Abstract

A solid immersion lens attached to a conventional objective increases the effective numerical aperture (NAeff) of an optical pickup and yields an areal recording density proportional to (NAeff)2. One version of this device, with an effective (NAeff) of 1.7, should be capable of very high density storage but would probably need a sealed system. Another simple configuration enables the use of this method for optical data storage in an unsealed environment and extends the spatial cutoff frequency 1.5 times. Experiments with these devices are compared with the full vector field theory of this type of imaging system.

© 1997 Optical Society of America

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References

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  1. E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, M. H. Kryder, C. H. Chang, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
    [CrossRef]
  2. S. M. Mansfield, G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 (1990).
    [CrossRef]
  3. S. M. Mansfield, W. R. Studenmund, G. S. Kino, K. Osato, “High-numerical-aperture lens system for optical storage,” Opt. Lett. 18, 305–307 (1993).
    [CrossRef]
  4. B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, G. S. Kino, “Near-field optical data storage using a solid immersion lens,” Appl. Phys. Lett. 65, 388–390 (1994).
    [CrossRef]
  5. H. J. Mamin, B. D. Terris, D. Rugar, “Near-field optical data storage,” Appl. Phys. Lett. 68, 141–143 (1996).
    [CrossRef]
  6. I. Ichimura, K. Osato, F. Maeda, H. Owa, H. Ooki, G. S. Kino, “High density optical disk system using a solid immersion lens,” in Optical Data Storage, G. R. Knight, H. Ooki, Y. Tyan, eds., Proc. SPIE2514, 176–181 (1995).
  7. E. Wolf, “Electromagnetic diffraction in optical systems: I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
    [CrossRef]
  8. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  9. M. Oka, S. Kubota, “Stable intracavity doubling of orthogonal linearly polarized modes in diode-pumped Nd:YAG lasers,” Opt. Lett. 13, 805–807 (1988).
    [CrossRef] [PubMed]
  10. M. Oka, S. Kubota, “Second-harmonic-generation green laser for high density optical disks,” Jpn. J. Appl. Phys. 31, 513–518 (1992).
    [CrossRef]
  11. I. Ichimura, Y. Sabi, Y. Takeshita, A. Fukumoto, M. Kaneko, H. Owa, “High density magneto-optical recording with a second-harmonic-generation green laser,” Jpn. J. Appl. Phys. 32, 5312–5316 (1993).
    [CrossRef]
  12. K. Balasubramanian, A. S. Marathay, A. Macleod, “Modeling magneto-optical thin film media for optical data storage,” Thin Solid Films 164, 391–403 (1988).
    [CrossRef]
  13. M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
    [CrossRef]
  14. M. Mansuripur, “Effects of high-numerical-aperture focusing on the state of polarization in optical and magneto-optic data storage systems,” Appl. Opt. 30, 3154–3162 (1991).
    [CrossRef] [PubMed]

1996 (1)

H. J. Mamin, B. D. Terris, D. Rugar, “Near-field optical data storage,” Appl. Phys. Lett. 68, 141–143 (1996).
[CrossRef]

1994 (1)

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, G. S. Kino, “Near-field optical data storage using a solid immersion lens,” Appl. Phys. Lett. 65, 388–390 (1994).
[CrossRef]

1993 (2)

S. M. Mansfield, W. R. Studenmund, G. S. Kino, K. Osato, “High-numerical-aperture lens system for optical storage,” Opt. Lett. 18, 305–307 (1993).
[CrossRef]

I. Ichimura, Y. Sabi, Y. Takeshita, A. Fukumoto, M. Kaneko, H. Owa, “High density magneto-optical recording with a second-harmonic-generation green laser,” Jpn. J. Appl. Phys. 32, 5312–5316 (1993).
[CrossRef]

1992 (2)

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, M. H. Kryder, C. H. Chang, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

M. Oka, S. Kubota, “Second-harmonic-generation green laser for high density optical disks,” Jpn. J. Appl. Phys. 31, 513–518 (1992).
[CrossRef]

1991 (1)

1990 (2)

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

S. M. Mansfield, G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 (1990).
[CrossRef]

1988 (2)

M. Oka, S. Kubota, “Stable intracavity doubling of orthogonal linearly polarized modes in diode-pumped Nd:YAG lasers,” Opt. Lett. 13, 805–807 (1988).
[CrossRef] [PubMed]

K. Balasubramanian, A. S. Marathay, A. Macleod, “Modeling magneto-optical thin film media for optical data storage,” Thin Solid Films 164, 391–403 (1988).
[CrossRef]

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems: I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Balasubramanian, K.

