Abstract

Forbidden gaps in Šolc folded-type anisotropic periodic structures are analyzed. The azimuth angle in the first quadrant and unequal thicknesses of alternating uniaxial layers are considered. Unit-cell translation matrices are explicitly given at oblique and normal incidences. Implicit dispersion relations are obtained. Birefringence and normal group velocities are determined. Numerical examples make the distinction between the gaps determined by one root and those determined by two roots of the eigenvalue equation. When layers have unequal thicknesses, the gap determined by one root is much wider, but inside the gap the reflectivity is lower than unity, breaks appear, and an increased birefringence results.

© 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 Development and Applications of Materials Exhibiting Photonic Band Gaps, J. Opt. Soc. Am. B 10, 279–413 (1993).
  2. G. Kurizki, J. W. Haus, eds., feature on Photonic Band Structures, J. Mod. Opt. 41, 171–404 (1994).
  3. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063 (1987).
    [CrossRef] [PubMed]
  4. I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
    [CrossRef]
  5. I. Šolc, “A new kind of double refracting filter,” Czech. J. Phys. 4, 53–66 (1954).
  6. I. Šolc, “Birefringent chain filters,” J. Opt. Soc. Am. 55, 621–625 (1965).
    [CrossRef]
  7. A. Lakhtakia, “Dielectric sculptured thin films as Šolc filters,” Opt. Eng. 37, 1870–1875 (1998).
    [CrossRef]
  8. A. Yariv, P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. 67, 438–448 (1977).
    [CrossRef]
  9. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  10. G. D. Landry, T. A. Maldonado, “Complete method to determine transmission and reflection characteristics at a planar interface between arbitrarily oriented biaxial madia,” J. Opt. Soc. Am. A 12, 2048–2063 (1995).
    [CrossRef]
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.
  12. C. Gu, P. Yeh, “Extended Jones matrix method. II,” J. Opt. Soc. Am. A 10, 966–973 (1993).
    [CrossRef]
  13. E. Cojocaru, “Explicit relations for the extraordinary-ray trajectory at the back of a rotating uniaxial birefringent plate,” Appl. Opt. 36, 8886–8888 (1997).
    [CrossRef]
  14. E. Cojocaru, “Forbidden gaps in periodic anisotropic layered media,” Appl. Opt. 39, 4641–4648 (2000).
    [CrossRef]
  15. J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
    [CrossRef]
  16. C. Gu, P. Yeh, “Form birefringence dispersion in periodic layered media,” Opt. Lett. 21, 504–506 (1996).
    [CrossRef] [PubMed]
  17. 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]

2000 (2)

I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
[CrossRef]

E. Cojocaru, “Forbidden gaps in periodic anisotropic layered media,” Appl. Opt. 39, 4641–4648 (2000).
[CrossRef]

1998 (1)

A. Lakhtakia, “Dielectric sculptured thin films as Šolc filters,” Opt. Eng. 37, 1870–1875 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

1994 (3)

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. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

G. Kurizki, J. W. Haus, eds., feature on Photonic Band Structures, J. Mod. Opt. 41, 171–404 (1994).

1993 (2)

C. M. Bowden, J. P. Dowling, H. O. Everitt, eds., feature on Development and Applications of Materials Exhibiting Photonic Band Gaps, J. Opt. Soc. Am. B 10, 279–413 (1993).

C. Gu, P. Yeh, “Extended Jones matrix method. II,” J. Opt. Soc. Am. A 10, 966–973 (1993).
[CrossRef]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063 (1987).
[CrossRef] [PubMed]

1979 (1)

1977 (1)

1965 (1)

1954 (1)

I. Šolc, “A new kind of double refracting filter,” Czech. J. Phys. 4, 53–66 (1954).

Abdulhalim, I.

I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
[CrossRef]

Bloemer, M. J.

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), Chap. 14, pp. 665–718.

Bowden, C. M.

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. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

Cojocaru, E.

Dowling, J. P.

J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

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]

Gu, C.

