phase singularity: Effects of misalignments,” Ukr. J. Phys. Opt. 14(4), 171–186 (2013).
    [Crossref]
  6. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45(6), 1102–1110 (2006).
    [Crossref] [PubMed]
  7. N. Verrier and M. Atlan, “Off-axis digital hologram reconstruction: some practical considerations,” Appl. Opt. 50(34), H136–H146 (2011).
    [Crossref] [PubMed]
  8. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  9. J. C. Li, “The Accurate Calculation of Fresnel Diffraction and Collins’ Formula by using the fast Fourier transform,” J. Opto-electronics Laser 12(5), 58–64 (2001).
  10. Y. Ohtake, T. Ando, N. Fukuchi, N. Matsumoto, H. Ito, and T. Hara, “Universal generation of higher-order multiringed Laguerre-Gaussian beams by using a spatial light modulator,” Opt. Lett. 32(11), 1411–1413 (2007).
    [Crossref] [PubMed]
  11. N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A 25(7), 1642–1651 (2008).
    [Crossref] [PubMed]
  12. G. J. Swanson, “Binary optics technology: The theory and design of multi-level diffractive optical elements,” Lincoln Laboratory Massachusetts Institute of Technology, Lexington, Massachusetts, Technical Report 854, (1989).
  13. G. J. Swanson, “Binary optics technology: Theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” Lincoln Laboratory Massachusetts Institute of Technology, Lexington, Massachusetts, Technical Report 914, (1991).
  14. S. F. Guo, K. Liu, H. X. Sun, J. X. Zhang, and J. R. Gao, “Generation of high-order Laguerre-Guassian beams by liquid crystal spatial light modulators,” J. Quantum Optics 21(1), 86–92 (2015).
  15. X. Q. Qi, C. Q. Gao, and Y. D. Liu, “Generation of helical beams with pre-determined energy distribution based on phase modulation gratings,” Wuli Xuebao 59(1), 264–270 (2010).
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

2018 (1)

G. J. Zhang, M. Zhang, and Y. Zhao, “Phase modulation characteristics of spatial light modulator and the system for its calibration,” J. Electr. Eng. 6, 193–205 (2018).

2015 (1)

S. F. Guo, K. Liu, H. X. Sun, J. X. Zhang, and J. R. Gao, “Generation of high-order Laguerre-Guassian beams by liquid crystal spatial light modulators,” J. Quantum Optics 21(1), 86–92 (2015).

2013 (1)

A. Bekshaev, S. Sviridova, A. Popov, A. Rimashevsky, and A. Tyurin, “Optical vortex generation by volume holographic elements with embedded phase singularity: Effects of misalignments,” Ukr. J. Phys. Opt. 14(4), 171–186 (2013).
[Crossref]

2012 (1)

L. Shi, J. Li, and T. Tao, “Micro-particles’ rotation by Laguerre-Gaussian beams produced by computer- generated holograms,” Laser Infrared 42(11), 1226–1229 (2012).

2011 (1)

2010 (1)

X. Q. Qi, C. Q. Gao, and Y. D. Liu, “Generation of helical beams with pre-determined energy distribution based on phase modulation gratings,” Wuli Xuebao 59(1), 264–270 (2010).

2008 (3)

N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A 25(7), 1642–1651 (2008).
[Crossref] [PubMed]

F. Li, C. Q. Gao, Y. D. Liu, and M. W. Tao, “Experimental study of the generation of Laguerre-Gaussian beam using a computer-generated amplitude grating [J],” Wuli Xuebao 57(2), 860–866 (2008).

A. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

2007 (1)

2006 (1)

2001 (1)

J. C. Li, “The Accurate Calculation of Fresnel Diffraction and Collins’ Formula by using the fast Fourier transform,” J. Opto-electronics Laser 12(5), 58–64 (2001).

Ando, T.

Atlan, M.

Bekshaev, A.

