Abstract

Periodic and quasi-periodic Cantor-like bandgap structures that bordered upon a medium of refractive index n 0 are analyzed. An immersion model is used with the assumption that each layer is embedded between two identical regions of refractive index n 0 and thickness d 0, where d 0 is set equal to zero. Transmittance and group velocity are determined. Their dependence on n 0 is emphasized. Relations for the midgap value of the normalized group velocity are given. By use of these relations, diagrams are completed at different values of n 0, showing the pairs of quarter-wave-layer refractive indices at which there is an apparent superluminal tunneling through the finite periodic and quasi-periodic Cantor-like bandgap structures.

© 2001 Optical Society of America

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References

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  1. C. M. Bowden, J. P. Dowling, H. O. Everitt, eds., feature on the development and applications of materials exhibiting photonic bandgaps, J. Opt. Soc. Am. B 10, 279–413 (1993).
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    [CrossRef]
  4. E. Yablonovitch, “Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filter,” Opt. Lett. 23, 1648–1649 (1998).
    [CrossRef]
  5. M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
    [CrossRef] [PubMed]
  6. M. Scalora, J. P. Dowling, C. M. Bowden, M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
    [CrossRef] [PubMed]
  7. P. Tran, “Optical limiting and switching of short pulses by use of a nonlinear photonic bandgap structure with a defect,” J. Opt. Soc. Am. B 14, 2589–2595 (1997).
    [CrossRef]
  8. J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
    [CrossRef]
  9. J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
    [CrossRef]
  10. A. M. Steinberg, P. G. Kwiat, R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
    [CrossRef] [PubMed]
  11. A. M. Steinberg, R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
    [CrossRef] [PubMed]
  12. M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, J. W. Haus, “Pulse propagation near highly reflective surfaces: applications to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
    [CrossRef] [PubMed]
  13. M. Bertolotti, P. Masciulli, C. Sibilia, “Spectral transmission properties of a self-similar optical Fabry–Perot resonator,” Opt. Lett. 19, 777–779 (1994).
    [CrossRef] [PubMed]
  14. C. Sibilia, I. S. Nefedov, M. Scalora, M. Bertolotti, “Electromagnetic mode density for finite quasi-periodic structures,” J. Opt. Soc. Am. B 15, 1947–1952 (1998).
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    [CrossRef]
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    [CrossRef]

1998 (3)

1997 (2)

N. Matuschek, F. X. Kärtner, U. Keller, “Exact coupled-mode theories for multilayer interference coatings with arbitrary strong index modulations,” IEEE J. Quantum Electron. 33, 295–302 (1997).
[CrossRef]

P. Tran, “Optical limiting and switching of short pulses by use of a nonlinear photonic bandgap structure with a defect,” J. Opt. Soc. Am. B 14, 2589–2595 (1997).
[CrossRef]

1996 (2)

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

1995 (1)

M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, J. W. Haus, “Pulse propagation near highly reflective surfaces: applications to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
[CrossRef] [PubMed]

1994 (5)

M. Bertolotti, P. Masciulli, C. Sibilia, “Spectral transmission properties of a self-similar optical Fabry–Perot resonator,” Opt. Lett. 19, 777–779 (1994).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, C. M. Bowden, M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

G. Kurizki, J. W. Haus, eds., feature on photonic band structures, J. Mod. Opt. 41, 171–404 (1994).

A. M. Steinberg, R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

1993 (2)

C. M. Bowden, J. P. Dowling, H. O. Everitt, eds., feature on the development and applications of materials exhibiting photonic bandgaps, J. Opt. Soc. Am. B 10, 279–413 (1993).

A. M. Steinberg, P. G. Kwiat, R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

1988 (1)

1977 (1)

1952 (1)

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Bertolotti, M.

Bloemer, M. J.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, C. M. Bowden, M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 1.6.

Bowden, C. M.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, J. W. Haus, “Pulse propagation near highly reflective surfaces: applications to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
[CrossRef] [PubMed]

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

Chiao, R. Y.

A. M. Steinberg, R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwiat, R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

Dowling, J. P.

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, J. W. Haus, “Pulse propagation near highly reflective surfaces: applications to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, C. M. Bowden, M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Epstein, L. I.

Fan, S.

Fink, Y.

Haus, J. W.

M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, J. W. Haus, “Pulse propagation near highly reflective surfaces: applications to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
[CrossRef] [PubMed]

Hong, C. S.

Joannopoulos, J. D.

Kärtner, F. X.

N. Matuschek, F. X. Kärtner, U. Keller, “Exact coupled-mode theories for multilayer interference coatings with arbitrary strong index modulations,” IEEE J. Quantum Electron. 33, 295–302 (1997).
[CrossRef]

Keller, U.

N. Matuschek, F. X. Kärtner, U. Keller, “Exact coupled-mode theories for multilayer interference coatings with arbitrary strong index modulations,” IEEE J. Quantum Electron. 33, 295–302 (1997).
[CrossRef]

Kwiat, P. G.

A. M. Steinberg, P. G. Kwiat, R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

Manka, A. S.

