Abstract

Optical disks are read out by focusing a beam of high numerical aperture (NA) through the substrate. Deviations of the thickness from the nominal value result in spherical aberration; tilting the substrate results in coma. Exact analytical expressions for the rms aberration per micrometer thickness mismatch (for spherical aberration) and per degree tilt (for coma) are derived. The paraxial estimates for these sensitivities proportional to NA4 (spherical aberration) and NA3 (coma) underestimate the exact values by a factor of ~2 for the value NA = 0.85, corresponding to the new Blu-ray disk format. Expansion of the aberration function in Zernike aberrations shows that the exact aberration functions are well described by the lowest-order Zernike spherical aberration (A40) and coma (A31) term for all but the very highest NA values.

© 2005 Optical Society of America

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References

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  1. M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
    [CrossRef]
  2. I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
    [CrossRef]
  3. M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
    [CrossRef]
  4. T. Sugaya, “A basic concept for next generation DVD: 0.6 mm substrate disk technology using blue-violet laser,” in Optical Data Storage 2003, M. O’Neill, N. Miyagawa, eds., Proc. SPIE5069, 278–280 (2003).
  5. W. T. Welford, Aberrations of Optical Systems (Hilger, London, 1986).
  6. G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, London, 1985).
  7. J. Braat, “Analytical expressions for the wave-front aberration coefficients of a tilted plane-parallel plate,” Appl. Opt. 36, 8459–8466 (1997).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge University, Cambridge, UK, 1980).
  9. S. Wolfram, Mathematica. A System for Mathematics by Computer, 2nd ed. (Addison-Wesley, Reading, Mass., 1991).
  10. W. J. Smith, Modern Optical Engineering. The Design of Optical Systems, 2nd ed. (McGraw-Hill, Boston, Mass., 1990).
  11. P. Asthana, B. I. Finkelstein, A. A. Fennema, “Rewritable optical disk drive technology,” IBM J. Res. Dev. 40, 543–558 (1996).
    [CrossRef]
  12. S. Stallinga, J. J. Vrehen, J. Wals, E. Stapert, E. Verstegen, “Liquid crystal aberration compensation devices,” in Optical Storage and Information Processing, H.-P. D. Shieh, T. D. Milster, eds., Proc. SPIE4081, 50–56 (2000).
    [CrossRef]
  13. S. Stallinga, J. J. Vrehen, “Advances in liquid crystal tilt compensation,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 465–473 (2001).
    [CrossRef]
  14. H. R. Stapert, J. Lub, E. J. K. Verstegen, B. M. I. van der Zande, S. Stallinga, “Photo replicated anisotropic liquid crystalline lenses for aberration control and dual layer read out of optical disks,” Adv. Functional Mater. 13, 732–738 (2003).
    [CrossRef]
  15. M. P. van der Meulen, Institute for Theoretical Physics, Valckenierstraat 65 1018 XE Amsterdam, The Netherlands (personal communication, 2001).

2003

H. R. Stapert, J. Lub, E. J. K. Verstegen, B. M. I. van der Zande, S. Stallinga, “Photo replicated anisotropic liquid crystalline lenses for aberration control and dual layer read out of optical disks,” Adv. Functional Mater. 13, 732–738 (2003).
[CrossRef]

1997

1996

P. Asthana, B. I. Finkelstein, A. A. Fennema, “Rewritable optical disk drive technology,” IBM J. Res. Dev. 40, 543–558 (1996).
[CrossRef]

Asthana, P.

P. Asthana, B. I. Finkelstein, A. A. Fennema, “Rewritable optical disk drive technology,” IBM J. Res. Dev. 40, 543–558 (1996).
[CrossRef]

Borg, H. J.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge University, Cambridge, UK, 1980).

Bouwhuis, G.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, London, 1985).

Braat, J.

J. Braat, “Analytical expressions for the wave-front aberration coefficients of a tilted plane-parallel plate,” Appl. Opt. 36, 8459–8466 (1997).
[CrossRef]

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, London, 1985).

Coene, W. M.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

Dekker, M. K.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

Fennema, A. A.

