Abstract

The focusing properties and resolving power of a device consisting of a tapered gradient-index (GRIN) lens with spherical input and output faces are investigated through the use of the ABCD formalism to achieve minimization of the Airy radius for the device. Diffractive elements, such as zone plates, can, with an appropriate choice of their parameters, increase the resolution of an imaging system compared with a conventional lens. We demonstrate that by combining both elements a hybrid refractive–diffractive–GRIN device can be designed that exhibits improved superresolution characteristics.

© 2008 Optical Society of America

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References

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  1. C. Di, C. Zhou, Y. Cao, and E. Dai, “Application of optical superresolution in read-only optical disk system,” in Conference on Lasers and Electro-Optics--Pacific Rim, 2007. CLEO/Pacific Rim 2007 (IEEE, 2007), pp. 1-2.
    [CrossRef]
  2. H. Liu, Y. Yan, D. Yi, and G. Jin, “Design of three-dimensional superresolution filters and limits of axial optical superresolution,” Appl. Opt. 42, 1463-1476 (2003).
    [CrossRef] [PubMed]
  3. H. Fukuda and R. Yamanaka, “A new pupil filter for annular illumination in optical lithography,” Jpn. J. Appl. Phys. 31, 4126-4130 (1992).
    [CrossRef]
  4. B. Lee, W. Y. Choi, and J. K. Walker, “Ultrahigh-resolution plastic graded-index fused image plates,” Opt. Lett. 25, 719-721 (2000).
    [CrossRef]
  5. R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field--I. Achievement of maximum central irradiance under an energy constraint,” J. Mod. Opt. 27, 587-610 (1980).
    [CrossRef]
  6. M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267-278 (1999).
    [CrossRef]
  7. H. Luo and C. Zhou, “Comparison of superresolution effects with annular phase and amplitude filters,” Appl. Opt. 43, 6242-6247 (2004).
    [CrossRef] [PubMed]
  8. T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14, 1637-1646 (1997).
    [CrossRef]
  9. H. Liu, Y. Yan, D. Yi, and G. Jin, “Design and experimental test of diffractive superresolution elements,” Appl. Opt. 45, 95-99 (2006).
    [CrossRef] [PubMed]
  10. C. J. R. Sheppard, G. Calvert, and M. Wheatland, “Focal distribution for superresolving Toraldo filters,” J. Opt. Soc. Am. A 15, 849-856 (1998).
    [CrossRef]
  11. J.Ojeda-Castañeda and C.Gómez-Reino, eds., Selected Papers on Zone Plates, Vol. MS 128 of SPIE Milestone Series (SPIE, 1996), and references therein.
  12. H. Liu, Y. Yan, D. Yi, and G. Jin, “Theories for the design of a hybrid refractive-diffractive superresolution lens with high numerical aperture,” J. Opt. Soc. Am. A 20, 913-924 (2003).
    [CrossRef]
  13. J. M. Rivas-Moscoso, C. Gómez-Reino, C. Bao, and M. V. Pérez, “Tapered gradient-index media and zone plates,” J. Mod. Opt. 47, 1549-1567 (2000).
  14. J. M. Rivas-Moscoso, D. Nieto, C. Gómez-Reino, and C. R. Fernández-Pousa, “Focusing of light by zone plates in Selfoc gradient-index lenses,” Opt. Lett. 28, 2180-2182 (2003).
    [CrossRef] [PubMed]
  15. C. Gómez-Reino, M. V. Pérez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).
  16. B. E. A. Saleh and M. C. T. Teich, Fundamentals of Photonics (Wiley, 1991).
    [CrossRef]
  17. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1997).
  18. J. M. Rivas-Moscoso, C. Gómez-Reino, M. V. Pérez Martín, and C. Bao Varela, “Marginal rays in tapered gradient-index lenses,” Opt. Eng. 41, 303-313 (2002).
    [CrossRef]
  19. J. M. Rivas-Moscoso and C. Gómez-Reino, “Resolving power of a hybrid zone-plate/gradient-index lens system,” Proc. SPIE 5622, 706-708 (2004).

