Abstract

A subwavelength grating structure made in an isotropic medium induces form birefringence effects, and the artificially produced optical axis is parallel to the grating vector. The phase shift between the two orthogonal electric-field components exiting this grating varies linearly with the thickness of the grating. When a grating with subwavelength period is formed on a uniaxial birefringent material with the grating vector aligned parallel to its natural optical axis, the total effect enhances the birefringence of the material. As a result, the thickness of the material can be reduced and still produce the same phase shift. If the natural optical axis and the induced optical axis lie within the surface plane and have an angular separation between them, the phase shift varies nonlinearly with the thickness of the grating.

© 1996 Optical Society of America

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References

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  1. C.-W. Han, E. W. Campbell, R. K. Kostuk, “Rigorous coupled-wave analysis of surface relief gratings in anisotropic media,” presented at the OSA Annual Meeting, Dallas, Texas, October 2–7, 1994.
  2. C.-W. Han, E. W. Campbell, R. K. Kostuk, “Diffraction properties of anisotropic surface relief gratings with arbitrary orientation of the optical axis,” submitted to J. Opt. Soc. Am. A.
  3. E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. A 4, 2061–1080 (1987).
    [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  5. A. Yariv, P. Yeh, Optical Waves in Crystals, Propagation and Control of Laser Radiation (Wiley, New York, 1984).
  6. K. Shiraishi, T. Sato, S. Kawakami, “Experimental verification of a form-birefringent polarization splitter,” Appl. Phys. Lett. 58, 211–212 (1991).
    [CrossRef]
  7. E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
    [CrossRef]
  8. D. H. Raguin, G. M. Morris, “Analysis of antireflection-structured surfaces with continuous one-dimensional surface profiles,” Appl. Opt. 32, 2582–2598 (1993).
    [CrossRef] [PubMed]
  9. E. B. Grann, M. G. Moharam, D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am. A 11, 2695–2703 (1994).
    [CrossRef]
  10. G. Campbell, R. K. Kostuk, “Effective-medium theory of sinusoidally modulated volume holograms,” J. Opt. Soc. Am. A 12, 1113–1117 (1995).
    [CrossRef]
  11. G. Campbell, L. Li, R. K. Kostuk, “Zeroth-order effective medium theory of zeroth-order volume grating,” in Application and Theory of Periodic Structures, T. Jannson, ed., Proc. SPIE2532, 33–43 (1995).
    [CrossRef]
  12. E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1989).

1995 (1)

1994 (1)

1993 (1)

1992 (1)

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
[CrossRef]

1991 (1)

K. Shiraishi, T. Sato, S. Kawakami, “Experimental verification of a form-birefringent polarization splitter,” Appl. Phys. Lett. 58, 211–212 (1991).
[CrossRef]

1987 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Campbell, E. W.

C.-W. Han, E. W. Campbell, R. K. Kostuk, “Rigorous coupled-wave analysis of surface relief gratings in anisotropic media,” presented at the OSA Annual Meeting, Dallas, Texas, October 2–7, 1994.

C.-W. Han, E. W. Campbell, R. K. Kostuk, “Diffraction properties of anisotropic surface relief gratings with arbitrary orientation of the optical axis,” submitted to J. Opt. Soc. Am. A.

Campbell, G.

G. Campbell, R. K. Kostuk, “Effective-medium theory of sinusoidally modulated volume holograms,” J. Opt. Soc. Am. A 12, 1113–1117 (1995).
[CrossRef]

G. Campbell, L. Li, R. K. Kostuk, “Zeroth-order effective medium theory of zeroth-order volume grating,” in Application and Theory of Periodic Structures, T. Jannson, ed., Proc. SPIE2532, 33–43 (1995).
[CrossRef]

Gaylord, T. K.

Gluch, E.

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
[CrossRef]

Glytsis, E. N.

Grann, E. B.

Haidner, H.

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
[CrossRef]

Han, C.-W.

C.-W. Han, E. W. Campbell, R. K. Kostuk, “Rigorous coupled-wave analysis of surface relief gratings in anisotropic media,” presented at the OSA Annual Meeting, Dallas, Texas, October 2–7, 1994.

