Abstract

The intensity patterns produced by a Gaussian (TEM00 mode) laser beam propagating in the vicinity of the optical axis of α-iodic acid have been photographed. Densitometric profiles for different polarization conditions have been obtained. The theory of Portigal and Burstein has been extended to permit a quantitative comparison between theory and experiment for conical refraction in noncentrosymmetric crystals.

© 1978 Optical Society of America

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References

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  1. A. J. Schell and N. Bloembergen, “Laser studies on internal conical diffraction I. Quantitative comparison of experimental and theoretical conical intensity distributions in aragonite,” J. Opt. Soc. Am. 68, 1093–1097 (1978). This paper will henceforth be referred to as I.
    [Crossref]
  2. O. Weder, “Die Lichtbewegung in zweiaxigen activen Krystallen,” Neues Jahrb. Mineral. Geol. Paläontol. 11, 1–45 (1897).
  3. P. Drude, The Theory of Optics (Longmans, Green, London, 1920).
  4. D. L. Portigal and E. Burstein, “Effect of optical activity or Faraday rotation on internal conical refraction,” J. Opt. Soc. Am. 62, 859–864 (1972).
    [Crossref]
  5. W. Voigt, “Über die Wellenflache zweiachsiger aktiver Kristalle und über ihre konische Refraktion,” Z. Phys. 6, 787–790 (1905) and W. Voigt, “Theoretisches und Experimentelles zur Aufklärung des optischen Verhaltens aktiver Kristalle,” Ann. Phys. 18, 645–694 (1905).
    [Crossref]
  6. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).
  7. A. J. Schell, “Laser studies of linear and second harmonic conical refraction,” Ph.D. thesis (Harvard University, 1977) (unpublished).
  8. S. K. Kurtz, T. T. Perry, and J. G. Bergmann, “Alpha-iodic acid: a solution-grown crystal for nonlinear optical studies and applications,” Appl. Phys. Lett. 12, 186–188 (1968).
    [Crossref]
  9. International Critical Tables, edited by E. Washburn (McGraw-Hill, New York, 1930), Vol. VII, p. 353.
  10. A. J. Schell and N. Bloembergen, “Second harmonic conical refraction,” Opt. Commun. 21, 150–153 (1977).
    [Crossref]
  11. S. G. Parker and J. E. Pinnell, “Growth of α-iodic acid crystals from aqueous solution,” J. Cryst. Growth 12, 277–280 (1972).
    [Crossref]
  12. The difference in sign of the terms in Eq. (21) due to σ1 and σ2′, respectively, is somewhat disturbing, as the homogeneous plane wave limit would indicate the same sign for these terms. This ambiguity has not been resolved theoretically, but the experimental results can be fitted in either case by a very small adjustment of the rotary power ŝ · Γ→ = 0.

1978 (1)

1977 (1)

A. J. Schell and N. Bloembergen, “Second harmonic conical refraction,” Opt. Commun. 21, 150–153 (1977).
[Crossref]

1972 (2)

S. G. Parker and J. E. Pinnell, “Growth of α-iodic acid crystals from aqueous solution,” J. Cryst. Growth 12, 277–280 (1972).
[Crossref]

D. L. Portigal and E. Burstein, “Effect of optical activity or Faraday rotation on internal conical refraction,” J. Opt. Soc. Am. 62, 859–864 (1972).
[Crossref]

1968 (1)

S. K. Kurtz, T. T. Perry, and J. G. Bergmann, “Alpha-iodic acid: a solution-grown crystal for nonlinear optical studies and applications,” Appl. Phys. Lett. 12, 186–188 (1968).
[Crossref]

1905 (1)

W. Voigt, “Über die Wellenflache zweiachsiger aktiver Kristalle und über ihre konische Refraktion,” Z. Phys. 6, 787–790 (1905) and W. Voigt, “Theoretisches und Experimentelles zur Aufklärung des optischen Verhaltens aktiver Kristalle,” Ann. Phys. 18, 645–694 (1905).
[Crossref]

1897 (1)

O. Weder, “Die Lichtbewegung in zweiaxigen activen Krystallen,” Neues Jahrb. Mineral. Geol. Paläontol. 11, 1–45 (1897).

Bergmann, J. G.