Lakhtakia, A.

A. Lakhtakia, “Dielectric sculptured thin films as Šolc filters,” Opt. Eng. 37, 1870–1875 (1998).
[CrossRef]

Landry, G. D.

Maldonado, T. A.

Scalora, M.

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]

Šolc, I.

I. Šolc, “Birefringent chain filters,” J. Opt. Soc. Am. 55, 621–625 (1965).
[CrossRef]

I. Šolc, “A new kind of double refracting filter,” Czech. J. Phys. 4, 53–66 (1954).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063 (1987).
[CrossRef] [PubMed]

Yariv, A.

Yeh, P.

Appl. Opt. (2)

Czech. J. Phys. (1)

I. Šolc, “A new kind of double refracting filter,” Czech. J. Phys. 4, 53–66 (1954).

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. (2)

J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

G. Kurizki, J. W. Haus, eds., feature on Photonic Band Structures, J. Mod. Opt. 41, 171–404 (1994).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

C. M. Bowden, J. P. Dowling, H. O. Everitt, eds., feature on Development and Applications of Materials Exhibiting Photonic Band Gaps, J. Opt. Soc. Am. B 10, 279–413 (1993).

Opt. Commun. (1)

I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
[CrossRef]

Opt. Eng. (1)

A. Lakhtakia, “Dielectric sculptured thin films as Šolc filters,” Opt. Eng. 37, 1870–1875 (1998).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063 (1987).
[CrossRef] [PubMed]

Other (1)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.

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Figures (8)

Fig. 1
Fig. 1

Šolc folded-type anisotropic periodic structure of alternating uniaxial layers with azimuth angles +ϕ and -ϕ and thicknesses d 1 = Λf and d 2 = Λ(1 - f), where Λ is the period and f is a subunitary number. Interfaces lie in the xy plane. The medium of incidence is isotropic of refractive index n 0; k 0 is the incident wave vector at angle φ0 with respect to the positive z axis.

Fig. 2
Fig. 2

Gap maps of the lowest forbidden bands in the (Ω, f) space when φ0 = 30° and ϕ = 40°. The lighter-shaded area represents the lowest gap that is determined by one root of eigenvalue equation (21). It includes the smaller dark-shaded area representing the gap determined by two roots. The curves represent Eqs. (40) with N = 1.

Fig. 3
Fig. 3

Gap maps of the lowest forbidden bands in the (Ω, f) space at normal incidence (φ0 = 0) when ϕ = 45°. The lighter- and the dark-shaded areas correspond to dispersion relations (31a) and (31b), respectively. The curves plotted on the center of either branch represent Eqs. (40) with ζ o and ζ eo replaced with n and n , respectively, and N = 1.

Fig. 4
Fig. 4

Width of the Ω interval, δΩ, in the lowest gap that is determined by one root, versus f and ϕ at normal incidence. The plot is symmetrical against axes f = 0.5 and ϕ = 45°. Only one quadrant of the symmetrical plot is shown. f is varied from 0.1 to 0.9 in increments of 0.01, ϕ from 5° to 85° in increments of 5°, and Ω from 0.2 to 0.3 in increments of 0.001.

Fig. 5
Fig. 5

Variation of R pt (dotted curve) and R st (solid curve) against Ω at normal incidence when ϕ = 40° at (a) f = 0.5 and (b) f = 0.27. The Ω interval in the gap that is determined by two roots of the eigenvalue equation is shown by the cross-hatched region, and the Ω interval in the gap that is determined by one root is simply hatched; Ω is varied from 0.2 to 0.4 in increments of 0.00075.

Fig. 6
Fig. 6

Values of Ω at which gap condition (27) is satisfied by one (×) and two (*) roots of the eigenvalue equation at various incidence angles φ0 when ϕ = 40° at (a) f = 0.5 and (b) f = 0.27. The curve plotted in (a) represents any of Eqs. (40) with f = 0.5 and N = 1; Ω is varied from 0.2 to 0.4 in increments of 0.00075, and φ0 from 0 to 90° in increments of 2.5°.