2007 (1)

C. Di, C. Zhou, Y. Cao, and E. Dai, “Application of optical superresolution in read-only optical disk system,” in Conference on Lasers and Electro-Optics--Pacific Rim, 2007. CLEO/Pacific Rim 2007 (IEEE, 2007), pp. 1-2.
[CrossRef]

2006 (1)

2004 (2)

J. M. Rivas-Moscoso and C. Gómez-Reino, “Resolving power of a hybrid zone-plate/gradient-index lens system,” Proc. SPIE 5622, 706-708 (2004).

H. Luo and C. Zhou, “Comparison of superresolution effects with annular phase and amplitude filters,” Appl. Opt. 43, 6242-6247 (2004).
[CrossRef] [PubMed]

2003 (3)

2002 (2)

J. M. Rivas-Moscoso, C. Gómez-Reino, M. V. Pérez Martín, and C. Bao Varela, “Marginal rays in tapered gradient-index lenses,” Opt. Eng. 41, 303-313 (2002).
[CrossRef]

C. Gómez-Reino, M. V. Pérez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

2000 (2)

J. M. Rivas-Moscoso, C. Gómez-Reino, C. Bao, and M. V. Pérez, “Tapered gradient-index media and zone plates,” J. Mod. Opt. 47, 1549-1567 (2000).

B. Lee, W. Y. Choi, and J. K. Walker, “Ultrahigh-resolution plastic graded-index fused image plates,” Opt. Lett. 25, 719-721 (2000).
[CrossRef]

1999 (1)

M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267-278 (1999).
[CrossRef]

1998 (1)

1997 (2)

1996 (1)

J.Ojeda-Castañeda and C.Gómez-Reino, eds., Selected Papers on Zone Plates, Vol. MS 128 of SPIE Milestone Series (SPIE, 1996), and references therein.

1992 (1)

H. Fukuda and R. Yamanaka, “A new pupil filter for annular illumination in optical lithography,” Jpn. J. Appl. Phys. 31, 4126-4130 (1992).
[CrossRef]

1991 (1)

B. E. A. Saleh and M. C. T. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

1980 (1)

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field--I. Achievement of maximum central irradiance under an energy constraint,” J. Mod. Opt. 27, 587-610 (1980).
[CrossRef]

Andrés, P.

M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267-278 (1999).
[CrossRef]

Bao, C.

C. Gómez-Reino, M. V. Pérez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

J. M. Rivas-Moscoso, C. Gómez-Reino, C. Bao, and M. V. Pérez, “Tapered gradient-index media and zone plates,” J. Mod. Opt. 47, 1549-1567 (2000).

Bao Varela, C.

J. M. Rivas-Moscoso, C. Gómez-Reino, M. V. Pérez Martín, and C. Bao Varela, “Marginal rays in tapered gradient-index lenses,” Opt. Eng. 41, 303-313 (2002).
[CrossRef]

Boivin, A.

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field--I. Achievement of maximum central irradiance under an energy constraint,” J. Mod. Opt. 27, 587-610 (1980).
[CrossRef]

Boivin, R.

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field--I. Achievement of maximum central irradiance under an energy constraint,” J. Mod. Opt. 27, 587-610 (1980).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1997).

Calvert, G.

Cao, Y.

C. Di, C. Zhou, Y. Cao, and E. Dai, “Application of optical superresolution in read-only optical disk system,” in Conference on Lasers and Electro-Optics--Pacific Rim, 2007. CLEO/Pacific Rim 2007 (IEEE, 2007), pp. 1-2.
[CrossRef]

Choi, W. Y.

Dai, E.

