Abstract

Huygens’s principle is used to derive equations for tracing the extraordinary ray in a uniaxial crystal when the crystal axis is normal to the refracting surface. Snell’s law is used to trace ordinary rays that lead to an array of ordinary spherical wavelets centered on the refracting surface. Each spherical wavelet is then replaced by an extraordinary wavelet in the shape of a rotationally symmetric ellipsoid whose major axis is in the direction of the crystal axis. The envelope of these is the extraordinary wave front; the extraordinary ray is a vector from the center of the wavelet to its point of contact with the extraordinary wave front. The inverse problem is also solved, yielding expressions for the ordinary ray in terms of the extraordinary ray, making it possible to trace the extraordinary ray out of the system by using a fictive ordinary ray. We found the geometrical invariant for the uniaxial crystals.

© 2002 Optical Society of America

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References

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  1. O. N. Stavroudis, “Ray tracing formulas for uniaxial crystals,” J. Opt. Soc. Am. 52, 187–191 (1962).
    [CrossRef]
  2. M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354–360 (1983).
    [CrossRef] [PubMed]
  3. M. C. Simon, “Image formation through monoaxial plane-parallel plates,” Appl. Opt. 27, 4176–4182 (1988).
    [CrossRef] [PubMed]
  4. E. Cojocaru, “Direction cosines and vectorial relations for extraordinary-wave propagation in uniaxial media,” Appl. Opt. 36, 302–306 (1997).
    [CrossRef] [PubMed]
  5. Quan-Ting Liang, “Simple ray tracing formulas for uniaxial optical crystal,” Appl. Opt. 29, 1008–1010 (1990).
    [CrossRef] [PubMed]
  6. Zhongxing Shao, Chen Yi, “Behavior of extraordinary rays in uniaxial crystals,” Appl. Opt. 33, 1209–1212 (1994).
    [CrossRef] [PubMed]
  7. G. Beyerle, I. S. McDermid, “Ray-tracing formulas for refraction and internal reflection in uniaxial crystals,” Appl. Opt. 37, 7947–7953 (1998).
    [CrossRef]
  8. O. N. Stavroudis, “Basic ray optics,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 15–22.
  9. P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, Orlando, Fla., 1985), Chap. 2.
  10. E. Goursat, A Course in Mathematical Analysis, translated by E. R. Hedrick (Ginn, Mass., 1904), pp. 459–461.
  11. D. H. Goldstein, “Anisotropic materials,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 847–878.

1998 (1)

1997 (1)

1994 (1)

1990 (1)

1988 (1)

1983 (1)

1962 (1)

Beyerle, G.

Cojocaru, E.

Goldstein, D. H.

D. H. Goldstein, “Anisotropic materials,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 847–878.

Goursat, E.

E. Goursat, A Course in Mathematical Analysis, translated by E. R. Hedrick (Ginn, Mass., 1904), pp. 459–461.

Lancaster, P.

P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, Orlando, Fla., 1985), Chap. 2.

Liang, Quan-Ting

McDermid, I. S.

Shao, Zhongxing

Simon, M. C.

Stavroudis, O. N.

O. N. Stavroudis, “Ray tracing formulas for uniaxial crystals,” J. Opt. Soc. Am. 52, 187–191 (1962).
[CrossRef]

O. N. Stavroudis, “Basic ray optics,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 15–22.

Tismenetsky, M.

P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, Orlando, Fla., 1985), Chap. 2.

Yi, Chen

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

Other (4)

O. N. Stavroudis, “Basic ray optics,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 15–22.

P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, Orlando, Fla., 1985), Chap. 2.

E. Goursat, A Course in Mathematical Analysis, translated by E. R. Hedrick (Ginn, Mass., 1904), pp. 459–461.

D. H. Goldstein, “Anisotropic materials,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 847–878.

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Figures (2)

Fig. 1
Fig. 1

Ordinary and extraordinary wavelets. wo and we are the ordinary and extraordinary wavelets, wfo and wfe are the corresponding wave fronts, and ro and re are the corresponding rays.

Fig. 2
Fig. 2

Invariance relation. Σ is the refracting surface, and N is its normal vector. θi is the angle of incidence, and θo is the ordinary ray angle of refraction. θe is the angle between the normal to the refracting surface and the extraordinary ray. ro and re are the ordinary and extraordinary rays, and wfo and wfe are the corresponding wave fronts. zo and ze represent distances along the two rays to intersection with their wave fronts. The invariance relation is ne2ze tan θe=no2zo tan θo.

Equations (74)