A. Bekshaev, S. Sviridova, A. Popov, A. Rimashevsky, and A. Tyurin, “Optical vortex generation by volume holographic elements with embedded phase singularity: Effects of misalignments,” Ukr. J. Phys. Opt. 14(4), 171–186 (2013).
[Crossref]

A. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

Fukuchi, N.

Gao, C. Q.

X. Q. Qi, C. Q. Gao, and Y. D. Liu, “Generation of helical beams with pre-determined energy distribution based on phase modulation gratings,” Wuli Xuebao 59(1), 264–270 (2010).

F. Li, C. Q. Gao, Y. D. Liu, and M. W. Tao, “Experimental study of the generation of Laguerre-Gaussian beam using a computer-generated amplitude grating [J],” Wuli Xuebao 57(2), 860–866 (2008).

Gao, J. R.

S. F. Guo, K. Liu, H. X. Sun, J. X. Zhang, and J. R. Gao, “Generation of high-order Laguerre-Guassian beams by liquid crystal spatial light modulators,” J. Quantum Optics 21(1), 86–92 (2015).

Guo, S. F.

S. F. Guo, K. Liu, H. X. Sun, J. X. Zhang, and J. R. Gao, “Generation of high-order Laguerre-Guassian beams by liquid crystal spatial light modulators,” J. Quantum Optics 21(1), 86–92 (2015).

Hara, T.

Inoue, T.

Ito, H.

Karamoch, A. I.

A. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

Li, F.

F. Li, C. Q. Gao, Y. D. Liu, and M. W. Tao, “Experimental study of the generation of Laguerre-Gaussian beam using a computer-generated amplitude grating [J],” Wuli Xuebao 57(2), 860–866 (2008).

Li, J.

L. Shi, J. Li, and T. Tao, “Micro-particles’ rotation by Laguerre-Gaussian beams produced by computer- generated holograms,” Laser Infrared 42(11), 1226–1229 (2012).

Li, J. C.

J. C. Li, “The Accurate Calculation of Fresnel Diffraction and Collins’ Formula by using the fast Fourier transform,” J. Opto-electronics Laser 12(5), 58–64 (2001).

Liu, K.

S. F. Guo, K. Liu, H. X. Sun, J. X. Zhang, and J. R. Gao, “Generation of high-order Laguerre-Guassian beams by liquid crystal spatial light modulators,” J. Quantum Optics 21(1), 86–92 (2015).

Liu, Y. D.

X. Q. Qi, C. Q. Gao, and Y. D. Liu, “Generation of helical beams with pre-determined energy distribution based on phase modulation gratings,” Wuli Xuebao 59(1), 264–270 (2010).

F. Li, C. Q. Gao, Y. D. Liu, and M. W. Tao, “Experimental study of the generation of Laguerre-Gaussian beam using a computer-generated amplitude grating [J],” Wuli Xuebao 57(2), 860–866 (2008).

Matsumoto, N.

Ohtake, Y.

Popov, A.

A. Bekshaev, S. Sviridova, A. Popov, A. Rimashevsky, and A. Tyurin, “Optical vortex generation by volume holographic elements with embedded phase singularity: Effects of misalignments,” Ukr. J. Phys. Opt. 14(4), 171–186 (2013).
[Crossref]

Qi, X. Q.

X. Q. Qi, C. Q. Gao, and Y. D. Liu, “Generation of helical beams with pre-determined energy distribution based on phase modulation gratings,” Wuli Xuebao 59(1), 264–270 (2010).

Rimashevsky, A.

A. Bekshaev, S. Sviridova, A. Popov, A. Rimashevsky, and A. Tyurin, “Optical vortex generation by volume holographic elements with embedded phase singularity: Effects of misalignments,” Ukr. J. Phys. Opt. 14(4), 171–186 (2013).
[Crossref]

Shen, F.

Shi, L.