C. Di, C. Zhou, Y. Cao, and E. Dai, “Application of optical superresolution in read-only optical disk system,” in Conference on Lasers and Electro-Optics--Pacific Rim, 2007. CLEO/Pacific Rim 2007 (IEEE, 2007), pp. 1-2.
[CrossRef]

Di, C.

C. Di, C. Zhou, Y. Cao, and E. Dai, “Application of optical superresolution in read-only optical disk system,” in Conference on Lasers and Electro-Optics--Pacific Rim, 2007. CLEO/Pacific Rim 2007 (IEEE, 2007), pp. 1-2.
[CrossRef]

Fernández-Pousa, C. R.

Fukuda, H.

H. Fukuda and R. Yamanaka, “A new pupil filter for annular illumination in optical lithography,” Jpn. J. Appl. Phys. 31, 4126-4130 (1992).
[CrossRef]

Gómez-Reino, C.

J. M. Rivas-Moscoso and C. Gómez-Reino, “Resolving power of a hybrid zone-plate/gradient-index lens system,” Proc. SPIE 5622, 706-708 (2004).

J. M. Rivas-Moscoso, D. Nieto, C. Gómez-Reino, and C. R. Fernández-Pousa, “Focusing of light by zone plates in Selfoc gradient-index lenses,” Opt. Lett. 28, 2180-2182 (2003).
[CrossRef] [PubMed]

J. M. Rivas-Moscoso, C. Gómez-Reino, M. V. Pérez Martín, and C. Bao Varela, “Marginal rays in tapered gradient-index lenses,” Opt. Eng. 41, 303-313 (2002).
[CrossRef]

C. Gómez-Reino, M. V. Pérez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

J. M. Rivas-Moscoso, C. Gómez-Reino, C. Bao, and M. V. Pérez, “Tapered gradient-index media and zone plates,” J. Mod. Opt. 47, 1549-1567 (2000).

Jin, G.

Kowalczyk, M.

M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267-278 (1999).
[CrossRef]

Lee, B.

Liu, H.

Luo, H.

Martínez-Corral, M.

M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267-278 (1999).
[CrossRef]

Morris, G. M.

Nieto, D.

Pérez, M. V.

C. Gómez-Reino, M. V. Pérez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

J. M. Rivas-Moscoso, C. Gómez-Reino, C. Bao, and M. V. Pérez, “Tapered gradient-index media and zone plates,” J. Mod. Opt. 47, 1549-1567 (2000).

Pérez Martín, M. V.

J. M. Rivas-Moscoso, C. Gómez-Reino, M. V. Pérez Martín, and C. Bao Varela, “Marginal rays in tapered gradient-index lenses,” Opt. Eng. 41, 303-313 (2002).
[CrossRef]

Rivas-Moscoso, J. M.

J. M. Rivas-Moscoso and C. Gómez-Reino, “Resolving power of a hybrid zone-plate/gradient-index lens system,” Proc. SPIE 5622, 706-708 (2004).

J. M. Rivas-Moscoso, D. Nieto, C. Gómez-Reino, and C. R. Fernández-Pousa, “Focusing of light by zone plates in Selfoc gradient-index lenses,” Opt. Lett. 28, 2180-2182 (2003).
[CrossRef] [PubMed]

J. M. Rivas-Moscoso, C. Gómez-Reino, M. V. Pérez Martín, and C. Bao Varela, “Marginal rays in tapered gradient-index lenses,” Opt. Eng. 41, 303-313 (2002).
[CrossRef]

J. M. Rivas-Moscoso, C. Gómez-Reino, C. Bao, and M. V. Pérez, “Tapered gradient-index media and zone plates,” J. Mod. Opt. 47, 1549-1567 (2000).

Saleh, B. E. A.

B. E. A. Saleh and M. C. T. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Sales, T. R. M.

Sheppard, C. J. R.

Teich, M. C. T.

B. E. A. Saleh and M. C. T. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Walker, J. K.

Wheatland, M.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1997).

Yamanaka, R.