K. Balasubramanian, A. S. Marathay, A. Macleod, “Modeling magneto-optical thin film media for optical data storage,” Thin Solid Films 164, 391–403 (1988).
[CrossRef]

Betzig, E.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, M. H. Kryder, C. H. Chang, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Chang, C. H.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, M. H. Kryder, C. H. Chang, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Finn, P. L.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, M. H. Kryder, C. H. Chang, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Fukumoto, A.

I. Ichimura, Y. Sabi, Y. Takeshita, A. Fukumoto, M. Kaneko, H. Owa, “High density magneto-optical recording with a second-harmonic-generation green laser,” Jpn. J. Appl. Phys. 32, 5312–5316 (1993).
[CrossRef]

Gyorgy, E. M.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, M. H. Kryder, C. H. Chang, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Ichimura, I.

I. Ichimura, Y. Sabi, Y. Takeshita, A. Fukumoto, M. Kaneko, H. Owa, “High density magneto-optical recording with a second-harmonic-generation green laser,” Jpn. J. Appl. Phys. 32, 5312–5316 (1993).
[CrossRef]

I. Ichimura, K. Osato, F. Maeda, H. Owa, H. Ooki, G. S. Kino, “High density optical disk system using a solid immersion lens,” in Optical Data Storage, G. R. Knight, H. Ooki, Y. Tyan, eds., Proc. SPIE2514, 176–181 (1995).

Kaneko, M.

I. Ichimura, Y. Sabi, Y. Takeshita, A. Fukumoto, M. Kaneko, H. Owa, “High density magneto-optical recording with a second-harmonic-generation green laser,” Jpn. J. Appl. Phys. 32, 5312–5316 (1993).
[CrossRef]

Kino, G. S.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, G. S. Kino, “Near-field optical data storage using a solid immersion lens,” Appl. Phys. Lett. 65, 388–390 (1994).
[CrossRef]

S. M. Mansfield, W. R. Studenmund, G. S. Kino, K. Osato, “High-numerical-aperture lens system for optical storage,” Opt. Lett. 18, 305–307 (1993).
[CrossRef]

S. M. Mansfield, G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 (1990).
[CrossRef]

I. Ichimura, K. Osato, F. Maeda, H. Owa, H. Ooki, G. S. Kino, “High density optical disk system using a solid immersion lens,” in Optical Data Storage, G. R. Knight, H. Ooki, Y. Tyan, eds., Proc. SPIE2514, 176–181 (1995).

Kryder, M. H.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, M. H. Kryder, C. H. Chang, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Kubota, S.

M. Oka, S. Kubota, “Second-harmonic-generation green laser for high density optical disks,” Jpn. J. Appl. Phys. 31, 513–518 (1992).
[CrossRef]

M. Oka, S. Kubota, “Stable intracavity doubling of orthogonal linearly polarized modes in diode-pumped Nd:YAG lasers,” Opt. Lett. 13, 805–807 (1988).
[CrossRef] [PubMed]

Macleod, A.

K. Balasubramanian, A. S. Marathay, A. Macleod, “Modeling magneto-optical thin film media for optical data storage,” Thin Solid Films 164, 391–403 (1988).
[CrossRef]

Maeda, F.

I. Ichimura, K. Osato, F. Maeda, H. Owa, H. Ooki, G. S. Kino, “High density optical disk system using a solid immersion lens,” in Optical Data Storage, G. R. Knight, H. Ooki, Y. Tyan, eds., Proc. SPIE2514, 176–181 (1995).

Mamin, H. J.

H. J. Mamin, B. D. Terris, D. Rugar, “Near-field optical data storage,” Appl. Phys. Lett. 68, 141–143 (1996).
[CrossRef]

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, G. S. Kino, “Near-field optical data storage using a solid immersion lens,” Appl. Phys. Lett. 65, 388–390 (1994).
[CrossRef]

Mansfield, S. M.

Mansuripur, M.