S. K. Kurtz, T. T. Perry, and J. G. Bergmann, “Alpha-iodic acid: a solution-grown crystal for nonlinear optical studies and applications,” Appl. Phys. Lett. 12, 186–188 (1968).
[Crossref]

Bloembergen, N.

Burstein, E.

Drude, P.

P. Drude, The Theory of Optics (Longmans, Green, London, 1920).

Kurtz, S. K.

S. K. Kurtz, T. T. Perry, and J. G. Bergmann, “Alpha-iodic acid: a solution-grown crystal for nonlinear optical studies and applications,” Appl. Phys. Lett. 12, 186–188 (1968).
[Crossref]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).

Parker, S. G.

S. G. Parker and J. E. Pinnell, “Growth of α-iodic acid crystals from aqueous solution,” J. Cryst. Growth 12, 277–280 (1972).
[Crossref]

Perry, T. T.

S. K. Kurtz, T. T. Perry, and J. G. Bergmann, “Alpha-iodic acid: a solution-grown crystal for nonlinear optical studies and applications,” Appl. Phys. Lett. 12, 186–188 (1968).
[Crossref]

Pinnell, J. E.

S. G. Parker and J. E. Pinnell, “Growth of α-iodic acid crystals from aqueous solution,” J. Cryst. Growth 12, 277–280 (1972).
[Crossref]

Portigal, D. L.

Schell, A. J.

A. J. Schell and N. Bloembergen, “Laser studies on internal conical diffraction I. Quantitative comparison of experimental and theoretical conical intensity distributions in aragonite,” J. Opt. Soc. Am. 68, 1093–1097 (1978). This paper will henceforth be referred to as I.
[Crossref]

A. J. Schell and N. Bloembergen, “Second harmonic conical refraction,” Opt. Commun. 21, 150–153 (1977).
[Crossref]

A. J. Schell, “Laser studies of linear and second harmonic conical refraction,” Ph.D. thesis (Harvard University, 1977) (unpublished).

Voigt, W.

W. Voigt, “Über die Wellenflache zweiachsiger aktiver Kristalle und über ihre konische Refraktion,” Z. Phys. 6, 787–790 (1905) and W. Voigt, “Theoretisches und Experimentelles zur Aufklärung des optischen Verhaltens aktiver Kristalle,” Ann. Phys. 18, 645–694 (1905).
[Crossref]

Weder, O.

O. Weder, “Die Lichtbewegung in zweiaxigen activen Krystallen,” Neues Jahrb. Mineral. Geol. Paläontol. 11, 1–45 (1897).

Appl. Phys. Lett. (1)

S. K. Kurtz, T. T. Perry, and J. G. Bergmann, “Alpha-iodic acid: a solution-grown crystal for nonlinear optical studies and applications,” Appl. Phys. Lett. 12, 186–188 (1968).
[Crossref]

J. Cryst. Growth (1)

S. G. Parker and J. E. Pinnell, “Growth of α-iodic acid crystals from aqueous solution,” J. Cryst. Growth 12, 277–280 (1972).
[Crossref]

J. Opt. Soc. Am. (2)

Neues Jahrb. Mineral. Geol. Paläontol. (1)

O. Weder, “Die Lichtbewegung in zweiaxigen activen Krystallen,” Neues Jahrb. Mineral. Geol. Paläontol. 11, 1–45 (1897).

Opt. Commun. (1)

A. J. Schell and N. Bloembergen, “Second harmonic conical refraction,” Opt. Commun. 21, 150–153 (1977).
[Crossref]

Z. Phys. (1)

W. Voigt, “Über die Wellenflache zweiachsiger aktiver Kristalle und über ihre konische Refraktion,” Z. Phys. 6, 787–790 (1905) and W. Voigt, “Theoretisches und Experimentelles zur Aufklärung des optischen Verhaltens aktiver Kristalle,” Ann. Phys. 18, 645–694 (1905).
[Crossref]

Other (5)

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).

A. J. Schell, “Laser studies of linear and second harmonic conical refraction,” Ph.D. thesis (Harvard University, 1977) (unpublished).

P. Drude, The Theory of Optics (Longmans, Green, London, 1920).

International Critical Tables, edited by E. Washburn (McGraw-Hill, New York, 1930), Vol. VII, p. 353.

