Abstract

Results of numerical computations pertaining to evanescent wave coupling for near-field magneto-optical and phase-change disks based on the concept of the solid immersion lens are presented. We investigated the relation between the coupling efficiency and the width of the air gap in terms of the throughput of the recording process and the resolution of the readout signal. The simulations show a drastic decrease with a widening air gap of the coupling efficiency by means of evanescent waves into the recording medium. In magneto-optical readout, loss of the signal may be attributed to the reduction of magneto-optical interaction, the rise of reflectance, and the variation of the relative phase between the two components of polarization. In the phase-change readout the reduced reflectivity contrast between crystalline and amorphous marks is the cause of signal reduction.

© 2000 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, W. R. Studenmund, G. S. Kino, K. Osato, “High numerical aperture lens system for optical storage,” Opt. Lett. 18, 305–307 (1993).
    [CrossRef]
  3. 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]
  4. I. Ichimura, S. Hayashi, G. S. Kino, “High-density optical recording using a solid immersion lens,” Appl. Opt. 36, 4339–4348 (1997).
    [CrossRef] [PubMed]
  5. C. W. Lee, K. H. Cho, C. S. Chung, J. H. Yoo, Y. H. Lee, “Feasibility study on near field optical memory using a catadioptric optical system,” in Optical Data Storage, Vol. 8 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), Paper WA4–1.
  6. K. Hirota, J. S. Jo, T. D. Milster, “High density phase change optical recording using a solid immersion lens,” in Optical Data Storage, S. Kubota, T. D. Milster, P. J. Wehrenberg, eds., Proc. SPIE3401, 34–39 (1998).
  7. S. M. Mansfield, G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 (1990).
    [CrossRef]
  8. DIFFRACT is a product of MM Research, Inc., Tucson, Arizona. The theoretical basis of this program is described in M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786–805 (1989)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]
  9. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  10. C. Peng, M. Mansuripur, W. M. Kim, S. G. Kim, “Edge detection in phase-change optical data storage,” Appl. Phys. Lett. 71, 2088–2090 (1997).
    [CrossRef]

1997

C. Peng, M. Mansuripur, W. M. Kim, S. G. Kim, “Edge detection in phase-change optical data storage,” Appl. Phys. Lett. 71, 2088–2090 (1997).
[CrossRef]

I. Ichimura, S. Hayashi, G. S. Kino, “High-density optical recording using a solid immersion lens,” Appl. Opt. 36, 4339–4348 (1997).
[CrossRef] [PubMed]

1994

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

1992

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]

1990

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

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]

Cho, K. H.

C. W. Lee, K. H. Cho, C. S. Chung, J. H. Yoo, Y. H. Lee, “Feasibility study on near field optical memory using a catadioptric optical system,” in Optical Data Storage, Vol. 8 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), Paper WA4–1.

Chung, C. S.

C. W. Lee, K. H. Cho, C. S. Chung, J. H. Yoo, Y. H. Lee, “Feasibility study on near field optical memory using a catadioptric optical system,” in Optical Data Storage, Vol. 8 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), Paper WA4–1.

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]

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]

Hayashi, S.

Hirota, K.

K. Hirota, J. S. Jo, T. D. Milster, “High density phase change optical recording using a solid immersion lens,” in Optical Data Storage, S. Kubota, T. D. Milster, P. J. Wehrenberg, eds., Proc. SPIE3401, 34–39 (1998).

Ichimura, I.

Jo, J. S.

K. Hirota, J. S. Jo, T. D. Milster, “High density phase change optical recording using a solid immersion lens,” in Optical Data Storage, S. Kubota, T. D. Milster, P. J. Wehrenberg, eds., Proc. SPIE3401, 34–39 (1998).

Kim, S. G.

C. Peng, M. Mansuripur, W. M. Kim, S. G. Kim, “Edge detection in phase-change optical data storage,” Appl. Phys. Lett. 71, 2088–2090 (1997).
[CrossRef]

Kim, W. M.

C. Peng, M. Mansuripur, W. M. Kim, S. G. Kim, “Edge detection in phase-change optical data storage,” Appl. Phys. Lett. 71, 2088–2090 (1997).
[CrossRef]

Kino, G. S.

I. Ichimura, S. Hayashi, G. S. Kino, “High-density optical recording using a solid immersion lens,” Appl. Opt. 36, 4339–4348 (1997).
[CrossRef] [PubMed]

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]

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]

Lee, C. W.