C.-W. Han, E. W. Campbell, R. K. Kostuk, “Diffraction properties of anisotropic surface relief gratings with arbitrary orientation of the optical axis,” submitted to J. Opt. Soc. Am. A.

Hecht, E.

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1989).

Kawakami, S.

K. Shiraishi, T. Sato, S. Kawakami, “Experimental verification of a form-birefringent polarization splitter,” Appl. Phys. Lett. 58, 211–212 (1991).
[CrossRef]

Kipfer, P.

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
[CrossRef]

Kostuk, R. K.

G. Campbell, R. K. Kostuk, “Effective-medium theory of sinusoidally modulated volume holograms,” J. Opt. Soc. Am. A 12, 1113–1117 (1995).
[CrossRef]

G. Campbell, L. Li, R. K. Kostuk, “Zeroth-order effective medium theory of zeroth-order volume grating,” in Application and Theory of Periodic Structures, T. Jannson, ed., Proc. SPIE2532, 33–43 (1995).
[CrossRef]

C.-W. Han, E. W. Campbell, R. K. Kostuk, “Rigorous coupled-wave analysis of surface relief gratings in anisotropic media,” presented at the OSA Annual Meeting, Dallas, Texas, October 2–7, 1994.

C.-W. Han, E. W. Campbell, R. K. Kostuk, “Diffraction properties of anisotropic surface relief gratings with arbitrary orientation of the optical axis,” submitted to J. Opt. Soc. Am. A.

Li, L.

G. Campbell, L. Li, R. K. Kostuk, “Zeroth-order effective medium theory of zeroth-order volume grating,” in Application and Theory of Periodic Structures, T. Jannson, ed., Proc. SPIE2532, 33–43 (1995).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Pommet, D. A.

Raguin, D. H.

Sato, T.

K. Shiraishi, T. Sato, S. Kawakami, “Experimental verification of a form-birefringent polarization splitter,” Appl. Phys. Lett. 58, 211–212 (1991).
[CrossRef]

Sheridan, J. T.

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
[CrossRef]

Shiraishi, K.

K. Shiraishi, T. Sato, S. Kawakami, “Experimental verification of a form-birefringent polarization splitter,” Appl. Phys. Lett. 58, 211–212 (1991).
[CrossRef]

Streibl, N.

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals, Propagation and Control of Laser Radiation (Wiley, New York, 1984).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals, Propagation and Control of Laser Radiation (Wiley, New York, 1984).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

K. Shiraishi, T. Sato, S. Kawakami, “Experimental verification of a form-birefringent polarization splitter,” Appl. Phys. Lett. 58, 211–212 (1991).
[CrossRef]

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

Opt. Commun. (1)

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
[CrossRef]

Other (6)

C.-W. Han, E. W. Campbell, R. K. Kostuk, “Rigorous coupled-wave analysis of surface relief gratings in anisotropic media,” presented at the OSA Annual Meeting, Dallas, Texas, October 2–7, 1994.

C.-W. Han, E. W. Campbell, R. K. Kostuk, “Diffraction properties of anisotropic surface relief gratings with arbitrary orientation of the optical axis,” submitted to J. Opt. Soc. Am. A.

G. Campbell, L. Li, R. K. Kostuk, “Zeroth-order effective medium theory of zeroth-order volume grating,” in Application and Theory of Periodic Structures, T. Jannson, ed., Proc. SPIE2532, 33–43 (1995).
[CrossRef]

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1989).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

A. Yariv, P. Yeh, Optical Waves in Crystals, Propagation and Control of Laser Radiation (Wiley, New York, 1984).

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Figures (9)

Fig. 1
Fig. 1

Light normally incident on a lamellar grating formed in an isotropic substrate.

Fig. 2
Fig. 2

(a) Light normally incident on a homogenous quartz substrate, (b) phase shift versus thickness for quartz substrate.

Fig. 3
Fig. 3

(a) Light normally incident on a subwavelength lamellar grating formed in quartz substrate, (b) phase shift versus thickness for a lamellar grating in quartz substrate with parallel natural and induced optical axes.

