Abstract

Detailed measurements of the conical intensity pattern aragonite have been carried out with an incident Gaussian beam (TEM00) of a helium neon laser. Quantitative agreement with theoretical calculations is found. It is shown that the integrated intensities on either side of the Poggendorff dark circle are not equal, unless the far-field condition for Fraunhofer diffraction is satisfied.

© 1978 Optical Society of America

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References

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  1. W. R. Hamilton, “Third Supplement to an Essay on the Theory of Systems of Rays,” Trans. R. Irish Acad. 17, 1–144 (1833).
  2. H. Lloyd, “On the Phenomena Presented by Light in its Passage along the Axes of Biaxial Crystals,” Trans. R. Irish Acad. 17, 145–157 (1833).
  3. J. C. Poggendorf, “Ueber die konische Refraction,” Ann. Phys. (Leipz.) 48, 461–463 (1839).
    [CrossRef]
  4. W. Voigt, “Bemerkung zur Theorie der konischen Refraction,” Phys. Z. 6, 672–673 (1905); “Nochmals die Theorie der konischen Refraction,” Phys. Z. 6, 818–820 (1905).
  5. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 14.
  6. R. W. Ditchburn, Light (Interscience, New York, 1963), Vol. II, Chap. 16.
  7. A. J. Schell, “Laser Studies of Linear and Second Harmonic Conical Refraction,” Ph.D. thesis (Harvard University, 1977) (unpublished).
  8. H. Juretschke, Crystal Physics: Macroscopic Physics of Anisotropic Solids (Benjamin, New York, 1974), Vol. 3 of the Modern Physics Monograph Series.
  9. D. L. Portigal and E. Burstein, “Internal conical refraction,” J. Opt. Soc. Am. 59, 1567–1573 (1969). (a)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]
  10. Éamon. Lalor, “The Angular Spectrum Representation of Electromagnetic Fields in Crystals: II. Biaxial Crystals,” J. Math Phys. 13, 443–449 (1972).
    [CrossRef]
  11. Éamon Lalor, “An Analytical Approach to the Theory of Internal Conical Refraction,” J. Math. Phys. 13, 449–454 (1972).
    [CrossRef]
  12. H. Shih and N. Bloembergen, “Conical Refraction in Second Harmonic Generation,” Phys. Rev. 184, 895–904 (1969).
    [CrossRef]
  13. W. Voigt, “Zur Deutung der Erscheinungen der sogenannten konischen Refraction,” Neues Jahrb. Mineral Geol. Paleontol. 1, 35–46 (1915).
  14. A. J. Schell and N. Bloembergen, “Second Harmonic Conical Refraction,” Opt. Commun. 21, 150–153 (1977).
    [CrossRef]
  15. Professor C. S. Hurlbut, Jr. generously loaned this crystal to us from his personal collection.
  16. See Ref. 5, Appendix III.
  17. SPSE Handbook of Photographic Science and Engineering, edited by Woodlief Thomas (Wiley, New York, 1973).
  18. Sidney Melmore, “Conical Refraction,” Nature 151, 620–621 (1943).
    [CrossRef]
  19. The plates were evaluated by Photometrics, Inc., 422 Marrett Rd., Lexington, Mass. 02173.

1977 (1)

A. J. Schell and N. Bloembergen, “Second Harmonic Conical Refraction,” Opt. Commun. 21, 150–153 (1977).
[CrossRef]

1972 (2)

Éamon. Lalor, “The Angular Spectrum Representation of Electromagnetic Fields in Crystals: II. Biaxial Crystals,” J. Math Phys. 13, 443–449 (1972).
[CrossRef]

Éamon Lalor, “An Analytical Approach to the Theory of Internal Conical Refraction,” J. Math. Phys. 13, 449–454 (1972).
[CrossRef]

1969 (2)

1943 (1)

Sidney Melmore, “Conical Refraction,” Nature 151, 620–621 (1943).
[CrossRef]

1915 (1)

W. Voigt, “Zur Deutung der Erscheinungen der sogenannten konischen Refraction,” Neues Jahrb. Mineral Geol. Paleontol. 1, 35–46 (1915).