P. Asthana, B. I. Finkelstein, A. A. Fennema, “Rewritable optical disk drive technology,” IBM J. Res. Dev. 40, 543–558 (1996).
[CrossRef]

Finkelstein, B. I.

P. Asthana, B. I. Finkelstein, A. A. Fennema, “Rewritable optical disk drive technology,” IBM J. Res. Dev. 40, 543–558 (1996).
[CrossRef]

Huijser, A.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, London, 1985).

Ichimura, I.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Kasami, Y.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Kawakubo, O.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Kuijper, M.

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

Lub, J.

H. R. Stapert, J. Lub, E. J. K. Verstegen, B. M. I. van der Zande, S. Stallinga, “Photo replicated anisotropic liquid crystalline lenses for aberration control and dual layer read out of optical disks,” Adv. Functional Mater. 13, 732–738 (2003).
[CrossRef]

Masuhara, S.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Meinders, E. R.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

Nakano, J.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Osato, K.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Pasman, J.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, London, 1985).

Pfeffer, N.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

Schep, K.

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

Schouhamer Immink, K.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, London, 1985).

Smith, W. J.

W. J. Smith, Modern Optical Engineering. The Design of Optical Systems, 2nd ed. (McGraw-Hill, Boston, Mass., 1990).

Spruijt, L.

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

Stallinga, S.

H. R. Stapert, J. Lub, E. J. K. Verstegen, B. M. I. van der Zande, S. Stallinga, “Photo replicated anisotropic liquid crystalline lenses for aberration control and dual layer read out of optical disks,” Adv. Functional Mater. 13, 732–738 (2003).
[CrossRef]

S. Stallinga, J. J. Vrehen, “Advances in liquid crystal tilt compensation,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 465–473 (2001).
[CrossRef]

S. Stallinga, J. J. Vrehen, J. Wals, E. Stapert, E. Verstegen, “Liquid crystal aberration compensation devices,” in Optical Storage and Information Processing, H.-P. D. Shieh, T. D. Milster, eds., Proc. SPIE4081, 50–56 (2000).
[CrossRef]

Stapert, E.

S. Stallinga, J. J. Vrehen, J. Wals, E. Stapert, E. Verstegen, “Liquid crystal aberration compensation devices,” in Optical Storage and Information Processing, H.-P. D. Shieh, T. D. Milster, eds., Proc. SPIE4081, 50–56 (2000).
[CrossRef]

Stapert, H. R.

H. R. Stapert, J. Lub, E. J. K. Verstegen, B. M. I. van der Zande, S. Stallinga, “Photo replicated anisotropic liquid crystalline lenses for aberration control and dual layer read out of optical disks,” Adv. Functional Mater. 13, 732–738 (2003).
[CrossRef]

Sugaya, T.

T. Sugaya, “A basic concept for next generation DVD: 0.6 mm substrate disk technology using blue-violet laser,” in Optical Data Storage 2003, M. O’Neill, N. Miyagawa, eds., Proc. SPIE5069, 278–280 (2003).

ter Meulen, J. M.

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

Ubbens, I. P.

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

van der Meulen, M. P.

M. P. van der Meulen, Institute for Theoretical Physics, Valckenierstraat 65 1018 XE Amsterdam, The Netherlands (personal communication, 2001).

van der Zande, B. M. I.

H. R. Stapert, J. Lub, E. J. K. Verstegen, B. M. I. van der Zande, S. Stallinga, “Photo replicated anisotropic liquid crystalline lenses for aberration control and dual layer read out of optical disks,” Adv. Functional Mater. 13, 732–738 (2003).
[CrossRef]

van Rosmalen, G.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, London, 1985).

Verstegen, E.

S. Stallinga, J. J. Vrehen, J. Wals, E. Stapert, E. Verstegen, “Liquid crystal aberration compensation devices,” in Optical Storage and Information Processing, H.-P. D. Shieh, T. D. Milster, eds., Proc. SPIE4081, 50–56 (2000).
[CrossRef]

Verstegen, E. J. K.