Fig. 4
Fig. 4

(a) Rotation of the natural optical axis in the XZ plane with respect to the induced optical axis, (b) phase shift versus thickness for 0°–40° optical axis rotation, (c) phase shift versus thickness for 0° and 50°–90° optical axis rotation.

Fig. 5
Fig. 5

(a) Phase shift versus thickness when the angle of incident is a 48.6°, at Bragg angle; (b) phase shift versus thickness when the angle of incident is at 40°, off-Bragg angle; (c) phase shift versus thickness when the angle of incident is at 20°, off-Bragg angle; (d) phase shift versus thickness when the angle of incident is at 10°, off-Bragg angle.

Fig. 6
Fig. 6

(a) Optical axis parallel to the grating vector, (b) optical axis perpendicular to the grating vector.

Fig. 7
Fig. 7

Fringe planes in a surface-relief hologram.

Fig. 8
Fig. 8

Phase shift versus thickness for rectangular, sinusoidal, and triangular surface relief profiles.

Fig. 9
Fig. 9

Boundaries for three layers.

Equations (14)

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ϕ = 2 π λ ( n E n O ) d ,
O ˆ = C ˆ × n ˆ | C ˆ × n ˆ | = C z x ˆ + C x z ˆ | C x 2 + C z 2 | , E ˆ = C ˆ = C x x ˆ + C z z ˆ .
T O i = | T O i | exp ( j θ O i ) , T E i = | T E i | exp ( j θ E i ) , | T O i | = ( T O i · O ˆ ) re 2 + ( T O i · O ˆ ) im 2 , | T E i | = ( T E i · E ˆ ) re 2 + ( T E i · E ˆ ) im 2 , θ O i = tan 1 ( ( T O i · O ˆ ) im ( T O i · O ˆ ) re ) , θ E i = tan 1 ( ( T E i · E ˆ ) im ( T E i · E ˆ ) re ) , phase shift = ϕ = θ E i θ O i ,
ϕ O A = 2 π λ ( n eff , E n O ) d .
ϕ OA = 2 π λ ( n E n eff , O ) d .
= f 1 1 + f 2 2 , = 1 2 f 1 2 + f 2 1 ,
ϕ O A = 2 π λ ( n n ) d = 68.31 ° ,
ϕ O A = 2 π λ ( n n ) d = 72.86 ° .
S x m l ( y ) exp [ j ( k x m x m k y l y + k z z ) ] , S x m ( l 1 ) ( y ) exp [ j ( k x m x m k y ( l 1 ) y + k z z ) ] ,
S xml ( ψ l ) exp [ j ( k x m x + m k y l ψ l + k z z ) ] = S x m ( l 1 ) ( ψ l ) × exp [ j ( k x m x + m k y ( l 1 ) ψ l + k z z ) ] , S x m l ( ψ l ) = S x m ( l 1 ) ( ψ l ) exp [ j m ( k y ( l 1 ) k y l ) ψ l ] .
V l ( ψ l ) = V ( l 1 ) ( ψ l ) exp [ j m ( k y ( l 1 ) k y l ) ψ l ] ,
V l ( y ) = W ˜ l { exp [ Λ ˜ l ( y + ψ l ) ] } C l , V ( l 1 ) ( y ) = W ˜ ( l 1 ) { exp [ Λ ˜ ( l 1 ) ( y + ψ ( l 1 ) ) ] } C ( l 1 ) .
V l ( ψ l ) = W ˜ l C l , V ( l 1 ) ( ψ l ) = W ˜ ( l 1 ) exp [ j Λ ˜ ( l 1 ) S ( l 1 ) ] C ( l 1 ) ,
W ˜ l C j = W ˜ ( l 1 ) exp [ j Λ ˜ ( l 1 ) S ( l 1 ) ] C ( l 1 ) × exp [ j m ( k y ( l 1 ) k y ) ψ l ] D ˜ ( l 1 ) W ˜ ( l 1 ) exp [ j Λ ˜ ( l 1 ) S ( l 1 ) ] × C ( l 1 ) W ˜ l C l = 0 ,

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