L. Shi, J. Li, and T. Tao, “Micro-particles’ rotation by Laguerre-Gaussian beams produced by computer- generated holograms,” Laser Infrared 42(11), 1226–1229 (2012).

Sun, H. X.

S. F. Guo, K. Liu, H. X. Sun, J. X. Zhang, and J. R. Gao, “Generation of high-order Laguerre-Guassian beams by liquid crystal spatial light modulators,” J. Quantum Optics 21(1), 86–92 (2015).

Sviridova, S.

A. Bekshaev, S. Sviridova, A. Popov, A. Rimashevsky, and A. Tyurin, “Optical vortex generation by volume holographic elements with embedded phase singularity: Effects of misalignments,” Ukr. J. Phys. Opt. 14(4), 171–186 (2013).
[Crossref]

Tao, M. W.

F. Li, C. Q. Gao, Y. D. Liu, and M. W. Tao, “Experimental study of the generation of Laguerre-Gaussian beam using a computer-generated amplitude grating [J],” Wuli Xuebao 57(2), 860–866 (2008).

Tao, T.

L. Shi, J. Li, and T. Tao, “Micro-particles’ rotation by Laguerre-Gaussian beams produced by computer- generated holograms,” Laser Infrared 42(11), 1226–1229 (2012).

Tyurin, A.

A. Bekshaev, S. Sviridova, A. Popov, A. Rimashevsky, and A. Tyurin, “Optical vortex generation by volume holographic elements with embedded phase singularity: Effects of misalignments,” Ukr. J. Phys. Opt. 14(4), 171–186 (2013).
[Crossref]

Verrier, N.

Wang, A.

Zhang, G. J.

G. J. Zhang, M. Zhang, and Y. Zhao, “Phase modulation characteristics of spatial light modulator and the system for its calibration,” J. Electr. Eng. 6, 193–205 (2018).

Zhang, J. X.

S. F. Guo, K. Liu, H. X. Sun, J. X. Zhang, and J. R. Gao, “Generation of high-order Laguerre-Guassian beams by liquid crystal spatial light modulators,” J. Quantum Optics 21(1), 86–92 (2015).

Zhang, M.

G. J. Zhang, M. Zhang, and Y. Zhao, “Phase modulation characteristics of spatial light modulator and the system for its calibration,” J. Electr. Eng. 6, 193–205 (2018).

Zhao, Y.

G. J. Zhang, M. Zhang, and Y. Zhao, “Phase modulation characteristics of spatial light modulator and the system for its calibration,” J. Electr. Eng. 6, 193–205 (2018).

Appl. Opt. (2)

J. Electr. Eng. (1)

G. J. Zhang, M. Zhang, and Y. Zhao, “Phase modulation characteristics of spatial light modulator and the system for its calibration,” J. Electr. Eng. 6, 193–205 (2018).

J. Opt. Soc. Am. A (1)

J. Opto-electronics Laser (1)

J. C. Li, “The Accurate Calculation of Fresnel Diffraction and Collins’ Formula by using the fast Fourier transform,” J. Opto-electronics Laser 12(5), 58–64 (2001).

J. Quantum Optics (1)

S. F. Guo, K. Liu, H. X. Sun, J. X. Zhang, and J. R. Gao, “Generation of high-order Laguerre-Guassian beams by liquid crystal spatial light modulators,” J. Quantum Optics 21(1), 86–92 (2015).

Laser Infrared (1)

L. Shi, J. Li, and T. Tao, “Micro-particles’ rotation by Laguerre-Gaussian beams produced by computer- generated holograms,” Laser Infrared 42(11), 1226–1229 (2012).