M. Mansuripur, “Effects of high-numerical-aperture focusing on the state of polarization in optical and magneto-optic data storage systems,” Appl. Opt. 30, 3154–3162 (1991).
[CrossRef] [PubMed]

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

Marathay, A. S.

K. Balasubramanian, A. S. Marathay, A. Macleod, “Modeling magneto-optical thin film media for optical data storage,” Thin Solid Films 164, 391–403 (1988).
[CrossRef]

Oka, M.

M. Oka, S. Kubota, “Second-harmonic-generation green laser for high density optical disks,” Jpn. J. Appl. Phys. 31, 513–518 (1992).
[CrossRef]

M. Oka, S. Kubota, “Stable intracavity doubling of orthogonal linearly polarized modes in diode-pumped Nd:YAG lasers,” Opt. Lett. 13, 805–807 (1988).
[CrossRef] [PubMed]

Ooki, H.

I. Ichimura, K. Osato, F. Maeda, H. Owa, H. Ooki, G. S. Kino, “High density optical disk system using a solid immersion lens,” in Optical Data Storage, G. R. Knight, H. Ooki, Y. Tyan, eds., Proc. SPIE2514, 176–181 (1995).

Osato, K.

S. M. Mansfield, W. R. Studenmund, G. S. Kino, K. Osato, “High-numerical-aperture lens system for optical storage,” Opt. Lett. 18, 305–307 (1993).
[CrossRef]

I. Ichimura, K. Osato, F. Maeda, H. Owa, H. Ooki, G. S. Kino, “High density optical disk system using a solid immersion lens,” in Optical Data Storage, G. R. Knight, H. Ooki, Y. Tyan, eds., Proc. SPIE2514, 176–181 (1995).

Owa, H.

I. Ichimura, Y. Sabi, Y. Takeshita, A. Fukumoto, M. Kaneko, H. Owa, “High density magneto-optical recording with a second-harmonic-generation green laser,” Jpn. J. Appl. Phys. 32, 5312–5316 (1993).
[CrossRef]

I. Ichimura, K. Osato, F. Maeda, H. Owa, H. Ooki, G. S. Kino, “High density optical disk system using a solid immersion lens,” in Optical Data Storage, G. R. Knight, H. Ooki, Y. Tyan, eds., Proc. SPIE2514, 176–181 (1995).

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Rugar, D.

H. J. Mamin, B. D. Terris, D. Rugar, “Near-field optical data storage,” Appl. Phys. Lett. 68, 141–143 (1996).
[CrossRef]

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, G. S. Kino, “Near-field optical data storage using a solid immersion lens,” Appl. Phys. Lett. 65, 388–390 (1994).
[CrossRef]

Sabi, Y.

I. Ichimura, Y. Sabi, Y. Takeshita, A. Fukumoto, M. Kaneko, H. Owa, “High density magneto-optical recording with a second-harmonic-generation green laser,” Jpn. J. Appl. Phys. 32, 5312–5316 (1993).
[CrossRef]

Studenmund, W. R.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, G. S. Kino, “Near-field optical data storage using a solid immersion lens,” Appl. Phys. Lett. 65, 388–390 (1994).
[CrossRef]

S. M. Mansfield, W. R. Studenmund, G. S. Kino, K. Osato, “High-numerical-aperture lens system for optical storage,” Opt. Lett. 18, 305–307 (1993).
[CrossRef]

Takeshita, Y.

I. Ichimura, Y. Sabi, Y. Takeshita, A. Fukumoto, M. Kaneko, H. Owa, “High density magneto-optical recording with a second-harmonic-generation green laser,” Jpn. J. Appl. Phys. 32, 5312–5316 (1993).
[CrossRef]

Terris, B. D.

H. J. Mamin, B. D. Terris, D. Rugar, “Near-field optical data storage,” Appl. Phys. Lett. 68, 141–143 (1996).
[CrossRef]

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, G. S. Kino, “Near-field optical data storage using a solid immersion lens,” Appl. Phys. Lett. 65, 388–390 (1994).
[CrossRef]

Trautman, J. K.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, M. H. Kryder, C. H. Chang, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems: I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

Wolfe, R.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, M. H. Kryder, C. H. Chang, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, M. H. Kryder, C. H. Chang, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

S. M. Mansfield, G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 (1990).
[CrossRef]

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, G. S. Kino, “Near-field optical data storage using a solid immersion lens,” Appl. Phys. Lett. 65, 388–390 (1994).
[CrossRef]