C. W. Lee, K. H. Cho, C. S. Chung, J. H. Yoo, Y. H. Lee, “Feasibility study on near field optical memory using a catadioptric optical system,” in Optical Data Storage, Vol. 8 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), Paper WA4–1.

Lee, Y. H.

C. W. Lee, K. H. Cho, C. S. Chung, J. H. Yoo, Y. H. Lee, “Feasibility study on near field optical memory using a catadioptric optical system,” in Optical Data Storage, Vol. 8 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), Paper WA4–1.

Mamin, H. J.

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.

C. Peng, M. Mansuripur, W. M. Kim, S. G. Kim, “Edge detection in phase-change optical data storage,” Appl. Phys. Lett. 71, 2088–2090 (1997).
[CrossRef]

Milster, T. D.

K. Hirota, J. S. Jo, T. D. Milster, “High density phase change optical recording using a solid immersion lens,” in Optical Data Storage, S. Kubota, T. D. Milster, P. J. Wehrenberg, eds., Proc. SPIE3401, 34–39 (1998).

Osato, K.

Peng, C.

C. Peng, M. Mansuripur, W. M. Kim, S. G. Kim, “Edge detection in phase-change optical data storage,” Appl. Phys. Lett. 71, 2088–2090 (1997).
[CrossRef]

Rugar, D.

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]

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]

Terris, B. D.

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.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

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]

Yoo, J. H.

C. W. Lee, K. H. Cho, C. S. Chung, J. H. Yoo, Y. H. Lee, “Feasibility study on near field optical memory using a catadioptric optical system,” in Optical Data Storage, Vol. 8 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), Paper WA4–1.

Appl. Opt.

Appl. Phys. Lett.

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]

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

C. Peng, M. Mansuripur, W. M. Kim, S. G. Kim, “Edge detection in phase-change optical data storage,” Appl. Phys. Lett. 71, 2088–2090 (1997).
[CrossRef]

Opt. Lett.

Other

DIFFRACT is a product of MM Research, Inc., Tucson, Arizona. The theoretical basis of this program is described in M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786–805 (1989)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]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

C. W. Lee, K. H. Cho, C. S. Chung, J. H. Yoo, Y. H. Lee, “Feasibility study on near field optical memory using a catadioptric optical system,” in Optical Data Storage, Vol. 8 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), Paper WA4–1.

K. Hirota, J. S. Jo, T. D. Milster, “High density phase change optical recording using a solid immersion lens,” in Optical Data Storage, S. Kubota, T. D. Milster, P. J. Wehrenberg, eds., Proc. SPIE3401, 34–39 (1998).

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Figures (26)

Fig. 1
Fig. 1

Beam of light is totally internally reflected from the rear facet of a glass prism. The electromagnetic field lurking in the free-space region behind the prism is evanescent; both its electric and magnetic components decay exponentially with distance from the interface, and the projection of its Poynting vector perpendicular to the interface is zero. The energy stored in the evanescent field is deposited there at the time when the light source is first turned on. In the steady state, energy is neither added nor removed from the evanescent field; all the incoming optical energy is reflected at the rear facet of the prism.

Fig. 2
Fig. 2

Ellipse of polarization is uniquely specified by E x and E y , the complex-valued electric field components along the X and the Y axes. The major axis of the ellipse makes an angle ρ with the X direction, and the angle η facing the minor axis represents polarization ellipticity. In general, -90° < ρ ≤ 90° and -45° ≤ η ≤ 45°.

Fig. 3
Fig. 3

Collimated beam of light, uniform, monochromatic (λ0 = 650 nm), and linearly polarized along X, enters an aplanatic 0.8 N.A. objective lens (f = 3750λ0). A glass hemisphere—also known as a SIL—of refractive index n = 2 is placed with its flat facet coinciding with the objective’s focal plane. The surfaces of the objective as well as the spherical surface of the SIL are antireflection coated, but the flat facet of the SIL is bare. The light reflected from this flat facet returns to the objective, is collimated by it, and appears at the exit pupil.

Fig. 4
Fig. 4

Various distributions of the reflected light at the exit pupil of the objective lens of Fig. 3. (a) Plot of reflected intensity corresponding to a 66% overall reflectivity at the flat facet of the hemisphere. (b) Logarithmic plot of the reflected intensity. (c) Intensity distribution for the x component of polarization, E x . (d) Intensity distribution for the y component of polarization, E y . (e) Polarization ellipticity η encoded by gray scale, covering a range from -37° (black) to +37° (white). (f) Polarization rotation angle ρ encoded by gray scale, covering a range from -90° (black) to +90° (white).