H. R. Stapert, J. Lub, E. J. K. Verstegen, B. M. I. van der Zande, S. Stallinga, “Photo replicated anisotropic liquid crystalline lenses for aberration control and dual layer read out of optical disks,” Adv. Functional Mater. 13, 732–738 (2003).
[CrossRef]

Vrehen, J. J.

S. Stallinga, J. J. Vrehen, “Advances in liquid crystal tilt compensation,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 465–473 (2001).
[CrossRef]

S. Stallinga, J. J. Vrehen, J. Wals, E. Stapert, E. Verstegen, “Liquid crystal aberration compensation devices,” in Optical Storage and Information Processing, H.-P. D. Shieh, T. D. Milster, eds., Proc. SPIE4081, 50–56 (2000).
[CrossRef]

Wals, J.

S. Stallinga, J. J. Vrehen, J. Wals, E. Stapert, E. Verstegen, “Liquid crystal aberration compensation devices,” in Optical Storage and Information Processing, H.-P. D. Shieh, T. D. Milster, eds., Proc. SPIE4081, 50–56 (2000).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Hilger, London, 1986).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge University, Cambridge, UK, 1980).

Wolfram, S.

S. Wolfram, Mathematica. A System for Mathematics by Computer, 2nd ed. (Addison-Wesley, Reading, Mass., 1991).

Yasuda, K.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Adv. Functional Mater.

H. R. Stapert, J. Lub, E. J. K. Verstegen, B. M. I. van der Zande, S. Stallinga, “Photo replicated anisotropic liquid crystalline lenses for aberration control and dual layer read out of optical disks,” Adv. Functional Mater. 13, 732–738 (2003).
[CrossRef]

Appl. Opt.

IBM J. Res. Dev.

P. Asthana, B. I. Finkelstein, A. A. Fennema, “Rewritable optical disk drive technology,” IBM J. Res. Dev. 40, 543–558 (1996).
[CrossRef]

Other

S. Stallinga, J. J. Vrehen, J. Wals, E. Stapert, E. Verstegen, “Liquid crystal aberration compensation devices,” in Optical Storage and Information Processing, H.-P. D. Shieh, T. D. Milster, eds., Proc. SPIE4081, 50–56 (2000).
[CrossRef]

S. Stallinga, J. J. Vrehen, “Advances in liquid crystal tilt compensation,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 465–473 (2001).
[CrossRef]

M. P. van der Meulen, Institute for Theoretical Physics, Valckenierstraat 65 1018 XE Amsterdam, The Netherlands (personal communication, 2001).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge University, Cambridge, UK, 1980).

S. Wolfram, Mathematica. A System for Mathematics by Computer, 2nd ed. (Addison-Wesley, Reading, Mass., 1991).

W. J. Smith, Modern Optical Engineering. The Design of Optical Systems, 2nd ed. (McGraw-Hill, Boston, Mass., 1990).

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

T. Sugaya, “A basic concept for next generation DVD: 0.6 mm substrate disk technology using blue-violet laser,” in Optical Data Storage 2003, M. O’Neill, N. Miyagawa, eds., Proc. SPIE5069, 278–280 (2003).

W. T. Welford, Aberrations of Optical Systems (Hilger, London, 1986).

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, London, 1985).

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Figures (5)

Fig. 1
Fig. 1

General ray path along P1P2P3 and chief ray path along A1A2A3 through a plate of thickness d tilted over an angle β. The reference point is the focus of the incident spherical wave O and may be changed to O′ in order to minimize the rms aberration (in that case, O′ is the diffraction focus).

Fig. 2
Fig. 2

Exact rms aberration for spherical aberration from thickness mismatch normalized by the thickness mismatch (solid curve), the rms aberration according to the paraxial approximation (small-dashed curve), the lowest-order spherical aberration contribution W40 (medium-dashed curve, nearly coinciding with the solid curve), and the higher-order spherical-aberration contributions W 60 2 + W 80 2 (long-dashed curve) as a function of NA for n1 = 1.0 and n2 = 1.6.

Fig. 3
Fig. 3

Ratio of the exact rms aberration and the paraxial NA4 estimate for spherical aberration from thickness mismatch (solid curve, left axis) and the ratio of the exact focal shift and the paraxial focal shift for spherical aberration from thickness mismatch (dashed curve, right axis) for n1 = 1.0 and n2 = 1.6.