H. Fukuda and R. Yamanaka, “A new pupil filter for annular illumination in optical lithography,” Jpn. J. Appl. Phys. 31, 4126-4130 (1992).
[CrossRef]

Yan, Y.

Yi, D.

Zapata-Rodríguez, C. J.

M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267-278 (1999).
[CrossRef]

Zhou, C.

C. Di, C. Zhou, Y. Cao, and E. Dai, “Application of optical superresolution in read-only optical disk system,” in Conference on Lasers and Electro-Optics--Pacific Rim, 2007. CLEO/Pacific Rim 2007 (IEEE, 2007), pp. 1-2.
[CrossRef]

H. Luo and C. Zhou, “Comparison of superresolution effects with annular phase and amplitude filters,” Appl. Opt. 43, 6242-6247 (2004).
[CrossRef] [PubMed]

Appl. Opt. (3)

J. Mod. Opt. (2)

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field--I. Achievement of maximum central irradiance under an energy constraint,” J. Mod. Opt. 27, 587-610 (1980).
[CrossRef]

J. M. Rivas-Moscoso, C. Gómez-Reino, C. Bao, and M. V. Pérez, “Tapered gradient-index media and zone plates,” J. Mod. Opt. 47, 1549-1567 (2000).

J. Opt. Soc. Am. A (3)

Jpn. J. Appl. Phys. (1)

H. Fukuda and R. Yamanaka, “A new pupil filter for annular illumination in optical lithography,” Jpn. J. Appl. Phys. 31, 4126-4130 (1992).
[CrossRef]

Opt. Commun. (1)

M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267-278 (1999).
[CrossRef]

Opt. Eng. (1)

J. M. Rivas-Moscoso, C. Gómez-Reino, M. V. Pérez Martín, and C. Bao Varela, “Marginal rays in tapered gradient-index lenses,” Opt. Eng. 41, 303-313 (2002).
[CrossRef]

Opt. Lett. (2)

Proc. SPIE (1)

J. M. Rivas-Moscoso and C. Gómez-Reino, “Resolving power of a hybrid zone-plate/gradient-index lens system,” Proc. SPIE 5622, 706-708 (2004).

Other (5)

J.Ojeda-Castañeda and C.Gómez-Reino, eds., Selected Papers on Zone Plates, Vol. MS 128 of SPIE Milestone Series (SPIE, 1996), and references therein.

C. Di, C. Zhou, Y. Cao, and E. Dai, “Application of optical superresolution in read-only optical disk system,” in Conference on Lasers and Electro-Optics--Pacific Rim, 2007. CLEO/Pacific Rim 2007 (IEEE, 2007), pp. 1-2.
[CrossRef]

C. Gómez-Reino, M. V. Pérez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

B. E. A. Saleh and M. C. T. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1997).

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Figures (10)

Fig. 1
Fig. 1

Schematic for the calculation of the Airy radius for an imaging system composed of two media with refractive indices n 1 and n 2 separated by a spherical boundary of radius of curvature R.

Fig. 2
Fig. 2

Geometry for the analysis of the resolving power of a tapered GRIN lens with spherical input and output faces of radii of curvature R 1 and R 2 .

Fig. 3
Fig. 3

Solid curves, Airy radius, r Ai , as a function of the GRIN lens length L for a Selfoc lens with (a) flat input and output faces, (b) a flat input face and an output curvature of 0.1 mm 1 , (e) an input curvature of 0.5 mm 1 and a flat output face, and (f) an input curvature of 0.5 mm 1 and an output curvature of 0.1 mm 1 . Dashed curves, distance from the output face of the lens to the image plane, z , as a function of the GRIN lens length L for a Selfoc lens with (c) flat input and output faces, (d) a flat input face and an output curvature of 0.1 mm 1 , (g) an input curvature of 0.5 mm 1 and a flat output face, and (h) an input curvature of 0.5 mm 1 and an output curvature of 0.1 mm 1 . Calculations were made for λ = 1.3 μm , d = 10 mm , a = 1 mm , n 0 = 1.5 , n 1 = n 3 = 1 , and g 0 = 0.2 mm 1 .