Opt. Commun. (1)

A. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

Opt. Lett. (1)

Ukr. J. Phys. Opt. (1)

A. Bekshaev, S. Sviridova, A. Popov, A. Rimashevsky, and A. Tyurin, “Optical vortex generation by volume holographic elements with embedded phase singularity: Effects of misalignments,” Ukr. J. Phys. Opt. 14(4), 171–186 (2013).
[Crossref]

Wuli Xuebao (2)

F. Li, C. Q. Gao, Y. D. Liu, and M. W. Tao, “Experimental study of the generation of Laguerre-Gaussian beam using a computer-generated amplitude grating [J],” Wuli Xuebao 57(2), 860–866 (2008).

X. Q. Qi, C. Q. Gao, and Y. D. Liu, “Generation of helical beams with pre-determined energy distribution based on phase modulation gratings,” Wuli Xuebao 59(1), 264–270 (2010).

Other (4)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

G. J. Swanson, “Binary optics technology: The theory and design of multi-level diffractive optical elements,” Lincoln Laboratory Massachusetts Institute of Technology, Lexington, Massachusetts, Technical Report 854, (1989).

G. J. Swanson, “Binary optics technology: Theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” Lincoln Laboratory Massachusetts Institute of Technology, Lexington, Massachusetts, Technical Report 914, (1991).

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

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Figures (23)

Fig. 1
Fig. 1 Coordinates used in discussion of Fresnel and Fraunhofer diffraction. Huygens-Fresnel principle (in Sommerfeld's first diffraction formula, if the distance from the observation point to the diffraction screen is much larger than the wavelength: λ < r)
Fig. 2
Fig. 2 The multilevel phase grating of one period.
Fig. 3
Fig. 3 Diffraction efficiency of multilevel phase grating and distribution of diffracted orders (top: N = 2; mid: N = 4; bottom: N = 8).
Fig. 4
Fig. 4 Photographs of diffraction patterns generated from multi-level phase holograms and binary amplitude holograms, respectively. (lower: from multi-level phase holograms, upper: from binarized amplitude holograms; The blaze number of holograms in both cases is n = 8.)
Fig. 5
Fig. 5 Simulation results of diffraction patterns generated from multi-level phase holograms and binary amplitude holograms, respectively. (upper: from multi-level phase holograms, lower: from binarized amplitude holograms. The blaze number of holograms in three cases are n = 12, 8 and 6 respectively.)
Fig. 6
Fig. 6 Simulation of the generation of LG beam using a computer-generated binary amplitude (AM) grating. The LG beam patterns and phase distributions. Set the magnification so that only the first one in all 16 diffraction orders is retained.
Fig. 7
Fig. 7 Simulated LG beam patterns and phase distributions, diffracted from the SLM with an amplitude mask. Set the propagation distance z to make it retain only the first two of the 16 diffraction orders.
Fig. 8
Fig. 8 The results of Fresnel diffraction integral for LG 5 5 mode output diffracted from blazed phase mask. Blaze number of 16 (left) and blaze number of 4 (right).
Fig. 9
Fig. 9 Extraction of amplitude information of diffraction field (order-cropping), and beam profile comparison between simulation and analytical value.
Fig. 10
Fig. 10 Simulated results of LG beam intensity and phase distributions, diffracted from the SLM with a multilevel phase mask.
Fig. 11
Fig. 11 Left: Simulated results fitting to theoretical value, the signal-to-noise ratio (SNR), Right: LG beam (p = 5, l = 5) profile, comparison between theoretical value and experimental value (inset:photo of beam pattern [1]).
Fig. 12
Fig. 12 Theoretical output mode purities as functions of parameter a = R02/w02.
Fig. 13
Fig. 13 Beam profile comparison between simulation and analytical value (p + l). (p,l: 3,7; 4,6; 5,5).
Fig. 14
Fig. 14 Beam profile comparison between simulation and analytical value (l). (p,l: 0,1; 1,1; 2,1).
Fig. 15
Fig. 15 Phase and intensity pattern of L G 3 7 mode, comparison between theoretical values (the left side of each pair) and simulation results of diffraction output from a phase only SLM.
Fig. 16
Fig. 16 The phase distribution of mode L G 5 5 , simulated vs. analytical results.
Fig. 17
Fig. 17 The phase distribution of mode LG (p = 3, l = 7), simulated and analytical results.
Fig. 18
Fig. 18 The phase distribution of mode LG (p = 3, l = 7), the diffraction outputs of the holographic masks with blazed number of 4,8 and 16, respectively.
Fig. 19
Fig. 19 The phase distribution of mode LG (p = 5, l = 10), simulated and analytical results.
Fig. 20
Fig. 20 The simulation results of the LG light beam’s intensity profile (of mode (p = 1, l = 1)) and its phase distribution, which is generated with a blazed phase gratings.
Fig. 21
Fig. 21 The phase and intensity distribution of the L G 2 10 beam, which is generated with a blazed phase grating.
Fig. 22
Fig. 22 Results of field simulation at different propagation distances.
Fig. 23
Fig. 23 The simulation results for L G 10 10 beam mode.