M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, J. W. Haus, “Pulse propagation near highly reflective surfaces: applications to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
[CrossRef] [PubMed]

Masciulli, P.

Matuschek, N.

N. Matuschek, F. X. Kärtner, U. Keller, “Exact coupled-mode theories for multilayer interference coatings with arbitrary strong index modulations,” IEEE J. Quantum Electron. 33, 295–302 (1997).
[CrossRef]

Nefedov, I. S.

Scalora, M.

C. Sibilia, I. S. Nefedov, M. Scalora, M. Bertolotti, “Electromagnetic mode density for finite quasi-periodic structures,” J. Opt. Soc. Am. B 15, 1947–1952 (1998).
[CrossRef]

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, J. W. Haus, “Pulse propagation near highly reflective surfaces: applications to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
[CrossRef] [PubMed]

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

Sibilia, C.

Spink, D. M.

Steinberg, A. M.

A. M. Steinberg, R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwiat, R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

Thomas, C. B.

Tocci, M. D.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

Tran, P.

Winn, J. N.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 1.6.

Yablonovitch, E.

Yariv, A.

Yeh, P.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

N. Matuschek, F. X. Kärtner, U. Keller, “Exact coupled-mode theories for multilayer interference coatings with arbitrary strong index modulations,” IEEE J. Quantum Electron. 33, 295–302 (1997).
[CrossRef]

J. Appl. Phys. (1)

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

J. Mod. Opt. (1)

G. Kurizki, J. W. Haus, eds., feature on photonic band structures, J. Mod. Opt. 41, 171–404 (1994).

J. Opt. Soc. Am. (2)

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

Opt. Lett. (3)

Phys. Rev. A (3)

A. M. Steinberg, R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, J. W. Haus, “Pulse propagation near highly reflective surfaces: applications to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
[CrossRef] [PubMed]

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

Phys. Rev. E (1)

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Phys. Rev. Lett. (2)

A. M. Steinberg, P. G. Kwiat, R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, C. M. Bowden, M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

Other (1)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 1.6.

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Figures (13)

Fig. 1
Fig. 1

Illustration of a plane wave of unit amplitude and zero phase at z = 0 that is normally incident upon a medium of variable refractive index n(z) and physical thickness d. This medium is bordered by two (hatched) regions of refractive index n 0. The complex amplitude transmission and reflection coefficients are t and r, respectively.

Fig. 2
Fig. 2

(a) Illustration of the N-period finite structure that results from repeating N times the unit-cell refractive-index variation n(z) of Fig. 1. The finite periodic structure is bordered by the (hatched) regions of refractive index n 0. (b) Illustration for the immersion model. Each unit cell is embedded between two (hatched) regions of refractive index n 0 and thickness d 0, where d 0 is set equal to zero. In both figures the complex amplitude transmission and reflection coefficients are t (N) and r (N), respectively.

Fig. 3
Fig. 3

Variations of T (N) relative to ω̅ for a finite periodic structure with N = 6 periods, where each period comprises two quarter-wave layers of refractive indices n 1 = 2.5 and n 2 = 1.5 when n 0 = 1 (solid curve) and n 0 = 1.5 (dotted curve). The ω̅ values that correspond to the forbidden gap are specified by a short, thick line that lies upon the ω̅ axis.

Fig. 4
Fig. 4

Variation of normalized Bloch group velocity V B /c relative to ω̅ for a periodic structure with alternating quarter-wave layers of refractive indices n 1 = 2.5 and n 2 = 1.5. The bandgap is specified by a short, thick line that lies upon the ω̅ axis. Only the real values of V B /c are taken into account.

Fig. 5
Fig. 5

Variation of V N /c relative to ω̅ for a finite periodic structure of N = 6 periods with alternating quarter-wave layers of refractive indices n 1 = 2.5 and n 2 = 1.5 when n 0 = 1 (solid curve) and n 0 = 1.5 (dotted curve). For comparison, the positive and real values of the normalized Bloch group velocity are also plotted against ω̅, and that curve is marked by asterisks. The bandgap is specified by the thick line that lies upon the ω̅ axis.

Fig. 6
Fig. 6

(a) Variation of T (N) and (b) variation of V N /c relative to ω̅ for a finite periodic structure of N = 14 periods with alternating quarter-wave layers of refractive indices n 1 = 2.5 and n 2 = 1.5 in air; n 0 = 1. The dotted horizontal line in (b) represents V N /c = 1. The bandgaps are specified by the thick lines that lie upon the ω̅ axes.

Fig. 7
Fig. 7

Variations of (a) BER normalized group velocity V N BER/c and (b) MG normalized group velocity V N MG/c relative to the number of periods N for a finite periodic structure with alternating quarter-wave layers of refractive indices n 1 = 2.5 and n 2 = 1.5 when n 0 = 1 (solid curves) and n 0 = 1.5 (dotted curves).

Fig. 8
Fig. 8

Regimes (shaded areas) of layer refractive indices n 1 and n 2 at which there is an apparent superluminal tunneling (V N MG/c > 1) through the finite periodic bandgap structure of N = 14 periods with alternating quarter-wave layers when (a) n 0 = 1 and (b) n 0 = 1.5.