Fig. 7
Fig. 7

Dispersion of birefringence in the case of oblique incidence when φ0 = 30° and ϕ = 40° at (a) f = 0.5 and (b) f = 0.27 and in the case of normal incidence (φ0 = 0) when ϕ = 45° at (c) f = 0.5 and (d) f = 0.27. At normal incidence, in (c) and (d), the phase-velocity index n K is determined with dispersion relations (31). The curve that is obtained with Eq. (31b) is marked in (d) with small circles. In (a)–(c), the Ω interval in the gap that is determined by two roots of the eigenvalue equation is represented by the cross-hatched region, whereas in (b) and (d), the Ω interval in the gap which is determined by one root is simply hatched.

Fig. 8
Fig. 8

Variation of the normal component of the group velocity normalized to the light velocity in vacuum, V gz /c, against Ω in the case of oblique incidence when φ0 = 30° and ϕ = 40° at (a) f = 0.5 and (b) f = 0.27 and in the case of normal incidence (φ0 = 0) when ϕ = 45° at (c) f = 0.5 and (d) f = 0.27. At normal incidence, in (c) and (d), V gz /c is determined by Eq. (39) and another expression that is derived from it when the subscripts o and eo are interchanged. The curve that represents Eq. (39) is marked in (d) by small circles. The significance of the simple-hatched and the cross-hatched regions is that given in Fig. 7.

Equations (76)

