Abstract

We present a study of volume index gratings operating in the form-birefringence regime and what we call the intermediate regime, defined as the regime between form birefringence and the onset of diffraction. The operating regime is characterized by the ratio between the measurement wavelength and the grating period: form birefringence regime below 10, diffraction around 3, and intermediate regime between 10 and 3. The behavior of the gratings in a given regime is studied by plotting the induced retardation as a function of the incidence angle. The shape of the curve fully characterizes the regime. We report on theoretical calculations of retardation by use of a rigorous theory and its experimental validation by measuring retardation of recorded gratings in photopolymers.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1998).
  2. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1994).
  3. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986).
  4. J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, “Thin-film birefringent devices based on form birefringence,” in Optical Thin Films IV: New developments,” J. D. Rancourt, ed., Proc. SPIE2262, 234–245 (1994).
  5. G. Campbell, R. Kostuk, “Effective medium theory of sinusoidally modulated volume holograms,” J. Opt. Soc. Am. A 12, 1113–1117 (1995).
    [CrossRef]
  6. T. J. Kim, G. Campbell, R. K. Kostuk, “Volume holographic phase retardation elements,” Opt. Lett. 20, 2030–2032 (1995).
    [CrossRef] [PubMed]
  7. C. Yang, P. Yeh, “Form birefringence of volume gratings in photopolymers,” Appl. Phys. Lett. 69, 3468–3470 (1996).
    [CrossRef]
  8. C. Yang, P. Yeh, “Artificial uniaxial and biaxial dielectrics with the use of photoinduced gratings,” J. Appl. Phys. 81, 23–29 (1997).
    [CrossRef]
  9. C. Joubert, J. C. Lehureau, L. Lee, F. Delauzun, B. Morbieu, “TN-LCD viewing angle compensation with holographic volume gratings,” in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar, ed., Proc. SPIE3635, 137–142 (1999).
  10. P. Lalanne, J. P. Hugonin, “High-order effective medium theory of subwavelength gratings in classical mounting: application to volume holograms,” J. Opt. Soc. Am. A 15, 1843–1851 (1998).
    [CrossRef]
  11. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  12. H. Kogelnik, “Coupled wave theory for thick hologram grating,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), pp. 694–699.
  14. A. M. Weber, W. K. Smothers, T. J. Trout, D. J. Mickish, in Practical Holography IV, S. A. Benton, ed., Proc. SPIE1212, 30–39 (1990).
  15. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), pp. 692–694.

1998 (1)

1997 (1)

C. Yang, P. Yeh, “Artificial uniaxial and biaxial dielectrics with the use of photoinduced gratings,” J. Appl. Phys. 81, 23–29 (1997).
[CrossRef]

1996 (1)

C. Yang, P. Yeh, “Form birefringence of volume gratings in photopolymers,” Appl. Phys. Lett. 69, 3468–3470 (1996).
[CrossRef]

1995 (2)

1983 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram grating,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Beedy, J.

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, “Thin-film birefringent devices based on form birefringence,” in Optical Thin Films IV: New developments,” J. D. Rancourt, ed., Proc. SPIE2262, 234–245 (1994).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), pp. 694–699.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), pp. 692–694.

Campbell, G.

Delauzun, F.

C. Joubert, J. C. Lehureau, L. Lee, F. Delauzun, B. Morbieu, “TN-LCD viewing angle compensation with holographic volume gratings,” in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar, ed., Proc. SPIE3635, 137–142 (1999).

Eblen, J. P.

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, “Thin-film birefringent devices based on form birefringence,” in Optical Thin Films IV: New developments,” J. D. Rancourt, ed., Proc. SPIE2262, 234–245 (1994).

Gaylord, T. K.

Gunning, W. J.

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, “Thin-film birefringent devices based on form birefringence,” in Optical Thin Films IV: New developments,” J. D. Rancourt, ed., Proc. SPIE2262, 234–245 (1994).

Hale, L.

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, “Thin-film birefringent devices based on form birefringence,” in Optical Thin Films IV: New developments,” J. D. Rancourt, ed., Proc. SPIE2262, 234–245 (1994).

Hugonin, J. P.

Joubert, C.

C. Joubert, J. C. Lehureau, L. Lee, F. Delauzun, B. Morbieu, “TN-LCD viewing angle compensation with holographic volume gratings,” in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar, ed., Proc. SPIE3635, 137–142 (1999).

Khoshnevisan, M.

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, “Thin-film birefringent devices based on form birefringence,” in Optical Thin Films IV: New developments,” J. D. Rancourt, ed., Proc. SPIE2262, 234–245 (1994).