Fig. 9
Fig. 9

(a) Illustration of the generation of a self-similar Cantor set13 at stages N = 0, 1, 2, 3. The structure at N = 0 is the initiator, and the one that corresponds to N = 1 is the generator. (b) Configuration of the three-stage Cantor-like multilayer when the initiator is a high-refractive-index layer (n 1 > n 2). Darker- and lighter-shaded areas correspond to the regions of high and low refractive indices, respectively. The complex amplitude transmission and reflection coefficients are t (N) and r (N), respectively, with N = 3. The finite quasi-periodic Cantor-like multilayer is bordered by the medium of refractive index n 0.

Fig. 10
Fig. 10

Variations of T (N) relative to ω̅ for a three-stage Cantor-like multilayer with layer refractive indices n 1 = 2.5 and n 2 = 1.5 when (a) n 0 = 1 and (b) n 0 = 1.5; ω̅ is varied in increments of 0.0001.

Fig. 11
Fig. 11

Variations of V N /c relative to ω̅ for a three-stage Cantor-like multilayer with layer refractive indices n 1 = 2.5 and n 2 = 1.5 when (a) n 0 = 1 and (b) n 0 = 1.5. Dotted lines represent V N /c = 1; ω̅ is varied in increments of 0.005.

Fig. 12
Fig. 12

Variations of V N MG/c relative to N for a Cantor-like multilayer with layer refractive indices n 1 = 2.5 and n 2 = 1.5 when n 0 = 1 and n 0 = 1.5 [curve marked by asterisks (*)].

Fig. 13
Fig. 13

Regimes (shaded areas) of layer refractive indices n 1 and n 2 at which there is an apparent superluminal tunneling (V N MG/c > 1) through the three-stage Cantor-like bandgap structure when (a) n 0 = 1 and (b) n 0 = 1.5.

Equations (47)

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

1/V=ζ/d1+ζ2,
=1/tr*/t*r/t1/t*,
μ2-2ξμ+1=0.
N= sin Nβ/sin β- sinN-1β/sin β,
N=-1N+1 sinh Nχ/sinh χ+ sinhN-1χ/sinh χ.
N=1/tNrN/tN*rN/tN1/tN*,
1/tN=ξN-jηN,  rN/tN=δN-jγN.
ζN=ζ tan Nβ cot β.
VN=ζN/D1+ζN2,
ti=T0iXi/1-R0iXi2,
ri=r0i1-Xi2/1-R0iXi2,
ϕi=2π/qiω¯.
ξ=cosϕ1+ϕ2-R12 cosϕ1-ϕ2/T12,
η=g1 sinϕ1+ϕ2+g2R12 sinϕ1-ϕ2/T12,
δ=r12cosϕ1-ϕ2-cosϕ1+ϕ2/T12,
γ=-r12g2 sinϕ1+ϕ2+g1 sinϕ1-ϕ2/T12,
g1=1+r01r02/1-r01r02,
g2=r01+r02/r01-r02.
ξ=cos πω¯-R12/T12,
η=g1 sin πω¯/T12,
δ=r121-cos πω¯/T12,
γ=-g2r12 sin πω¯/T12.
1/TN=1+1/T-1sin2 Nβ/sin2 β
1/TN=1+1/T-1sinh2 Nχ/sinh2 χ
1/TNMG=1+1/4n1/n2N-n2/n1N2.
VB/c=1/21/n1+1/n2T122-cos πω¯-R1221/2/sin πω¯,
VNBER/c=2/g1n1+n21+R12 cot2π/2N;
VNMG/c=-2N/g1n1+n2sin βMG/tan NβMG,
βMG=arccos-n1/n2+n2/n1/2.
D=2Nd1+3N-2Nd2,
Lopt=3Nλ0/4.
Nϕ=N-1ϕ23N-1ϕN-1ϕ.
ξ1=cos 3ϕ-R12 cos ϕ/T12,
η1=c1sin 3ϕ+c3 sin ϕ/T12,
δ1=0,  γ1=c4 sin 3ϕ+c5 sin ϕ,
ci=1+R0i/T0i,  i=1, 2,
c3=4r01r12/1+R01-R12,
c4=2r01/T01T12,
c5=2r121+R01-r01r12/T01T12,
c6=2r02/1+R02.
ξN=2ξN-12-1cos3N-1ϕ+2c2c6γN-1-ηN-1ξN-1 sin3N-1ϕ,
ηN=2ηN-1ξN-1 cos3N-1ϕ+c21+2c6γN-1-ηN-1ηN-1sin3N-1ϕ,
γN=2γN-1ξN-1 cos3N-1ϕ-c2c61-2/TN-1+2γN-1ηN-1sin3N-1ϕ,
1/TN-1=ξN-12+ηN-12.
1/TNMG=1+1/4n0/NN-NN/n02,
NN=n1κ/n2κ-1,  κ=2N.
VNMG/c=-GηN/ξdNMG,

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