Fig. 4
Fig. 4

Exact rms aberration for coma from disk tilt normalized by the product of substrate thickness and disk tilt angle (solid curve), the rms aberration according to the paraxial approximation (small-dashed curve), the lowest-order coma contribution W31 (medium-dashed curve, nearly coinciding with the solid curve) and the higher-order coma contributions W 51 2 + W 71 2 (long-dashed curve) as a function of NA for n1 = 1.0 and n2 = 1.6.

Fig. 5
Fig. 5

Ratio of the exact rms aberration and the paraxial NA3 estimate for coma from disk tilt (solid curve, left axis) and the ratio of the exact focal shift and the paraxial focal shift for coma from disk tilt (dashed curve, right axis) as a function of NA for n1 = 1.0 and n2 = 1.6.

Tables (1)

Tables Icon

Table 1 Aberration Sensitivities of Optical Disk Formats

Equations (70)

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

W = [ O P 1 P 2 P 3 ] - [ O A 1 A 2 A 3 ] ,
[ O P 1 P 2 P 3 ] = g 1 · r 1 + g 2 · ( r 2 - r 1 ) + g 1 · ( r 3 - r 2 ) = ( g 2 - g 1 ) · ( r 2 - r 1 ) + g 1 · r 3 = ( g 2 - g 1 ) d ,
W O = ( g 2 - g 1 ) d - ( g 2 - g 1 ) d ,
[ P 3 P 3 ] = g 1 · ( r 3 - r 3 ) = g 1 · r 3 = g 1 · [ Δ r + ( r 3 - Δ r ) ] = g 1 · Δ r .
W O = ( g 2 - g 1 ) d + g 1 · Δ r - ( g 2 - g 1 ) d - g 1 · Δ r ,
g j x = - NA ρ cos θ ,
g j y = - NA ρ sin θ ,
g j z = n j 2 - NA 2 ρ 2 ,
g 1 · Δ r - g 1 · Δ r = - Δ x NA ρ cos θ - Δ y NA ρ sin θ + Δ z ( n 1 2 - NA 2 ρ 2 - n 1 ) .
W ( ρ ,     θ ) = n = 0 m = 0 [ A n , m Z n m ( ρ ) cos ( m θ ) + A n ,     - m Z n m ( ρ ) sin ( m θ ) ] ,
Z m + 2 k m ( ρ ) = ρ m P k 0 m ( 2 ρ 2 - 1 ) ,
A n , m = n + 1 π P d 2 ρ W ( ρ ,     θ ) Z n m ( ρ ) cos ( m θ ) ,
A n ,     - m = n + 1 π P d 2 ρ W ( ρ ,     θ ) Z n m ( ρ ) sin ( m θ ) ,
W rms 2 = n = 0 m = 0 A n , m 2 + A n ,     - m 2 2 ( n + 1 ) ,
W O = Δ d n 2 2 - NA 2 ρ 2 - Δ d n 1 2 - NA 2 ρ 2 - Δ d ( n 2 - n 1 ) .
W O = Δ d [ n 2 2 - NA 2 ρ 2 + α n 1 2 - NA 2 ρ 2 - ( n 2 + α n 1 ) ] ,
α = Δ z Δ d - 1.