1905 (1)

W. Voigt, “Bemerkung zur Theorie der konischen Refraction,” Phys. Z. 6, 672–673 (1905); “Nochmals die Theorie der konischen Refraction,” Phys. Z. 6, 818–820 (1905).

1839 (1)

J. C. Poggendorf, “Ueber die konische Refraction,” Ann. Phys. (Leipz.) 48, 461–463 (1839).
[CrossRef]

1833 (2)

W. R. Hamilton, “Third Supplement to an Essay on the Theory of Systems of Rays,” Trans. R. Irish Acad. 17, 1–144 (1833).

H. Lloyd, “On the Phenomena Presented by Light in its Passage along the Axes of Biaxial Crystals,” Trans. R. Irish Acad. 17, 145–157 (1833).

Bloembergen, N.

A. J. Schell and N. Bloembergen, “Second Harmonic Conical Refraction,” Opt. Commun. 21, 150–153 (1977).
[CrossRef]

H. Shih and N. Bloembergen, “Conical Refraction in Second Harmonic Generation,” Phys. Rev. 184, 895–904 (1969).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 14.

Burstein, E.

Ditchburn, R. W.

R. W. Ditchburn, Light (Interscience, New York, 1963), Vol. II, Chap. 16.

Hamilton, W. R.

W. R. Hamilton, “Third Supplement to an Essay on the Theory of Systems of Rays,” Trans. R. Irish Acad. 17, 1–144 (1833).

Juretschke, H.

H. Juretschke, Crystal Physics: Macroscopic Physics of Anisotropic Solids (Benjamin, New York, 1974), Vol. 3 of the Modern Physics Monograph Series.

Lalor, Éamon

Éamon Lalor, “An Analytical Approach to the Theory of Internal Conical Refraction,” J. Math. Phys. 13, 449–454 (1972).
[CrossRef]

Lalor, Éamon.

Éamon. Lalor, “The Angular Spectrum Representation of Electromagnetic Fields in Crystals: II. Biaxial Crystals,” J. Math Phys. 13, 443–449 (1972).
[CrossRef]

Lloyd, H.

H. Lloyd, “On the Phenomena Presented by Light in its Passage along the Axes of Biaxial Crystals,” Trans. R. Irish Acad. 17, 145–157 (1833).

Melmore, Sidney

Sidney Melmore, “Conical Refraction,” Nature 151, 620–621 (1943).
[CrossRef]

Poggendorf, J. C.

J. C. Poggendorf, “Ueber die konische Refraction,” Ann. Phys. (Leipz.) 48, 461–463 (1839).
[CrossRef]

Portigal, D. L.

Schell, A. J.

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).

Shih, H.

H. Shih and N. Bloembergen, “Conical Refraction in Second Harmonic Generation,” Phys. Rev. 184, 895–904 (1969).
[CrossRef]

Voigt, W.

W. Voigt, “Zur Deutung der Erscheinungen der sogenannten konischen Refraction,” Neues Jahrb. Mineral Geol. Paleontol. 1, 35–46 (1915).

W. Voigt, “Bemerkung zur Theorie der konischen Refraction,” Phys. Z. 6, 672–673 (1905); “Nochmals die Theorie der konischen Refraction,” Phys. Z. 6, 818–820 (1905).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 14.

Ann. Phys. (Leipz.) (1)

J. C. Poggendorf, “Ueber die konische Refraction,” Ann. Phys. (Leipz.) 48, 461–463 (1839).
[CrossRef]

J. Math Phys. (1)

Éamon. Lalor, “The Angular Spectrum Representation of Electromagnetic Fields in Crystals: II. Biaxial Crystals,” J. Math Phys. 13, 443–449 (1972).
[CrossRef]

J. Math. Phys. (1)

Éamon Lalor, “An Analytical Approach to the Theory of Internal Conical Refraction,” J. Math. Phys. 13, 449–454 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

Nature (1)

Sidney Melmore, “Conical Refraction,” Nature 151, 620–621 (1943).
[CrossRef]

Neues Jahrb. Mineral Geol. Paleontol. (1)

W. Voigt, “Zur Deutung der Erscheinungen der sogenannten konischen Refraction,” Neues Jahrb. Mineral Geol. Paleontol. 1, 35–46 (1915).

