Abstract

The modifications to the holodiagram concept to describe free propagation (the extraordinary ray) inside birefringent materials are described. Holodiagrams are graphs showing the loci where the sum or the difference in the optical path from a generic point to two foci is the same. The holodiagrams obtained in this way give the shape of the surfaces that satisfy Fermat’s principle, conjugate by reflection of one focus into the other, and represent the interference fringes obtained if both points are coherent sources. The reflection law in birefringent media is investigated in relation to this diagram. One direction for the optical axis is considered: parallel to the line joining the source and the observation point. Quartz-type and calcite-type crystals are studied.

© 2003 Optical Society of America

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References

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  1. N. Abramson, “The holodiagram, a practical device for the making and evaluation of holograms,” Appl. Opt. 8, 1235–1240 (1969).
    [CrossRef] [PubMed]
  2. N. Abramson, Light in Flight or The Holodiagram, Columbi Egg in Optics (SPIE, Bellingham, Wash., 1996).
  3. N. Abramson, The Making and Evaluation of Holograms (Academic, London, 1981).
  4. H. Rabal, “The holodiagram with virtual sources,”Optik (Stuttgart) 112, 487–492 (2001).
    [CrossRef]
  5. G. Baldwin, F. De Zela, H. Rabal, “Refraction holodiagrams,” Optik (Stuttgart) 112, 555–560 (2001).
    [CrossRef]
  6. M. Simon, K. Gottschalk, “Optical path in birefringent media and Fermat’s principle,” Pure Appl. Opt. 7, 1403–1410 (1998).
    [CrossRef]
  7. K. V. Gottschalk, “Brewster angle and the behavior of the electric polarization vector in the optics of birefringent crystals,” Ph.D. dissertation (Universidad de Buenos Aires, Argentina, 2001), pp. 103–109.

2001

H. Rabal, “The holodiagram with virtual sources,”Optik (Stuttgart) 112, 487–492 (2001).
[CrossRef]

G. Baldwin, F. De Zela, H. Rabal, “Refraction holodiagrams,” Optik (Stuttgart) 112, 555–560 (2001).
[CrossRef]

1998

M. Simon, K. Gottschalk, “Optical path in birefringent media and Fermat’s principle,” Pure Appl. Opt. 7, 1403–1410 (1998).
[CrossRef]

1969

Abramson, N.

N. Abramson, “The holodiagram, a practical device for the making and evaluation of holograms,” Appl. Opt. 8, 1235–1240 (1969).
[CrossRef] [PubMed]

N. Abramson, Light in Flight or The Holodiagram, Columbi Egg in Optics (SPIE, Bellingham, Wash., 1996).

N. Abramson, The Making and Evaluation of Holograms (Academic, London, 1981).

Baldwin, G.

G. Baldwin, F. De Zela, H. Rabal, “Refraction holodiagrams,” Optik (Stuttgart) 112, 555–560 (2001).
[CrossRef]

De Zela, F.

G. Baldwin, F. De Zela, H. Rabal, “Refraction holodiagrams,” Optik (Stuttgart) 112, 555–560 (2001).
[CrossRef]

Gottschalk, K.

M. Simon, K. Gottschalk, “Optical path in birefringent media and Fermat’s principle,” Pure Appl. Opt. 7, 1403–1410 (1998).
[CrossRef]

Gottschalk, K. V.

K. V. Gottschalk, “Brewster angle and the behavior of the electric polarization vector in the optics of birefringent crystals,” Ph.D. dissertation (Universidad de Buenos Aires, Argentina, 2001), pp. 103–109.

Rabal, H.

G. Baldwin, F. De Zela, H. Rabal, “Refraction holodiagrams,” Optik (Stuttgart) 112, 555–560 (2001).
[CrossRef]

H. Rabal, “The holodiagram with virtual sources,”Optik (Stuttgart) 112, 487–492 (2001).
[CrossRef]

Simon, M.

M. Simon, K. Gottschalk, “Optical path in birefringent media and Fermat’s principle,” Pure Appl. Opt. 7, 1403–1410 (1998).
[CrossRef]

Appl. Opt.