W rms = W O 2 - W O 2 ,
A = 1 π P d 2 ρ A ( ρ ) ,
f j = n j 2 - ρ 2 NA 2
W O = Δ d ( f 2 + α f 1 ) ,
W O = Δ d ( f 2 + α f 1 ) ,
W O 2 = ( Δ d ) 2 ( f 2 2 + 2 α f 1 f 2 + α 2 f 1 2 ) .
f j = 2 3 NA 2 [ n j 3 - ( n j 2 - N A 2 ) 3 / 2 ] ,
f j 2 = n j 2 - 1 2 NA 2 ,
f 1 f 2 = 1 4 NA 2 { n 1 n 2 ( n 1 2 + n 2 2 ) - ( n 1 2 + n 2 2 - 2 NA 2 ) n 1 2 - NA 2 n 2 2 - NA 2 + ( n 1 2 - n 2 2 ) 2 log ( n 1 2 - NA 2 + n 2 2 - NA 2 n 1 + n 2 ) } .
α = - f 1 f 2 - f 1 f 2 f 1 2 - f 1 2 .
( W rms Δ d ) 2 = f 2 2 - f 2 2 - [ f 1 f 2 - f 1 f 2 ] 2 f 1 2 - f 1 2 .
Δ z par = n 2 - n 1 n 2 d ,
Δ z = Δ z par ( 1 + n 1 + n 2 4 n 1 n 2 2 NA 2 ) ,
W rms , par = 1 6 5 n 2 2 - n 1 2 8 n 2 3 n 1 2 Δ d NA 4 ,
W rms = W rms , par ( 1 + 3 n 1 2 + 2 n 2 2 4 n 1 2 n 2 2 NA 2 ) ,
W O = Δ d ( n 2 2 - NA 2 ρ 2 - n 1 n 2 n 1 2 - NA 2 ρ 2 - n 2 2 - n 1 2 n 2 ) .
W par = n 2 2 - n 1 2 8 n 2 3 n 1 2 Δ d NA 4 ρ 4 .
A 2 k , 0 = 2 k + 1 π P d 2 ρ W O ( ρ ) Z 2 k 0 ( ρ ) = 2 ( 2 k + 1 ) 0 1 d ρ ρ W O ( ρ ) Z 2 k 0 ( ρ ) = Δ d n 2 ξ k ( NA n 2 ) + α Δ d n 1 ξ k ( NA n 1 ) ,
ξ k ( ) = 2 ( 2 k + 1 ) 0 1 d ρ ρ 1 - 2 ρ 2 P k ( 2 ρ 2 - 1 ) .
ξ 0 ( ) = 1 - 1 - 2 ( 1 - 2 ) 3 2 ,
ξ 1 ( ) = ( 8 - 10 2 ) - 1 - 2 ( 8 - 6 2 - 2 4 ) 15 4 ,
ξ 2 ( ) = ( 96 - 168 2 + 70 4 ) - 1 - 2 ( 96 - 120 2 + 22 4 + 2 6 ) 105 6 ,
ξ 3 ( ) = ( 640 - 1440 2 + 1008 4 - 210 6 ) 315 8 - 1 - 2 ( 640 - 1120 2 + 528 4 - 46 6 - 2 8 ) 315 8 .
W = Δ z ( n 1 2 - NA 2 ρ 2 - n 1 ) .
( W rms Δ z ) 2 = f 1 2 - f 1 2 .
( W rms Δ z ) 2 = NA 2 48 n 1 2 .
g 1 x = g 2 x = cos β g 1 x + sin β g 1 z = - cos β NA ρ cos θ + sin β n 1 2 - NA 2 ρ 2 ,
g 1 y = g 2 y = - NA ρ sin θ .
g 1 z = - sin β g 1 x + cos β g 1 z = sin β NA ρ cos θ + cos β n 1 2 - NA 2 ρ 2 ,
g 2 z = n 2 2 - g 1 x 2 - g 1 y 2 = [ n 2 2 - ( cos β NA ρ cos θ - sin β n 1 2 - NA 2 ρ 2 ) + NA 2 ρ 2 sin 2 θ ] 1 / 2 .
W O = d [ n 2 2 - ( cos β NA ρ cos θ - sin β n 1 2 - NA 2 ρ 2 ) 2 + NA 2 ρ 2 sin 2 θ ] 1 / 2 - d ( sin β NA ρ cos θ + cos β n 1 2 - NA 2 ρ 2 ) = d n 2 2 - n 1 2 sin 2 β 1 - t - d ( sin β NA ρ cos θ + cos β n 1 2 - NA 2 ρ 2 ) ,