Opt. Commun. (1)

A. J. Schell and N. Bloembergen, “Second Harmonic Conical Refraction,” Opt. Commun. 21, 150–153 (1977).
[CrossRef]

Phys. Rev. (1)

H. Shih and N. Bloembergen, “Conical Refraction in Second Harmonic Generation,” Phys. Rev. 184, 895–904 (1969).
[CrossRef]

Phys. Z. (1)

W. Voigt, “Bemerkung zur Theorie der konischen Refraction,” Phys. Z. 6, 672–673 (1905); “Nochmals die Theorie der konischen Refraction,” Phys. Z. 6, 818–820 (1905).

Trans. R. Irish Acad. (2)

W. R. Hamilton, “Third Supplement to an Essay on the Theory of Systems of Rays,” Trans. R. Irish Acad. 17, 1–144 (1833).

H. Lloyd, “On the Phenomena Presented by Light in its Passage along the Axes of Biaxial Crystals,” Trans. R. Irish Acad. 17, 145–157 (1833).

Other (8)

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 14.

R. W. Ditchburn, Light (Interscience, New York, 1963), Vol. II, Chap. 16.

A. J. Schell, “Laser Studies of Linear and Second Harmonic Conical Refraction,” Ph.D. thesis (Harvard University, 1977) (unpublished).

H. Juretschke, Crystal Physics: Macroscopic Physics of Anisotropic Solids (Benjamin, New York, 1974), Vol. 3 of the Modern Physics Monograph Series.

Professor C. S. Hurlbut, Jr. generously loaned this crystal to us from his personal collection.

See Ref. 5, Appendix III.

SPSE Handbook of Photographic Science and Engineering, edited by Woodlief Thomas (Wiley, New York, 1973).

The plates were evaluated by Photometrics, Inc., 422 Marrett Rd., Lexington, Mass. 02173.

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Figures (7)

FIG. 1
FIG. 1

Polar coordinates of the wave vector and the directions of the Poynting vectors in the vicinity of the optical v ˆ axis.

FIG. 2
FIG. 2

Experimental setup for linear conical refraction studies. The focal lengths of the lenses are: L1, 10 cm; L2, 5 cm; and L3, 10 cm.

FIG. 3
FIG. 3

Internal conical refraction patterns in aragonite. The polarization of the incident light is parallel to the û direction in A and to the y axis in B.

FIG. 4
FIG. 4

Isodensitometric map of the conical diffraction patterns. (a). Incident electric field vector perpendicular to y axis. Background OD = 0.07; max density = 1.05; OD contour interval = 0.07; intensity contour interval = 0.07. (b). Incident electric field vector parallel to y axis. Background OD = 0.07; max density = 0.98; OD contour interval = 0.09; intensity contour interval = 0.09.

FIG. 5
FIG. 5

Intensity profile on the crystal exit face along the û direction. (a). Incident electric field vector perpendicular to y axis. (b). Incident electric field vector parallel to y axis.

FIG. 6
FIG. 6

Far-field intensity profiles for w0 = 34 μm. Solid lines are calculated by the method of stationary phase [Eq. (14)]. Dashed line is calculated by numerical integration of Eq. (9) for L = 10 cm. (a). Profile along the û axis with incident electric field vector perpendicular to y axis. (b). Profile along the û axis with incident electric field vector parallel to y axis. (c). Intensity profile along ŷ axis.

FIG. 7
FIG. 7

Intensity profiles, along û axis with incident electric field vector perpendicular to y axis, for three crystal lengths. The solid lines are calculated numerically from Eq. (9), with w0 = 34 μm. The dashed lines result from each of the two (±) terms separately. (a). L = 1 cm. (b). L = 3 cm. (c). L = 10 cm.