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P±=cos ϕ±sin ϕ0001±sin ϕ-cos ϕ0.
kνkxxˆ+kνzzˆ=ω/cξxˆ+ζνzˆ,  ν=0, o, eo,
ξ=nν sin φν,  ζν=nν cos φν,  ν=0, o, eo.
ζo=n2-ξ21/2,
ζeo=n2-ξ21+sin2 ϕn2/n2-11/2.
E0=E0ssˆ+E0ppˆexpjk0r-ωt,
cos ηeo=n2+n2-n2cos2 βeo/n4+n4-n4×cos2 βeo1/2,
tan ηeo=n2-n2sin βeo cos βeo/n2+n2-n2×cos2 βeo,
cos βν=sin ϕ sin φν,sin βν=1-cos2 βν1/2, ν=o, eo.
E0sif, E0sib, E0pif, E0pibT=C±Eoif, Eoib, Eeoif, EeoibT
Ē0i=C±Ēi,
Ē0i+1=C±XiĒi,
Xi=diagXoi, Xoi*, Xeoi, Xeoi*,
Xνi=expjkνzdi, ν=o, eo.
T=C+-1C-X2C--1C+X1.
T=T11T12T13T14T12*T11*-T14*-T13*T31T32T33T34-T32*-T31*T34*T33*,
T11=Xo1cos τeo2+αo1Ao2+Aeo2/2+jAoAeo sin τo2+g0+ sin τeo2,
T12=-Xo1*αo1Ao2-Aeo2/2+jg0- sin τeo2,
T13=μXeo1αo1g1++jαo2g2+,
T14=μXeo1*αo1g1-+jαg2-,
T31=-Xo1/μαeo1g3++jαeo2g4+,
T32=Xo1*/μαeo1g3-+jαg4-,
T33=Xeo1cos τo2+αeo1Ao2+Aeo2/2+jAoAeo sin τeo2+g0+ sin τo2,
T34=-Xeo1*αeo1Aeo2-Ao2/2+jg0- sin τo2.
τνi=ω/cζνdi,  ν=o, eo,  i=1, 2,
α=sin τo2+sin τeo2,
αo1=cos τo2-cos τeo2,  αeo1=-αo1,
αo2=sin τo2-sin τeo2,  αeo2=-αo2,
Aν=aν-1/aν+1,  ν=o, eo,
aν=ρν tan2 ϕ cos2 φν,  ν=o, eo,
ρo=1, ρeo=1+tan ηeo sin 2βeo/2 sin2 ϕ cos2 φeo,
μ=-noζeo cos ηeo sin βo/ζo tan ϕ sin βeo,
g0±=1+Aeo1-Aoζeo/ζo±1-Aeo×1+Aoζo/ζeo/2,
g1±=Ao1+Ao/ζeo±Aeo1+Aeo/ζo/2,
g2±=Aeo1+Ao/ζeo±Ao1+Aeo/ζo/2,
g3±=Ao1-Aoζeo±Aeo1-Aeoζo/2,
g4±=Aeo1-Aoζeo±Ao1-Aeoζo/2.
T11=Xo1cos2 2ϕ cos τo2+sin2 2ϕ cos τeo2+jcos2 2ϕ sin τo2+g+ sin2 2ϕ sin τeo2,
T12=-jXo1*g- sin2 2ϕ sin τeo2,
T13=Xeo1neo+no/4nosin 4ϕαo1+jαo2,
T14=-Xeo1*neo-no/4nosin 4ϕαo1+jα,
T31=-Xo1neo+no/4neosin 4ϕαeo1+jαeo2,
T32=Xo1*neo-no/4neosin 4ϕαeo1+jα,
T33=Xeo1cos2 2ϕ cos τeo2+sin2 2ϕ cos τo2+jcos2 2ϕ sin τeo2+g+ sin2 2ϕ sin τo2,
T34=-jXeo1*g- sin2 2ϕ sin τo2,
EKx, z, t=EKzexpjKzexpjkxx-ωt.
K=kxxˆ+Kzˆ.
detT-γI=0,
γ4-c1+c2γ3+c1c2+c3γ2-c1+c2γ+1=0,
c1=T11+T11*, c2=T33+T33*, c3=T11T11*+T33T33*-T12T12*-T34T34*+2 realT14T32*-T13T31.
γ2+b+γ+1γ2+b-γ+1=0,
b±=-c1+c2±c1-c22-4c3-21/2/2.
exp-jKΛ=-b±±b±2-41/2/2.
|realexp-jKΛ|>1.
ΩωΛ/2πc=Λ/λ,
c1=2cos2 2ϕ cosτo1+τo2+sin2 2ϕFo1,
c2=2cos2 2ϕ cosτeo1+τeo2+sin2 2ϕFeo1,
c3=2-sin2 4ϕ1-F1-F2+Fo2+Feo2/2,
Fo1=cos τo1 cos τeo2-g+ sin τo1 sin τeo2,
Feo1=cos τeo1 cos τo2-g+ sin τeo1 sin τo2,
Fo2=cos τeo1 cos2τo2+τo1-g+ sin τeo1 sin2τo2+τo1,
Feo2=cos τo1 cos2τeo2+τeo1-g+ sin τo1 sin2τeo2+τeo1,
F1=cos τo2 cos τeo2+g+ sin τo2 sin τeo2,
F2=cosτo1+τo2cosτeo1+τeo2-g+ sinτo1+τo2×sinτeo1+τeo2.
cosKΛFo1=g++1cosτo1+τeo2-g+-1cosτo1-τeo2/2,
cosKΛFeo1=g++1cosτeo1+τo2-g+-1cosτeo1-τo2/2.
K=j/Λ ln γ.
Vp=ω/kx2+K21/2.
nK=realξ2+c/ωK21/2.
ω, Kexp-jKΛ+b±+b±2-41/2/2=0.
Vgz=real-/Kω//ωK.
Vgz/c=real-jb±2-41/2/b±c/Λ,
realω, KcosKΛ-Fν1=0,  ν=o, eo.
Vgz/c=τeo1/neo+τo2/no16no2neo2-no+neo2×cosτeo1+τo2-no-neo2 cosτeo1-τo221/2/no+neo2τeo1+τo2sinτeo1+τo2-no-neo2τeo1-τo2sinτeo1-τo2.
Ω=N/2ζeo+fζo-ζeo,
Ω=N/2ζo+fζeo-ζo,

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