t = [ - sin ( 2 β ) NA ρ cos θ n 1 2 - NA 2 ρ 2 + cos ( 2 β ) NA 2 ρ 2 cos 2 θ + cos 2 β NA 2 ρ 2 sin 2 θ ] / ( n 2 2 - n 1 2 sin 2 β ) .
W O = ( n 1 2 - NA 2 ρ 2 n 2 2 - NA 2 ρ 2 - 1 ) d β NA ρ cos θ .
W O = d β ( h 2 + μ h 1 ) ,
h 1 = NA ρ cos θ ,
h 2 = NA ρ cos θ n 1 2 - NA 2 ρ 2 n 2 2 - NA 2 ρ 2 ,
μ = Δ x d β - 1.
h 1 = h 2 = 0 ,
h 1 2 = NA 2 4 ,
h 2 2 = 1 4 NA 2 [ ( 2 n 2 2 - 2 n 1 2 + NA 2 ) NA 2 + 2 n 2 2 ( n 2 2 - n 1 2 ) log ( n 2 2 - NA 2 n 2 2 ) ] ,
h 1 h 2 = 1 8 NA 2 [ n 1 n 2 ( 3 n 2 2 - n 1 2 ) - ( 3 n 2 2 - n 1 2 + 2 NA 2 ) n 1 2 - NA 2 n 2 2 - NA 2 + ( 3 n 2 2 + n 1 2 ) ( n 2 2 - n 1 2 ) log ( n 1 2 - NA 2 + n 2 2 - NA 2 n 1 + n 2 ) ] .
μ = - h 1 h 2 - h 1 h 2 h 1 2 - h 1 2 = - h 1 h 2 h 1 2 .
( W rms d β ) 2 = h 2 2 - h 2 2 - [ h 1 h 2 - h 1 h 2 ] 2 h 1 2 - h 1 2 = h 2 2 - h 1 h 2 2 h 1 2 .
Δ x par = n 2 - n 1 n 2 d β ,
Δ x = Δ x par ( 1 + n 1 + n 2 3 n 1 n 2 2 NA 2 ) ,
W rms , par = 1 3 8 n 2 2 - n 1 2 2 n 2 3 n 1 d β NA 3 .
W rms = W rms , par [ 1 + 3 3 ( 3 n 1 2 + n 2 2 ) 10 n 1 2 n 2 2 NA 2 ] ,
W O = ( n 1 2 - NA 2 ρ 2 n 2 2 - NA 2 ρ 2 - n 1 n 2 ) d β NA ρ cos θ .
W par = - n 2 2 - n 1 2 2 n 2 3 n 1 d β NA 3 ρ 3 cos θ .
A 2 k + 1 , 1 = 4 k + 4 π P d 2 ρ W O ( ρ ) Z 2 n + 1 1 ( ρ ) cos θ = d β NA n 1 n 2 ζ k ( NA n 1 ,     NA n 2 ) + μ d β NA δ k 0 ,
ζ k ( 1 ,     2 ) = 4 ( k + 1 ) 0 1 d ρ     ρ 3 1 - 1 2 ρ 2 1 - 2 2 ρ 2 P k 01 ( 2 ρ 2 - 1 ) .
ζ 0 ( 1 ,     2 ) = 1 8 1 3 2 5 [ 1 2 ( 3 1 2 - 2 2 ) - 1 2 ( 3 1 2 + 2 1 2 2 2 + 2 2 ) ( 1 - 1 2 ) ( 1 - 2 2 ) + ( 1 2 - 2 2 ) ( 3 1 2 + 2 2 ) log ( 1 1 - 1 2 + 2 1 - 1 2 1 + 2 ) ] ,
ζ 1 ( 1 ,     2 ) = 1 16 1 5 2 7 [ 1 2 ( 15 1 4 - 3 2 2 + 4 1 2 2 2 - 12 1 4 2 2 + 4 1 2 4 2 ) - 1 2 ( 15 1 4 - 3 2 2 - 4 1 2 2 2 - 2 1 4 2 2 + 2 1 2 4 2 ) ( 1 - 1 2 ) ( 1 - 2 2 ) + ( 1 2 - 2 2 ) ( 15 1 4 + 3 2 2 + 6 1 2 2 2 - 12 1 4 2 2 - 4 1 2 4 2 ) log ( 1 1 - 2 2 + 2 1 - 1 2 1 + 2 ) ] .

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