Fig. 4
Fig. 4

Minimum Airy radius, r Ai , as a function of the gradient function, g 0 . Calculations were made for λ = 1.3 μm , d = 10 mm , a = 1 mm , n 0 = 1.5 , n 1 = n 3 = 1 , and R 1 = R 2 .

Fig. 5
Fig. 5

Minimum Airy radius, r Ai , as a function of (a)  1 / R 1 with 1 / R 2 = 0 and g 0 = 0.2 mm 1 , (b)  1 / R 1 with 1 / R 2 = 0.5 mm 1 and g 0 = 0.2 mm 1 , (c)  1 / R 2 with g 0 = 0.2 mm 1 , (d)  1 / R 2 with g 0 = 0.3 mm 1 . Calculations were made for λ = 1.3 μm , d = 10 mm , a = 1 mm , n 0 = 1.5 , and n 1 = n 3 = 1 .

Fig. 6
Fig. 6

Minimum Airy radius, r Ai , as a function of (a)  n 1 = n 0 = n 3 with 1 / R 2 = 0 , (b)  n 3 with n 0 = 1.5 and 1 / R 2 = 0.07 mm 1 , (c)  n 0 with n 3 = 1.5 and 1 / R 2 = 0.07 mm 1 , (d)  n 0 with n 3 = 1 and 1 / R 2 = 0.07 mm 1 . Calculations were made for λ = 1.3 μm , d = 10 mm , a = 1 mm , and g 0 = 0.2 mm 1 .

Fig. 7
Fig. 7

Geometry for the study of superresolution with a hybrid refractive–diffractive–GRIN structure.

Fig. 8
Fig. 8

Normalized irradiance distribution in terms of the transversal variable r and parameter C 0 for a Selfoc lens (a) in the absence of a ZP with a eff = 0.949 mm ; (b) with a phase Fresnel ZP of period p = 0.232 mm 2 with N = 8 , ZP diffraction order m = 1 , and a eff ZP = 0.964 mm ; (c) with a negative amplitude Fresnel ZP of period p = 0.232 mm 2 with N = 8 , m = 1 , and a eff ZP = 0.963 mm ; (d) with a negative amplitude Fresnel ZP of period p = 0.9 mm 2 with N = 2 , m = 1 , and a eff ZP = 0.949 mm ; (e) with a negative amplitude Fresnel ZP of period p = 0.9 mm 2 with N = 2 , m = 0 , and a eff ZP = 0.948 mm . Calculations were made for λ = 1.3 μm , d = 10 mm , a = 1 mm , n 0 = 1.5 , n 1 = n 3 = 1 , R 1 = R 2 , g 0 = 0.2 mm 1 . Inset, enlargement of the irradiance patterns.

Fig. 9
Fig. 9

C 0 as a function of the number of zones N for a positive amplitude Fresnel ZP, a negative amplitude Fresnel ZP, and a phase Fresnel ZP.

Fig. 10
Fig. 10

Minimum Airy radius, r Ai , as a function of length L for a Selfoc lens (a) with a phase ZP (or in the absence of a ZP), (b) with a negative amplitude ZP with N = 8 , and (c) with a negative amplitude ZP with N = 2 , in all cases with R 1 = R 2 . (d) The same as (c) but with 1 / R 2 = 0.1 mm 1 . Calculations were made for λ = 1.3 μm , d = 10 mm , a = 1 mm , n 0 = 1.5 , n 1 = n 3 = 1 , g 0 = 0.2 mm 1 , and ZP diffraction order m = 1 .