The difference in sign of the terms in Eq. (21) due to σ1 and σ2′, respectively, is somewhat disturbing, as the homogeneous plane wave limit would indicate the same sign for these terms. This ambiguity has not been resolved theoretically, but the experimental results can be fitted in either case by a very small adjustment of the rotary power ŝ · Γ→ = 0.

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Figures (13)

FIG. 1
FIG. 1

Direction of the Poynting vector t ˆ 1 , as a function of the polar angles θ and ϕ of the wave normal. Circle a for θ = 10−4; b for θ = 10−3; c for θ = 2 × 10−3; and d for θ = 10−2. The dashed circles would have been obtained in the absence of natural activity, ŝ · Γ = 0.

FIG. 2
FIG. 2

Direction of the Poynting vector t ˆ 2 , as a function of θ and ϕ. Circle a for θ = 10−4; b for θ = 10−3; c for θ = 3 × 10−3; d for θ = 5 × 10−3; e for θ = 7.5 × 10−3. Dashed circles would have been obtained in the absence of natural activity.

FIG. 3
FIG. 3

Ellipticity as a function of Poynting vector directions. The small letters a, b, and c mark the points where ϕ is 0 and θ is 0.001, 0.05, and 0.01, respectively.

FIG. 4
FIG. 4

Poynting vector directions as a function of θ, for ϕ = 0 and π, respectively. The dashed lines would have been obtained in the absence of natural activity.

FIG. 5
FIG. 5

Conical refraction pattern for α-iodic acid. (a) Incident E field perpendicular to y axis; (b) Incident E field parallel to y axis. Crystal length 1.4 cm, incident Gaussian spot size w0 = 60 μm, magnification 102.

FIG. 6
FIG. 6

Same as Fig. 5, but now crystal length is 2.5 cm, w0 = 30 μm, and magnification 72.

FIG. 7
FIG. 7

Same as Fig. 6(b). The incident E field is parallel to the y axis. The polarization of the transmitted light is analyzed: (a) perpendicular to the y axis; (b) parallel to the y axis.

FIG. 8
FIG. 8

Same as Fig. 6(a). The incident field is perpendicular to y axis. (a) Transmitted field is perpendicular to y axis; (b) transmitted field is parallel to y axis.

FIG. 9
FIG. 9

Conical refraction pattern for tightly focused incident beam. Crystal length 2.5 cm, spot size w0 = 12 μm, magnification 30. (a) Incident E field perpendicular to y axis; (b) incident E field parallel to y axis.

FIG. 10
FIG. 10

Intensity profile along u axis for geometry of Fig. 7(a). The dashed curve is calculated from the theory.

FIG. 11
FIG. 11

Intensity profile along u axis for geometry of Fig. 8(a). The dashed curve is calculated from the theory.

FIG. 12
FIG. 12

Intensity profile along u axis for geometry of Fig. 8(b). The dashed curve is calculated from the theory.

FIG. 13
FIG. 13

Intensity profile along u axis for geometry of Fig. 9(b). The dashed curve is calculated from the theory.

Tables (1)

Tables Icon

TABLE I Refractive indexa and optical rotary powerb of α-HIO3.

Equations (25)