Tables (2)

Tables Icon

Table 1 Analysis of the fitting errors (case of Blazed numbers 16, LG55 mode)

Tables Icon

Table 2 Error analysis of beam intensity fitting

Equations (48)

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U( ξ,η )= e jkz jλz U( x,y )exp{ jk 2z [ ( ξx ) 2 + ( ηy ) 2 ] }dxdy ,
h z ( x,y )=exp[ jk 2z ( x 2 + y 2 ) ]
U( x,y ) e jk 2z [ x 2 + y 2 ] E rec ( ξ,η )= e jkz jλz F 1 { F[ U 0 ( x,y ) ]F[ h z ( x,y ) ] }
A 0 ( f x , f y ,0 )= U( x,y,0 )exp{ j2π( x f x +y f y ) }dxdy
A( cosα λ , cosβ λ ,z )=A( cosα λ , cosβ λ ,0 )exp{ jkz 1 cos 2 α cos 2 β }
U( ξ,η,z )= A 0 ( f x , f y ,0 )exp{ j2π λ z[ 1 ( λ f x ) 2 ( λ f y ) 2 ] }d f x d f y
E rec ( ξ,η )= F 1 { F[ U 0 ( x,y ) ]exp( j 2π λ z 1 ( λ f x ) 2 ( λ f y ) 2 ) }.
F{ U( ξ,η ) }=F{ U( x,y ) }F{ exp(jkd) jλd exp( j k 2d ( ξ 2 + η 2 ) ) }==F[ U( x,y ) ]exp( jkd[ 1 λ 2 2 ( f x 2 + f y 2 ) ] )
F{ U( x,y ) }=F{ U( ξ,η ) }exp( jkd[ 1 λ 2 2 ( f x 2 + f y 2 ) ] )=F[ U( x,y ) ]F{ exp(jkd) jλd exp( j k 2d ( x 2 + y 2 ) ) }
Δ x = L x /M; Δ y = L y /N;Δ f x =1/( M Δ x );Δ f y =1/( N Δ y );
x={ M/2,M/2+1,,M/21,M/2 } Δ x ;y={ N/2,N/2+1,,N/21,N/2 } Δ y
f x ( i )={ M/2,M/2+1,,M/21,M/2 }Δ f x ; f y ( i )={ N/2,N/2+1,,N/21,N/2 }Δ f y
( ξ i , η i )=( m i 2 M x i , n i 2 N x i )λz=( f x , f y )λz
Z= ξ λ f x = η λ f y FFT one period ξ max λ f xmax = M Δ ξ max λ( 1/ Δ x ) Δ ξ max = Δ x M max Δ x 2 λ
u p l ( r,φ,z )= ( 1 ) p ω 2 π p! ( p+| l | )! ( 2 r 2 ω 2 ) | l |/2 L p | l | ( 2 r 2 ω 2 )exp( ilφ i r 2 z ω 2 z R )exp[ i( 2p+| l |+1 ) tan 1 ( z z R ) ]
ϕ h ( r,φ,z )=arg[ u p l ( r,φ,z ) ],
ϕ b ( r,φ,0 )=m 2π Λ rcosφ
z 0 = x max Λ m max λ = M Δ pix 2 λ .
R'( x )= k=0 N1 rect( xkd' d' ) e ik 2π λ 2 Δ h .