Fig. 5
Fig. 5

Collimated beam of light, uniform, monochromatic (λ0 = 650 nm), and linearly polarized along X, enters an aplanatic 0.8 N.A. objective lens (f = 3750λ0). The two glass hemispheres of refractive index n = 2 are separated by an air gap. Both hemispheres are antireflection coated on their spherical surfaces but are left bare on their flat surfaces. The light reflected at the air gap returns to the objective, is collimated by it, and appears at the exit pupil.

Fig. 6
Fig. 6

Various distributions of the reflected light at the exit pupil of the objective lens of Fig. 5; the gap width is fixed at 100 nm. (a) Plot of reflected intensity corresponding to a 42% overall reflectivity at the air gap. (b) Logarithmic plot of the reflected intensity. (c) Intensity distribution for E x . (d) Intensity distribution for E y . (e) Polarization ellipticity η encoded by gray scale, covering a range from -29° (black) to +29° (white). (f) Polarization rotation angle ρ encoded by gray scale, covering a range from -90° (black) to +90° (white).

Fig. 7
Fig. 7

Computed plot of reflectance versus gap width in the system of Fig. 5, when a circular mask is placed in the incident beam’s path to block the rays that arrive at the interface between the hemispheres at or below the critical TIR angle.

Fig. 8
Fig. 8

Quadrilayer structure of a typical MO disk (denoted as stack A) used in conjunction with the SIL. The local state of magnetization (up or down) represents the state of the recorded bit (0 or 1). The size of the focused spot directed through the SIL at the magnetic layer determines the minimum mark size that can be recorded and read out.

Fig. 9
Fig. 9

Various distributions of the reflected light at the exit pupil of the objective lens of Fig. 3, when the MO stack of Fig. 8 is placed in front of the SIL with a 100-nm air gap. (a) Intensity distribution for E x containing 35% of the incident optical power. (b) Phase distribution for E x ; the gray scale covers the range from -180° (black) to +180° (white). (c) Intensity distribution for E y containing 11% of the incident power. (d) Phase distribution for E y ; the phase difference between adjacent quadrants is nearly 180°. (e) Polarization ellipticity η ranging from -45° (black) to +45° (white). (f) Polarization rotation angle ρ ranging from -90° (black) to +90° (white).

Fig. 10
Fig. 10

Plots of intensity and phase for the MO contribution to the reflected light, ΔE y , at the exit pupil of the objective lens of Fig. 3. The multilayer stack of Fig. 8 is assumed to be in front of the SIL with a 100-nm air gap. (a) Intensity distribution containing a fraction 0.37 × 10-4 of the incident optical power. (b) Phase distribution ranging from -50° (black) to +252° (white).

Fig. 11
Fig. 11

Total reflectance (solid curve) and the integrated intensity of the MO signal (dotted curve) at the exit pupil as functions of the gap width. These calculations correspond to the system of Fig. 3 in conjunction with the quadrilayer MO stack of Fig. 8, when a mask blocks the central region of the beam.

Fig. 12
Fig. 12

Schematic diagram of a differential detection module consisting of a Wollaston prism and two identical photodetectors. By being flexible to rotate around the optical axis Z, the module may be oriented arbitrarily relative to the ellipse of polarization of the incident beam. In particular, if the original linear polarization of the laser beam is along the X axis, and the magneto-optically generated component of polarization is along the Y axis, the module may be aligned such that the transmission axes of the Wollaston prism make 45° angles with X and Y.

Fig. 13
Fig. 13

Effective N.A. and power transmission efficiency versus r o /RA, representing the ratio of the 1/e amplitude radius (r o ) of the Gaussian beam to the aperture radius (RA) of the objective, for a Gaussian incident beam in conjunction with a 0.8-N.A. objective lens. For instance, the effective N.A. is 0.74, and the throughput power is 86% for a 1/e radius of r o = RA.

Fig. 14
Fig. 14

Optimum designs of the quadrilayer MO stack used in conjunction with the SIL. A Gaussian beam with a 1/e radius of r o = RA is assumed in the optical system of Fig. 3. Stacks B and C are optimized for the fixed gap widths of 100 and 200 nm, respectively. The search range of the optimum layer thickness is from 20 to 50 nm with a step of 2 nm for the inner dielectric layer, from 16 to 30 nm with a step of 2 nm for the magnetic layer, and from 50 to 200 nm with a step of 10 nm for the upper dielectric layer.