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ξo=μξ,ηo=μη,ζo=[μ2ζ2+(1-μ2)]1/2,
Po=H+(λ/no)So
xo=h+ξoλ/no,yo=k+ηoλ/no,
zo=ζoλ/no,
λ=nozo/ζo.
Po·So=xoξo+yoηo+zoζo=0,
zo=-(xoξo+yoηo)/ζo.
(xo-h)ξo+(yo-k)ηo+zoζo=λ/no,
λ=-no(hξo+kηo)=nozo/ζo=-no(xoξo+yoηo)/ζo2.
(Po-H)2=λ2/no2,
F(xo-h)2+(yo-k)2+zo2-(hξo+kηo)2=0(Po-H)2-(H·So)2=0.
F=xo-hyo-kTxo-hyo-k+zo2-hkTξoηo2=0.
Fh-2(xo-h)-2ξo(hξo+kηo)=0,
Fk-2(yo-k)-2ηo(hξo+kηo)=0,
h(1-ξo2)-kξoηo=xo,
-hξoηo+k(1-ηo2)=yo,
1-ξo2-ξoηo-ξoηo1-ηo2hk=(I-Em)hk=xoyo,
Em=ξo2ξoηoξoηoηo2.
(1-ξo2)(1-ηo2)-ξo2ηo2=1-ξo2-ηo2=ζo2,
hk=1ζo2(I-En)xoyo,
En=ηo2-ξoηo-ξoηoξo2.
xo-hyo-k=-1ζo2Emxoyo,
xo-hyo-kTxo-hyo-k=1-ζo2ζo4(ξoxo+ηoyo)2.
hkTξoηo=1ζo2xoyoT(I-En)ξoηo=1ζo2(ξoxo+ηoyo),
(xoξo+yoηo)2(1-ζo2)+zo2ζo4=(xoξo+yoηo)2,
(xoξo+yoηo)2-zo2ζo2=0.
(xoξo+yoηo+zoζo)(xoξo+yoηo-zoζo)=0.
(xe-h)2+(ye-k)2(λ/ne)2+ze2(λ/no)2=1,
ne2[(xe-h)2+(ye-k)2]+no2ze2=λ2.
Gne2[(xe-h)2+(ye-k)2]+no2ze2-no2(hξo+kηo)2=0.
Gne2{Z×[(Pe-H)×Z]}2+no2[(Pe-H)·Z]2-no2(So·H)2=0.
Gne2xe-hye-kTxe-hye-k+no2ze2-no2hkTξoηo2=0.
Gh-2ne2(xe-h)-2no2ξo(hξo+kηo)=0,
Gk-2ne2(ye-k)-2no2ηo(hξo+kηo)=0,
ne2-no2ξo2-no2ξoηo-no2ξoηone2-no2ηo2hk=(ne2I-no2Em)hk=ne2xeye,
(ne2-no2ξo2)(ne2-no2ηo2)-no4ξo2ηo2=ne2δo2,
δo2=ne2-no2(1-ζo2).
hk=1δo2(ne2I-no2En)xeye,
EnEm=EmEn=0,ξoηoTEn=Enξoηo=0.
ξoηoTEm=(1-ζo2)ξoηoT,
Em2=(1-ζo2)Em.
(1-ζo2)I-En=Em.
xe-hye-k=-no2δo2Emxeye.
xe-hye-kTxe-hye-k=ηo4(1-ζo2)δo4(xeξo+yeηo)2.
hkTξoηo=1δo2xeyeT(ne2I-no2En)ξoηo=ne2δo2(xeξo+yeηo).
δo2ze2-ne2(xeξo+yeηo)2=0,
[zeδo+ne(xeξo+yeηo)][zeδo-ne(xeξo+yeηo)]=0,
zeδo+ne(xeξo+yeηo)=0.
Pe=H+τSe,
xeye=hk+zeζeξeηe.
hk=1ζo2(I-En)xoyo=1δo2(ne2I-no2En)xeye.
(ne2I-no2En)-1=1ne2δo2(ne2I-ηo2Em).
xeye=δo2ζo2(ne2I-no2En)-1(I-En)xoyo=1ne2ζo2[ne2(I-En)-no2Em]xoyo.
xeyeTξoηo=1ne2ζo2xoyoT[ne2(I-En)-no2Em]ξoηo=1ne2ζo2xoyoT[ne2-no2(1-ζo2)]ξoηo=δo2ne2ζo2(xoξo+yoηo)=-δo2ne2ζozo,
zezo=δoneζo.
xeye=1δo2(ne2I-no2En)xeye+zeζeξeηe,
ξeηe=-no2ζeδo2zeEmxeye.
ξeηe=-no2ζene2ζo2δo2zeEm[ne2(I-En)-no2Em]xoyo=-no2ζene2ζo2zeEmxoyo,
Emxoyo=(xoξo+yoηo)ξoηo=-zoζoξoηo,
ξeηe=no2ζezone2ζozeξoηo.
1-ζe2=no2ζezone2ζoze2(1-ζo2).
ne2ze1-ζe2ζe=no2zo1-ζo2ζo.
ne2ze tan θe=no2zo tan θo,
ξeηe=no2ζeneδoξoηo.
1-ζe2=no4ζe2ne2δo2(1-ζo2),
ζe=neδoδe,
δe2=ne4-no2(ne2-no2)(1-ζo2).
ξeηeζe=1δeno2000ne2000neξoηoδo=1δeno2000no2000neδo/ζoξoηoζo.
ne2δo2(1-ζe2)=ne2[ne2-no2(1-ζo2)](1-ζe2)=no4ζe2(1-ζo2),
1-ζo2=ne4(1-ζe2)no2[ne2-(ne2-no2)ζe2].
ζ¯o2=-ne2(ne2-no2)+[ne4-no2(ne2-no2)]ζe2no2[ne2-(ne2-no2)ζe2].
δ¯o2=ne2no2ζe2ne2-(ne2-no2)ζe2,
δ¯e2=ne4no2ne2-(ne2-no2)ζe2.
ξoηoζo=δ¯e1/no20001/no2000ζ¯o/neδ¯oξeηeζe,

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