Equations (42)

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ψ ( r ; z ) = 0 2 π 0 a ψ ( r 0 ) K ( r , r 0 , Ω , Ω 0 ; z ) r 0 d r 0 d Ω 0 ,
K ( r , r 0 , Ω , Ω 0 ; z ) = i k 2 π B exp ( i k n 2 z ) exp { i k 2 B [ D r 2 + A r 0 2 2 r r 0 cos ( Ω Ω 0 ) ] } .
n 1 d + A B = n 2 n 1 R ,
ψ ( r ; z ) = i k B d exp [ i k ( n 1 d + n 2 z ) ] exp [ i k D 2 B r 2 ] 0 a J 0 ( k B r r 0 ) r 0 d r 0 ,
I ( r ; z ) = k 2 a 4 4 B 2 [ 2 J 1 ( k a r / B ) k a r / B ] 2 ,
r spot = 1.22 λ 2 a | B | .
n 2 2 ( r , z ) = n 0 2 [ 1 g 2 ( z ) r 2 ] ,
M 2 = [ A 2 B 2 C 2 D 2 ] = [ 1 z 0 1 ] [ 1 0 n 3 n 0 n 3 R 2 n 0 n 3 ] [ H 2 ( L ) H 1 ( L ) H ˙ 2 ( L ) H ˙ 1 ( L ) ] ,
H 1 ( z , z 0 ) = [ g ( z 0 ) g ( z ) ] 1 / 2 sin [ z 0 z g ( z ˜ ) d z ˜ ] = [ g ( z 0 ) g ( z ) ] 1 H ˙ 2 ( z , z 0 ) ,
H 2 ( z , z 0 ) = [ g ( z 0 ) / g ( z ) ] 1 / 2 cos [ z 0 z g ( z ˜ ) d z ˜ ] = g ( z 0 ) g ( z ) H ˙ 1 ( z , z 0 ) ,
r spot = 1.22 λ 2 a eff | H 1 ( L ) + n 0 z [ ( 1 n 0 1 n 3 ) H 1 ( L ) R 2 + H ˙ 1 ( L ) n 3 ] | ,
[ r ( z ) r ˙ ( z ) ] = [ H 2 ( z ) H 1 ( z ) H ˙ 2 ( z ) H ˙ 1 ( z ) ] [ r ( 0 ) r ˙ ( 0 ) ] ,
M 1 = [ A 1 B 1 C 1 D 1 ] = [ 1 0 ( n 0 n 1 ) n 0 R 1 n 1 n 0 ] [ 1 d 0 1 ] ,
A ( R 1 ) = n 1 d n 0 n 1 R 1 ,
a = a eff { H 2 ( z m ) + [ A ( R 1 ) / n 0 ] H 1 ( z m ) } ,
0 = a eff { H ˙ 2 ( z m ) + [ A ( R 1 ) / n 0 ] H ˙ 1 ( z m ) } ,
tan [ 0 z m g ( z ˜ ) d z ˜ ] = A ( R 1 ) / ( n 0 g 0 ) ,
a eff = [ g 0 g ( z m ) n 0 2 g 0 2 d 2 + ( n 1 n 0 n 1 R 1 d ) 2 ] 1 / 2 n 0 d a .
z = n 3 R 2 F ( R 1 , L ) ( n 3 n 0 ) F ( R 1 , L ) + n 0 R 2 F ˙ ( R 1 , L ) ,
F ( · ) ( R 1 , L ) = H 2 ( · ) ( L ) + H 1 ( · ) ( L ) n 0 A ( R 1 ) .
tan [ 0 L g ( z ˜ ) d z ˜ ] = ( n 3 n 0 ) A ( R 1 ) n 0 2 g 0 g ( L ) R 2 n 0 [ g 0 ( n 3 n 0 ) + g ( L ) R 2 A ( R 1 ) ] ,
tan [ 0 L g ( z ˜ ) d z ˜ ] = n 0 g 0 A ( R 1 ) .
L = 1 g 0 tan 1 { ( n 3 n 0 ) A ( R 1 ) n 0 2 g 0 2 R 2 n 0 g 0 [ ( n 3 n 0 ) + R 2 A ( R 1 ) ] } ,