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ρ = 2 A = [ ( x - 1 - z - 1 ) / y - 1 ] ( sin 2 η ) / 2 ,
η = arctan [ ( x - 1 - y - 1 ) / ( y - 1 - z - 1 ) ] 1 / 2 .
ζ = π n 3 / 2 ( ŝ · Γ ) / λ 0 ,
θ y ( ŝ · Γ ) / 8 A .
θ [ 5 ( ŝ · Γ ) / A ] y .
A > θ > [ 5 ( ŝ · Γ ) / A ] y .
E = - 1 D - i ( Γ × D )
B = H .
υ p 4 - υ p 2 ( υ p + 2 + υ p - 2 ) + υ p + 2 υ p - 2 - c 4 ( ŝ · Γ ) 2 = 0 ,
υ p 1 , 2 2 = υ y 2 - [ ( υ x 2 - υ z 2 ) / 2 ] θ sin 2 η cos ϕ [ ( υ x 2 - υ z 2 ) 2 θ 2 sin 2 2 η + 4 c 4 ( ŝ · Γ ) 2 ] 1 / 2 .
d ˆ 1 = α 1 d ˆ + + β 1 d ˆ - , d ˆ 2 = α 2 d ˆ + + β 2 d ˆ - .
α 1 = i sin ( δ / 2 ) ,             β 1 = cos ( δ / 2 ) , α 2 = i cos ( δ / 2 ) ,             β 2 = - sin ( δ / 2 ) ,
cos δ = ρ θ / [ ρ 2 θ 2 + y 2 ( ŝ · Γ ) 2 ] 1 / 2 .
t 1 u / t 1 v = ( ρ / 2 ) ( - 1 + cos δ cos ϕ ) + θ cos ϕ ,
t 1 y / t 1 v = ( ρ / 2 ) cos δ sin ϕ + θ sin ϕ ,
t 2 u / t 2 v = ( ρ / 2 ) ( - 1 - cos δ cos ϕ ) + θ cos ϕ ,
t 2 y / t 2 v = - ( ρ / 2 ) cos δ sin ϕ + θ sin ϕ .
e ˆ 1 = ( i sin δ 2 cos ϕ 2 - cos δ 2 sin ϕ 2 ,             i sin δ 2 sin ϕ 2 + cos δ 2 cos ϕ 2 ,             0 )
e ˆ 2 = ( i cos δ 2 cos ϕ 2 + sin δ 2 sin ϕ 2 ,             i cos δ 2 sin ϕ 2 - sin δ 2 cos ϕ 2 ,             0 )
E ( R ) = y 0 π / 2 0 2 π [ y 0 ( cos δ 2 cos ϕ 2 - i sin δ 2 sin ϕ 2 ) + u 0 ( - i sin ϕ 2 cos δ 2 - i sin δ 2 cos ϕ 2 ) ] e ˆ 1 e i ( ω / v p 1 ) ( ŝ · R ) θ d θ d ϕ + y 0 π / 2 0 2 π [ y 0 ( - i sin δ 2 cos ϕ 2 - i cos δ 2 sin ϕ 2 ) + u 0 ( sin δ 2 sin ϕ 2 - i cos δ 2 cos ϕ 2 ) ] × e ˆ 2 e i ( ω / v p 2 ) ( ŝ · R ) θ d θ d ϕ .
i 0 = E i 0 ( k / 2 π ) 2 w 0 2 π exp [ - k 2 w 0 2 y θ 2 / 4 ] ,             i = u ˆ , ŷ
k R Δ n 5 θ .
v p i - 1 ( ŝ · R ) u = v p i - 1 ( ŝ · R ) y = 0 ,             ( i = 1 , 2 )
E ( R u , R y , R v ) = 2 π i σ 1 y Δ 1 1 / 2 [ y 0 ( θ 1 ) ( cos δ 1 2 cos ϕ 1 2 - i sin δ 1 2 sin ϕ 1 2 ) + u 0 ( θ 1 ) ( - sin ϕ 1 2 cos δ 1 2 - i sin + δ 1 2 cos ϕ 1 2 ) ] e ˆ 1 ( θ 1 , ϕ 1 ) e i ( ω / v p 1 ) ( ŝ 1 · R ) k R + 2 π i σ 2 y Δ 2 1 / 2 { y 0 ( θ 2 ) ( - sin δ 2 2 cos ϕ 2 2 - i cos δ 2 2 sin ϕ 2 2 ) + u 0 ( θ 2 ) ( - sin δ 2 2 sin ϕ 2 2 - i cos ϕ 2 2 cos δ 2 2 ) } e ˆ 2 ( θ 2 , ϕ 2 ) e i ( ω / v p 2 ) ( ŝ 2 · R ) k R + 2 π i σ 2 y Δ 2 1 / 2 [ y 0 ( θ 2 ) ( - sin δ 2 2 cos ϕ 2 2 - i cos ϕ 2 2 sin ϕ 2 2 ) + E u 0 ( θ 2 ) ( sin δ 2 2 sin ϕ 2 2 - i cos δ 2 2 cos ϕ 2 2 ) ] e ˆ 2 ( θ 2 , ϕ 2 ) e i ( ω / v p 2 ) ( ŝ 2 · R ) k R ,
Δ = [ 2 Φ u 2 2 Φ y 2 - ( 2 Φ u y ) 2 ] 1 k 2 R 2