R'( x )= k=0 N1 rect( xkd' d' ) e i2kπd' u 0
F[ R'( x ) ]= e i( N1 )πd'( u 0 u ) d'sinc( d'u ) sin[ Nπ( u u 0 )d' ] sin[ π( u u 0 )d' ]
R( x )= m δ( xmd ) R'( x )=δ( x )*R'( x )
E ˜ =F[ A 0 R( x ) ]= A 0 d' m sinc( d'u ) sin[ Nπ( u u 0 )d' ] sin[ π( u u 0 )d' ] 1 d δ( u m d )
E ˜ = m sin( mπ/N ) mπ sin[ π( m 2dsinγ λ ) ] sin[ π( m 2dsinγ λ )/N ] δ( u m d )
I= [ sin( mπ/N ) mπ sin[ π( m 2dsinγ λ ) ] sin[ π( m 2dsinγ λ )/N ] δ( u m d ) ] 2
2π λ 2 Δ h = 2π N .
η= [ sin( mπ/N ) mπ sin[ π( m1 ) ] sin[ π( m1 )/N ] ] 2
η= [ sin( π/N ) π/N ] 2
ϕ( ρ,φ,z )=lφ+kxsinθ
a( ϕ )={ 1 mod(ϕ,2π)<π 0 other
a( ϕ )= n= A n e inϕ
A n = sin( nπ/2 ) nπ e inπ/2
u in ( ρ,φ )= 2 π 1 ω 0 exp[ ( ρ ω 0 ) 2 ]a( ϕ )= u 00 A n e inφ
u far ( ρ',φ' )= n= A n F[ u 00 ( ρ,φ ) e inlφ e inkxsinθ ]= n= A n F[ u 00 e inlϕ ] F[ e inkxsinθ ]
u far ( ρ',φ' )=F[ u in ( ρ,φ ) ]=F[ u 00 n= A n e inϕ ]= n= A n F[ u 00 e inϕ ]
u far ( ρ',φ' )= n= A n p c p i 2p+| nl | u p,nl ( ρ',φ' )δ( v x + nksinθ 2π )
c p = 0 2π 0 u 00 ( ρ,φ ) e inlφ u p,nl * ( ρ,φ )ρdρdφ
SNR= n I the ( n ) / n | I the ( n ) I cal ( n ) |
R1=1 Q/ n I the 2 ( n )
R2=1 Q/ n I cal 2 ( n )
MAE= n | I the ( n ) I cal ( n ) | /N
RMSE= n ( I the ( n ) I cal ( n ) ) 2 /N
ϕ(r,φ,z)=lφ+πΘ[ L p | l | ( 2 r 2 / ω 0 2 ) ]
A( r )= A 0 Θ( R 0 r )
0 2π 0 A 2 ( r ) rdrdφ=1
u p l ( r,φ,0 )=A( r )exp( ilφ )= 1 π R 0 Θ( R 0 r ) e ilφ+iπΘ[ L p || l | ( 2 r 2 / ω 0 2 ) ]
c q k = u q k ( r,φ,0 )| ( u p l ( r,φ,0 ) ) * = 1 π R 0 0 2π 0 R 0 u q k ( r,φ,0 ) e ilφ { 2Θ[ L p | l | ( 2 r 2 / ω 0 2 ) ]1 }rdrdφ
c q k = ( 1 ) q 2 a q! ( q+| l | )! 0 2 a 2 ζ | l |/2 e ζ/2 L p | l | ( ζ ){ 2Θ[ L p | l | ( ζ ) ]1 }dζ

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