Kim, T. J.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram grating,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Kostuk, R.

Kostuk, R. K.

Lalanne, P.

Lee, L.

C. Joubert, J. C. Lehureau, L. Lee, F. Delauzun, B. Morbieu, “TN-LCD viewing angle compensation with holographic volume gratings,” in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar, ed., Proc. SPIE3635, 137–142 (1999).

Lehureau, J. C.

C. Joubert, J. C. Lehureau, L. Lee, F. Delauzun, B. Morbieu, “TN-LCD viewing angle compensation with holographic volume gratings,” in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar, ed., Proc. SPIE3635, 137–142 (1999).

Mickish, D. J.

A. M. Weber, W. K. Smothers, T. J. Trout, D. J. Mickish, in Practical Holography IV, S. A. Benton, ed., Proc. SPIE1212, 30–39 (1990).

Moharam, M. G.

Morbieu, B.

C. Joubert, J. C. Lehureau, L. Lee, F. Delauzun, B. Morbieu, “TN-LCD viewing angle compensation with holographic volume gratings,” in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar, ed., Proc. SPIE3635, 137–142 (1999).

Smothers, W. K.

A. M. Weber, W. K. Smothers, T. J. Trout, D. J. Mickish, in Practical Holography IV, S. A. Benton, ed., Proc. SPIE1212, 30–39 (1990).

Taber, D.

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, “Thin-film birefringent devices based on form birefringence,” in Optical Thin Films IV: New developments,” J. D. Rancourt, ed., Proc. SPIE2262, 234–245 (1994).

Trout, T. J.

A. M. Weber, W. K. Smothers, T. J. Trout, D. J. Mickish, in Practical Holography IV, S. A. Benton, ed., Proc. SPIE1212, 30–39 (1990).

Weber, A. M.

A. M. Weber, W. K. Smothers, T. J. Trout, D. J. Mickish, in Practical Holography IV, S. A. Benton, ed., Proc. SPIE1212, 30–39 (1990).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), pp. 692–694.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), pp. 694–699.

Yang, C.

C. Yang, P. Yeh, “Artificial uniaxial and biaxial dielectrics with the use of photoinduced gratings,” J. Appl. Phys. 81, 23–29 (1997).
[CrossRef]

C. Yang, P. Yeh, “Form birefringence of volume gratings in photopolymers,” Appl. Phys. Lett. 69, 3468–3470 (1996).
[CrossRef]

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1994).

Yeh, P.

C. Yang, P. Yeh, “Artificial uniaxial and biaxial dielectrics with the use of photoinduced gratings,” J. Appl. Phys. 81, 23–29 (1997).
[CrossRef]

C. Yang, P. Yeh, “Form birefringence of volume gratings in photopolymers,” Appl. Phys. Lett. 69, 3468–3470 (1996).
[CrossRef]

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1998).

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1994).

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, “Thin-film birefringent devices based on form birefringence,” in Optical Thin Films IV: New developments,” J. D. Rancourt, ed., Proc. SPIE2262, 234–245 (1994).

Appl. Phys. Lett. (1)

C. Yang, P. Yeh, “Form birefringence of volume gratings in photopolymers,” Appl. Phys. Lett. 69, 3468–3470 (1996).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram grating,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

J. Appl. Phys. (1)

C. Yang, P. Yeh, “Artificial uniaxial and biaxial dielectrics with the use of photoinduced gratings,” J. Appl. Phys. 81, 23–29 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Lett. (1)

Other (8)

C. Joubert, J. C. Lehureau, L. Lee, F. Delauzun, B. Morbieu, “TN-LCD viewing angle compensation with holographic volume gratings,” in Liquid Crystal Materials, Devices, and Applications VII, R. Shashidhar, ed., Proc. SPIE3635, 137–142 (1999).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1998).

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1994).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986).

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, “Thin-film birefringent devices based on form birefringence,” in Optical Thin Films IV: New developments,” J. D. Rancourt, ed., Proc. SPIE2262, 234–245 (1994).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), pp. 694–699.

A. M. Weber, W. K. Smothers, T. J. Trout, D. J. Mickish, in Practical Holography IV, S. A. Benton, ed., Proc. SPIE1212, 30–39 (1990).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), pp. 692–694.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1
Fig. 1

Definition of substrate and grating related referential and grating parameters.

Fig. 2
Fig. 2

Characterization of the illuminating wave.

Fig. 3
Fig. 3

Description of the three different regimes of a volume index grating as a function of F = λ/Λ.