Optik (Stuttgart)

H. Rabal, “The holodiagram with virtual sources,”Optik (Stuttgart) 112, 487–492 (2001).
[CrossRef]

G. Baldwin, F. De Zela, H. Rabal, “Refraction holodiagrams,” Optik (Stuttgart) 112, 555–560 (2001).
[CrossRef]

Pure Appl. Opt.

M. Simon, K. Gottschalk, “Optical path in birefringent media and Fermat’s principle,” Pure Appl. Opt. 7, 1403–1410 (1998).
[CrossRef]

Other

K. V. Gottschalk, “Brewster angle and the behavior of the electric polarization vector in the optics of birefringent crystals,” Ph.D. dissertation (Universidad de Buenos Aires, Argentina, 2001), pp. 103–109.

N. Abramson, Light in Flight or The Holodiagram, Columbi Egg in Optics (SPIE, Bellingham, Wash., 1996).

N. Abramson, The Making and Evaluation of Holograms (Academic, London, 1981).

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Figures (4)

Fig. 1
Fig. 1

Geometric parameters involved in the HD calculations.

Fig. 2
Fig. 2

Sum holodiagrams: (a) for calcite, (b) for vaterite.

Fig. 3
Fig. 3

Difference holodiagrams: (a) calcite, (b) vaterite.

Fig. 4
Fig. 4

Geometric parameters in the calculation of the reflection law.

Equations (42)

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l1=d1ν1, l2=d2ν2,
ν12=1ne2+no2-ne2cos2 φ1,
ν22=1ne2+no2-ne2cos2 φ2.
cos φ1=x+ad1, cos φ2=x-ad2,
d1=y2+x+a2, d2=y2+x-a2.
l1=ne2y2+no2x2+no2a2+2no2ax, l2=ne2y2+no2x2+no2a2-2no2ax.
l=l1+l2.
y2no2l2-4no2a2+x2l2ne2=14no2ne2.
y2B2+x2A2=1, A2=l24no2, B2=l2-4no2a24ne2.
e=A2-B2A2+B2=l2ne2-no2+4a2no4l2ne2+no2-4a2no4.
g2=A2-B2,
g2=4no4a2-l2no2-ne24ne2no2.
lc=2ano2no2-ne2.
r=a1-ne2no2.
g=B2-A2;
g=l2no2-ne2-4no4a24ne2no2.
ene=no=2a2no2l2-2a2no2.
x2A2-y2B2=1, A2=l24no2, B2=4no2a2-l24ne2.
g2=A2+B2=4no4a2-l2no2-ne24ne2no2,
g=±noane.
g=±a.
α=σ-π2,
tan σ=dydx.
y2=B21-x2A2.
tan α=yxA2B2.
ρ=φ1-α,
ρe=φ2-α.
tan ρ=tan φ1-tan α1+tan φ1 tan α, tan ρe=tan φ2-tan α1+tan φ2 tan α,
tan φ2=yx-a, tan φ1=yx+a.
B2x2+A2y2=A2B2,
tan ρe=B2-A2xy+A2ayB2A2-ax, tan ρ=B2-A2xy-A2ayB2A2+ax.
tan ρ+tan ρe=2A2B2-A2+a2xyB2A4-a2x2.
x2=A4 cos2 αA2 cos2 α+B2 sin2 α,
y2=B4 sin2 αA2 cos2 α+B2 sin2 α,
tan ρ+tan ρe=2B2-A2+a2sin α cos αA2+B2 sin2 α-a2 cos2 α.
tan ρ+tan ρe=2 sin α cos αno2-ne2ne2 cos2 α+no2 sin2 α.
tan ρ+tan ρe=2no2-ne2sin θ cos θne2 sin2 θ+no2 cos2 θ.
-sin γeue=sin γu;
u2c2=1ne2+ne2-no2ne2sin2θ-γ, ue2c2=1ne2+ne2-no2ne2sin2θ-γe,
tan γe= -no2 cos2 θ+ne2 sin2 θtan γ2no2-ne2sin θ cos θ+no2 cos2 θ+ne2 sin2 θ.
tan γ=no2-ne2sin θ cos θ-no2 cos2 θ+ne2 sin2 θtan ρno2-ne2sin θ cos θ tan ρ-no2 sin2 θ+ne2 cos2 θ,
tan ρ+tan ρe=2no2-ne2sin θ cos θne2 sin2 θ+no2 cos2 θ,

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