H. J. Mamin, B. D. Terris, D. Rugar, “Near-field optical data storage,” Appl. Phys. Lett. 68, 141–143 (1996).
[CrossRef]

J. Appl. Phys. (1)

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

Jpn. J. Appl. Phys. (2)

M. Oka, S. Kubota, “Second-harmonic-generation green laser for high density optical disks,” Jpn. J. Appl. Phys. 31, 513–518 (1992).
[CrossRef]

I. Ichimura, Y. Sabi, Y. Takeshita, A. Fukumoto, M. Kaneko, H. Owa, “High density magneto-optical recording with a second-harmonic-generation green laser,” Jpn. J. Appl. Phys. 32, 5312–5316 (1993).
[CrossRef]

Opt. Lett. (2)

Proc. R. Soc. London Ser. A (2)

E. Wolf, “Electromagnetic diffraction in optical systems: I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Thin Solid Films (1)

K. Balasubramanian, A. S. Marathay, A. Macleod, “Modeling magneto-optical thin film media for optical data storage,” Thin Solid Films 164, 391–403 (1988).
[CrossRef]

Other (1)

I. Ichimura, K. Osato, F. Maeda, H. Owa, H. Ooki, G. S. Kino, “High density optical disk system using a solid immersion lens,” in Optical Data Storage, G. R. Knight, H. Ooki, Y. Tyan, eds., Proc. SPIE2514, 176–181 (1995).

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Figures (17)

Fig. 1
Fig. 1

Solid immersion lens. (a) Hemispherical SIL focusing at the bottom surface. (b) Stigmatic focusing SIL.

Fig. 2
Fig. 2

Truncated hemispherical SIL with the same refractive index as the disk substrate.

Fig. 3
Fig. 3

Definition of electric-field coordinates for an imaging system.

Fig. 4
Fig. 4

Amplitude transmission coefficients between two n = 1.9 media with various air gaps: (a) p-polarized component, (b) s-polarized component.

Fig. 5
Fig. 5

Point-spread functions with air gaps for a stigmatic SIL (NAeff = 1.7).

Fig. 6
Fig. 6

Phase shift caused by air gaps between two n = 1.5 media.

Fig. 7
Fig. 7

Point-spread functions with air gaps for a truncated hemispherical SIL (NAeff = 0.83).

Fig. 8
Fig. 8

Mechanical configuration of the SIL optics.

Fig. 9
Fig. 9

Schematic diagram of the optical block.

Fig. 10
Fig. 10

Measured modulation transfer function with calculated results for various air gaps.

Fig. 11
Fig. 11

Experimental and theoretical edge responses.

Fig. 12
Fig. 12

EFM eye pattern for a high-density CD ROM (CDx6).

Fig. 13
Fig. 13

Cross-sectional structures of MO disks: (a) air-incident configuration for λ = 830 nm, (b) glass-incident configuration for λ = 532 nm.

Fig. 14
Fig. 14

Reflection coefficients of the MO disks: (a) air-incident configuration, (b) glass-incident configuration.

Fig. 15
Fig. 15

Point-spread functions in MO recording for a stigmatic SIL (NAeff = 1.5).

Fig. 16
Fig. 16

Point-spread functions in MO recording for a truncated hemispherical SIL (NAeff = 0.83).

Fig. 17
Fig. 17

Eye pattern of a high density (1,7) RLL mark length recording (CDx6).

Equations (46)