Equations (17)

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ρ = 2 A = [ ( v x 2 - v z 2 ) / v y 2 ] ( sin 2 η ) / 2.
η = arctan [ ( v x 2 - v y 2 ) / ( v y 2 - v z 2 ) ] 1 / 2
v ˆ = x ˆ sin η + z ˆ cos η , u ˆ = x ˆ cos η - z ˆ sin η .
s x = cos θ sin η + sin θ cos ϕ cos η , s y = sin θ sin ϕ , s z = cos θ cos η - sin θ cos ϕ sin η .
t u ± / t v ± = - A ( 1 ± cos ϕ ) + θ cos ϕ θ ( 1 ± cos ϕ ) 2 × { [ v x 2 - v z 2 ) / 4 v y 2 ] cos 2 η + A 2 } ,
t y ± / t v ± = A sin ϕ + θ sin ϕ - θ sin ϕ × [ ( v x 2 - v z 2 ) 4 v y 2 cos 2 η + ( 1 ± cos ϕ ) ( ( v x 2 - v z 2 ) 4 v y 2 cos 2 η + A 2 ) ] .
v p ± 2 = v y 2 + [ ( v x 2 - v z 2 ) / 2 ] { θ sin 2 η ( - cos ϕ 1 ) + θ 2 cos 2 η ( - 1 cos ϕ ) } .
w ( L ) = w 0 [ 1 + ( λ L / π w 0 2 n ) 2 ] 1 / 2 L A
L > 64 λ / A 2 n π .
E u ˆ , ŷ , v ˆ ( x , y , z ) = y 2 0 0 2 π [ y 0 + sin η sin ( ϕ + / 2 ) - z 0 + cos ( ϕ + / 2 ) ] × { cos ( ϕ + / 2 ) , sin ( ϕ + / 2 ) , [ ( v x 2 - v y 2 ) / v y 2 ] sin η cos η cos ( ϕ + / 2 ) } × e i ω v + - 1 ( s x x + s y y + s z z ) θ + d θ + d ϕ + + y 2 0 0 2 π [ y 0 - sin η cos ( ϕ - / 2 ) + z 0 - sin ( ϕ - / 2 ) ] × { - sin ( ϕ - / 2 ) , cos ( ϕ - / 2 ) , - [ ( v x 2 - v z 2 ) / v y 2 ] sin η cos η sin ( ϕ - / 2 ) } × e i ω v - - 1 ( s x x + s y y + s z z ) θ - d θ - d ϕ - .
E i ( 0 , y , z ) = E 0 i e - { z 2 ( 1 - y cos 2 η ) + y 2 } / w 0 2 e i k z y 1 / 2 cos η .
i 0 + = ( ω 2 π c ) 2 E i 0 w 0 2 π ( 1 - y cos 2 η ) 1 / 2 exp { - ( ω / 2 c ) 2 w 0 2 y ( θ ± ) 2 × [ sin 2 η 1 - y cos 2 η ( - z + y 2 z cos ϕ ± ± z - y 2 z ) 2 + sin 2 ϕ ± ] } .
Φ ± = w v ± - 1 ( s x x + s y y + s z z )
Φ ± / u = Φ ± / y = 0.
Δ ± = 1 k 2 R 2 [ 2 Φ ± u 2 2 Φ ± y 2 - ( 2 Φ ± u y ) 2 ] .
E u ˆ , ŷ , v ˆ ( t ˆ + R ) = 2 π i σ + y Δ + 1 / 2 k R ( + y 0 sin ϕ + 2 sin η - + z 0 cos ϕ + 2 ) e i Φ + × [ cos ϕ + 2 , sin ϕ + 2 , v x 2 - v z 2 v y 2 sin η cos η cos ϕ + 2 ] .
σ + = + 1 for Δ + > 0 and 2 Φ + / u 2 > 0 = - 1 for Δ + > 0 and 2 ϕ + / u 2 < 0 = - i for Δ + < 0.