Fig. 15
Fig. 15

Computed plots of optical absorption within the MO disk versus gap width for different multilayer stacks of Figs. 8 and 14. Both uniform and Gaussian beams are assumed for stack A, whereas only the Gaussian beam is used for stacks B and C. (a) Results obtained for full aperture illumination. (b) Results obtained for evanescent coupling, where an annular mask is assumed to block the central region of the incoming laser beam, thus eliminating all the rays below the critical angle.

Fig. 16
Fig. 16

Computed plots of the normalized differential signal, (S 1 - S 2)/(S 1 + S 2), versus the gap width for the quadrilayer MO stacks of Figs. 8 and 14. These results correspond to the system of Fig. 3 in conjunction with the differential detector of Fig. 12. At 100-nm gap width, the signal derived from stack B (curve with filled circles) drops by as much as 40% from its peak.

Fig. 17
Fig. 17

Computed plots of the normalized differential signal, (S 1 - S 2)/(S 1 + S 2), versus the gap width for the quadrilayer MO stacks of Figs. 8 and 14. These results correspond to the system of Fig. 3 in conjunction with the differential detector of Fig. 12. An annular aperture is used to eliminate all the rays below the critical angle.

Fig. 18
Fig. 18

Quadrilayer structure of a typical PC disk (stack D) used in conjunction with the SIL. The local state of the PC layer, namely, crystalline or amorphous, represents the state of the recorded bit (0 or 1).

Fig. 19
Fig. 19

Optical readout system used in conjunction with a PC disk. A collimated beam of light, uniform, monochromatic (λ0 = 650 nm), and linearly polarized along X, is converted to a circularly polarized beam by a quarter-wave plate before it enters an aplanatic 0.8-N.A. objective lens (f = 3750λ0). As the disk spins, the SIL with reflective index n = 2 rides on an air cushion, which separates the two by a fixed gap width. The PBS selects the component of reflected light with polarization along the Y direction and sends it to the detector.

Fig. 20
Fig. 20

Computed distributions of the reflected beam at the detector of Fig. 19, when the PC stack of Fig. 18 is placed in front of the SIL with a 100-nm air gap. (a) Intensity distribution for the crystalline state, exhibiting an overall reflectance of 22.2%. (b) Intensity distribution for the amorphous state (overall reflectance = 12.5%). (c) Difference of the intensity distributions in (a) and (b).

Fig. 21
Fig. 21

Cross section of the difference of the intensity distributions between crystalline and amorphous states observed at the plane of the detector. The reflectivity difference in the region of evanescent wave coupling (effective N.A. > 1) is sharply reduced as the gap width widens. Owing to interference effects, the reflectivity difference is reserved at some incident angle.

Fig. 22
Fig. 22

Optimum designs of the quadrilayer PC stack used in conjunction with the SIL. A Gaussian beam with 1/e radius of r o = RA is assumed in the optical system of Fig. 19. Stacks E and F are optimized for the fixed gap widths of 100 and 200 nm, respectively. The search range for optimum layer thickness is from 20 to 60 nm with a step of 2 nm for the inner dielectric layer, from 20 to 60 nm with a step of 2 nm for the PC layer, and from 50 to 200 nm with a step of 10 for the upper dielectric layer.

Fig. 23
Fig. 23

Computed plots of optical absorption within the PC disk versus gap width for different multilayer stacks of Figs. 18 and 22. Both uniform and Gaussian beams are assumed for stack D, whereas only the Gaussian beam is used for stacks E and F. These results are obtained for the full-aperture illumination. (a) PC layer in the crystalline state. (b) PC layer in the amorphous state.

Fig. 24
Fig. 24

Computed plots of optical absorption within the PC disk versus gap width for different multilayer stacks of Figs. 18 and 22. These results are obtained for the case of evanescent coupling only, where an annular mask is used to block the central region of the incident beam. (a) PC layer in the crystalline state. (b) PC layer in the amorphous state.

Fig. 25
Fig. 25

Computed plots of the contrast at the detector versus the gap width for the quadrilayer PC stacks of Figs. 18 and 22. At 100-nm gap width, the contrast for stack E (curve with filled circles) is down 30% from its peak.

Fig. 26
Fig. 26

Computed plots of the contrast at the detector versus the gap width for the quadrilayer PC stacks of Figs. 18 and 22. The central region of the beam is blocked to confine attention to evanescently coupled rays only.

Equations (2)

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=0-000.
ΔS=S1-S2=12 |Ex+Ey|2-12 |Ex-Ey|2dxdy=2|ExEy|cos(ϕx-ϕy)dxdy.

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