Fig. 4
Fig. 4

Description of the ordinary plane of incidence.

Fig. 5
Fig. 5

Phase volume grating behavior when in accordance with the effective medium theory.

Fig. 6
Fig. 6

Induced retardation R of a slanted volume grating in accordance with the effective medium theory for two specific angles of incidence in the medium in the extraordinary plane: α = Φ and α = Φ ± 90°.

Fig. 7
Fig. 7

Theoretical behavior of R (α or β, respectively) cos α (or cos β, resp.) as a function of α or resp. β, respectively for a uniaxial medium: (a) extraordinary plane, square sine law; (b) ordinary plane, constant.

Fig. 8
Fig. 8

Theoretical behavior of retardation versus α of a grating with Φ = 56° and R0 = -52 nm, for different values of F between 10 and 2.9. (a) extraordinary plane, progressively away from the square sine law when F decreases; (b) ordinary plane, constant whatever the F value.

Fig. 9
Fig. 9

Simulated typical volume holographic grating recorded in photopolymers (n 1 = 0.056 and d = 24.8 µm). Evolution of retardation versus incidence in the extraordinary plane for four values of F: (crosses) F = 10 (square sine law), (solid curve) F = 4.63, (dashed curve) F = 3.77, (dotted curve) F = 3.35. (a), Slant angle Φ = 0°; (b), slant angle Φ = 45°; (c), slant angle Φ = 90°.

Fig. 10
Fig. 10

Classical recording of index modulation gratings in three different geometries.

Fig. 11
Fig. 11

Experimental setup for the measurement of the retardation introduced by the index modulation grating at different incidences i with a Babinet Soleil compensator.

Fig. 12
Fig. 12

Comparison between experimental and theoretical curves of retardation versus incidence in the extraordinary plane for recorded index grating H34 (Φ = -34°): (dashed curve), square sine law, (solid curve) rigorous theory of coupled waves, (filled squares) experimental data. (a) Characterization wavelength λc = 628.8 nm (red), F = 4.69; (b) characterization wavelength λc = 514 nm (green), F = 3.57; (c) characterization wavelength λc = 458 nm (blue), F = 3.18.

Fig. 13
Fig. 13

Comparison between experimental and theoretical curves of retardation versus incidence in the extraordinary plane for recorded index grating H45 (Φ = -45°): symbols same as Fig. 12.

Fig. 14
Fig. 14

Comparison between experimental and theoretical curves of retardation versus incidence in the extraordinary plane for recorded index grating H90 (Φ = -90°): symbols same as Fig. 12, values (a) characterization wavelength λc = 628.8 nm (red), F = 3.3; (b) characterization wavelength λc = 514 nm (green), F = 2.69; (c) characterization wavelength λc = 458 nm (blue), F = 2.4.

Fig. 15
Fig. 15

Comparison between experimental and theoretical curves of retardation versus incidence in the ordinary plane for the recorded index grating H90 (Φ = -90°) for the three characterization wavelength λc. λc = 628.8 nm (red), F = 3.3 and R0 = -53 nm, solid line, rigorous theory of coupled waves, filled circles, experimental data; λc = 514 nm (green), F = 2.69 and R0 = -61 nm, dashed line, rigorous theory of coupled waves, solid squares, experimental data; λc = 458 nm (blue), F = 2.4 and R0 = -63 nm, gray line, rigorous theory of coupled waves, open triangles, experimental data.

Tables (2)

Tables Icon

Table 1 Grating Parameters (Λ Period, Φ Slant Angle)a

Tables Icon

Table 2 Parameters of Recorded Gratings Used in the Calculation

Equations (17)

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

Rα, δ=Δnα, δ · eα, δ
Δnα=ne-nor · sin2Φ-α.
Rα=Δnα · eα=ne-nor · sin2Φ-α · dcos α,
Rαcos α=d · ne-nor · sin2Φ-α=R0 · sin2Φ-α.
cos α=sin Φ cos β,
tan δ=-tan β/cos Φ.
Δnβ=ne-nor
eβ=dsin Φ · cos β.
Rβcos β=ne-nordsin Φ=R0sin Φ
Rβcos α=R0.
ne-nor=- n12n0.
2n0ΛλcosΦ-α=1,
2n0FcosΦ-α=1.
Rα, δ=λ2πφTMα, δ-φTEα, δ.
tan Ψ=tan Φ · sin α · sin δ,
Λth=λ02n0 sinθ1-θ2/2,
Φth=θ1+θ22+90° · sign θ1-θ2.

Metrics