L = 1 g 0 tan 1 [ n 0 g 0 A ( R 1 ) ] .
M 1 = [ A 1 B 1 C 1 D 1 ] = [ 1 0 2 m λ p 1 ] [ H 2 ( L ) H 1 ( L ) H ˙ 2 ( L ) H ˙ 1 ( L ) ] [ 1 0 ( n 0 n 1 ) n 0 R 1 n 1 n 0 ] [ 1 d 0 1 ] ,
M 2 = [ A 2 B 2 C 2 D 2 ] = [ 1 z 0 1 ] [ 1 0 n 3 n 0 n 3 R 2 n 0 n 3 ] [ H 2 ( L , L ) H 1 ( L , L ) H ˙ 2 ( L , L ) H ˙ 1 ( L , L ) ] ,
z = n 3 R 2 F m ( R 1 , L , L ) ( n 3 n 0 ) F m ( R 1 , L , L ) + n 0 R 2 F ˙ m ( R 1 , L , L ) ,
F m ( · ) ( R 1 , L , L ) = H 2 ( · ) ( L , L ) + H 1 ( · ) ( L , L ) A m ( R 1 , L ) ,
A m ( R 1 , L ) = F ˙ ( R 1 , L ) F ( R 1 , L ) 2 m λ p .
a = a eff [ H 2 ( z m , L ) + A m ( R 1 , L ) H 1 ( z m , L ) ] ,
0 = a eff [ H ˙ 2 ( z m , L ) + A m ( R 1 , L ) H ˙ 1 ( z m , L ) ] ,
tan [ L z m g ( z ˜ ) d z ˜ ] = A m ( R 1 , L ) / g ( L ) .
a eff = [ g ( L ) g ( z m ) g 2 ( L ) + A m 2 ( R 1 , L , L ) ] 1 / 2 a .
ψ ( r ; z ) = ρ 1 ρ 2 i k B 2 d exp [ i k ( n 1 d + n 0 z ) ] exp [ i k 2 d n 1 n 0 r 0 2 ] exp [ i k 2 R 1 ( n 0 n 1 ) r 0 2 ] exp [ i k 2 B 2 ( D 2 r 2 + A 2 r 0 2 ) ] J 0 ( k B 2 r r 0 ) r 0 d r 0 ,
ρ ( 1 2 ) = [ ( N 1 N 2 ) ( a eff ZP ) 2 N ] 1 / 2 .
a eff ZP = h 1 ( a eff 2 h 1 2 ) 1 / 2 if    a eff 2 h 1 2 is an odd number , a eff ZP = a eff otherwise
a eff ZP = h 1 ( a eff 2 h 1 2 ) 1 / 2 if a eff 2 h 1 2 is an even number , a eff ZP = a eff otherwise
r spot = C 0 λ 2 a eff | H 1 ( L , L ) + n 0 z [ ( 1 n 0 1 n 3 ) H 1 ( L , L ) R 2 + H ˙ 1 ( L , L ) n 3 ] | ,
tan [ L L g ( z ˜ ) d z ˜ ] = ( n 3 n 0 ) A m ( R 1 , L ) n 0 g ( L ) g ( L ) R 2 g ( L ) ( n 3 n 0 ) + n 0 g ( L ) R 2 A m ( R 1 , L ) ,
tan [ L L g ( z ˜ ) d z ˜ ] = g ( L ) A m ( R 1 , L ) .
L = L + 1 g 0 tan 1 { ( n 3 n 0 ) A m ( R 1 , L ) n 0 g 0 2 R 2 g 0 [ ( n 3 n 0 ) + n 0 R 2 A m ( R 1 , L ) ] } ,
L = L 1 g 0 tan 1 [ g 0 A m ( R 1 , L ) ] .

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