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

apas=TppTpsTspTssap0as0,
TppTpsTspTss=rppMOrpsMOrspMOrssMOtpag00tsag×tpSIL00tsSILtpob00tsobcos1/2θ,
ap0as0=cos ϕsin ϕ-sin ϕcos ϕax0ay0.
axayaz=cos θ001sin θ0apas,
axayaz=cos ϕ-sin ϕ0sin ϕcos ϕ0001axayaz.
ax=12Tss+Tpp cos θ-Tss-Tpp cos θcos 2ϕ-Tsp+Tps cos θsin 2ϕax0-Tsp-Tps cos θ-Tsp+Tps cos θcos 2ϕ+Tss-Tpp cos θsin 2ϕay0,
ay=12Tsp-Tps cos θ+Tsp+Tps cos θcos 2ϕ-Tss-Tpp cos θsin 2ϕax0+Tss+Tpp cos θ+Tss-Tpp cos θcos 2ϕ+Tsp+Tps cos θsin 2ϕay0,
az=Tpp cos ϕ-Tps sin ϕsin θax0+Tps cos ϕ+Tpp sin ϕsinθ ay0.
ep=-jλ0α sinθ·dθ 02π;dϕ expjks·p·aθ, ϕ,
ŝ=sin θ cos ϕ, cos θ sin ϕ,cos θ.
02π cos nϕ expjρ cosϕ-γdϕ=2πjnJnρcos nγ,
02π sin nϕ expjρ cosϕ-γdϕ=2πjnJnρsin nγ,
s·p=rp sin θ cosϕ-ϕp+zp cos θ,
xprp cos ϕp,   yprp sin ϕp.
ex=-jπλI0de+I2de cos 2ϕp+I2ce sin 2ϕpax0+-I0ce-I2ce cos 2ϕp+I2de sin 2ϕpay0,
ey=-jπλI0ce-I2ce cos 2ϕp+I2de sin 2ϕpax0+I0de-I2de cos 2ϕp-I2ce sin 2ϕpay0,
ez=-2πλI1de cos ϕp-I1ce sin ϕpax0+I1ce cos ϕp+I1de sin ϕay0,
I0de=0αTss+Tpp cos θsin θ J0krp sin θ×expjkzp cos θdθ,
I0ce=0αTsp-Tps cos θsin θ J0krp sin θ×expjkzp cos θdθ,
I1de=0αTpp sin2 θ J1krp sin θ×expjkzp cos θdθ,
I1ce=0αTps sin2 θ J1krp sin θ×expjkzp cos θdθ,
I2de=0αTss-Tpp cos θsin θ J2krp sin θ×expjkzp cos θdθ,
I2ce=0αTsp+Tps cos θsin θ J2krp sin θ×expjkzp cos θdθ.
b=s×aη,
bx=12ηTps-Tsp cos θ-Tps+Tsp cos θcos 2ϕ-Tpp-Tss cos θsin 2ϕax0-Tpp+Tss cos θ-Tpp-Tss cos θcos 2ϕ+Tps+Tsp cos θsin 2ϕay0,
by=12ηTpp+Tss cos θ+Tpp-Tss cos θcos 2ϕ-Tps+Tsp cos θsin 2ϕax0+Tps-Tsp cos θ+Tps+Tsp cos θcos 2ϕ+Tpp-Tss cos θsin 2ϕay0,
bz=1η-Tsp cos ϕ+Tss sin ϕsin θ ax0-Tss cos ϕ+Tsp sin ϕsin θ ay0.
hx=-jπληI0ch+I2ch cos 2ϕp+I2dh sin 2ϕpax0+-I0dh-I2dh cos 2ϕp+I2ch sin 2ϕpay0,
hy=-jπληI0dh-I2dh cos 2ϕp+I2ch sin 2ϕpax0+I0ch-I2ch cos 2ϕp-I2dh sin 2ϕpay0,
hz=2πληI1ch cos ϕp-I1dh sin ϕpax0+I1dh cos ϕp+I1ch sin ϕpay0,
I0dh=0αTpp+Tss cos θsin θ J0krp sin θ×expjkzp cos θdθ,
I0ch=0αTps-Tsp cos θsin θ J0krp sin θ×expjkzp cos θdθ,
I1dh=0αTss sin2 θ J1krp sin θexpjkzp cos θdθ,
I1ch=0αTsp sin2 θ J1krp sin θexpjkzp cos θdθ,
I2dh=0αTpp-Tss cos θsin θ J2krp sin θ×expjkzp cos θdθ,
I2ch=0αTps+Tsp cos θsin θ J2krp sin θ×expjkzp cos θdθ.
P=12Re SExHy*-EyHx*ds,
PSFdirect=KReexhy*-eyhx*,
Sa=sEx+EyHy*-Hx*ds,
Sb=SEx-EyHy*+Hx*ds,
PSFrotated=K Reeyhy*-exhx*,
ε=εdεc0-εcεd000εd.
kzko2+Bkzko2+D=0,
B=-2εd-S2,
D=S4-2εd2+εc2εdS2+εd2+εc2,
S=nSIL sin θconstant for all layers.

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