Abstract

We continue and extend Paper I [M. Avendaño-Alejo et al., J. Opt. Soc. Am. A 19, 1669–1674 (2002)] to the more general case where the crystal axis is not parallel to the surface normal. The procedures and the derivations follow those of Paper I. The result enables one to trace extraordinary rays in and out of a prism as well as through a crystal whose refracting surfaces are not planar.

© 2002 Optical Society of America

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References

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  1. M. Avendaño-Alejo, O. N. Stavroudis, A. R. Boyain y Goitial, “Huygens’s principle and rays in uniaxial anisotropic media. I. Crystal axis normal to refracting surface,” J. Opt. Soc. Am. A 19, 1669–1674 (2002).
  2. G. Beyerle, I. S. McDermid, “Ray-tracing formulas for refraction and internal reflection in uniaxial crystals,” Appl. Opt. 37, 7947–7953 (1998).
    [CrossRef]
  3. Quan-Ting Liang, “Simple ray tracing formulas for uniaxial optical crystal,” Appl. Opt. 29, 1008–1010 (1990).
    [CrossRef] [PubMed]

2002

M. Avendaño-Alejo, O. N. Stavroudis, A. R. Boyain y Goitial, “Huygens’s principle and rays in uniaxial anisotropic media. I. Crystal axis normal to refracting surface,” J. Opt. Soc. Am. A 19, 1669–1674 (2002).

1998

1990

Avendaño-Alejo, M.

M. Avendaño-Alejo, O. N. Stavroudis, A. R. Boyain y Goitial, “Huygens’s principle and rays in uniaxial anisotropic media. I. Crystal axis normal to refracting surface,” J. Opt. Soc. Am. A 19, 1669–1674 (2002).

Beyerle, G.

Boyain y Goitial, A. R.

M. Avendaño-Alejo, O. N. Stavroudis, A. R. Boyain y Goitial, “Huygens’s principle and rays in uniaxial anisotropic media. I. Crystal axis normal to refracting surface,” J. Opt. Soc. Am. A 19, 1669–1674 (2002).

Liang, Quan-Ting

McDermid, I. S.

Stavroudis, O. N.

M. Avendaño-Alejo, O. N. Stavroudis, A. R. Boyain y Goitial, “Huygens’s principle and rays in uniaxial anisotropic media. I. Crystal axis normal to refracting surface,” J. Opt. Soc. Am. A 19, 1669–1674 (2002).

Appl. Opt.

J. Opt. Soc. Am. A

M. Avendaño-Alejo, O. N. Stavroudis, A. R. Boyain y Goitial, “Huygens’s principle and rays in uniaxial anisotropic media. I. Crystal axis normal to refracting surface,” J. Opt. Soc. Am. A 19, 1669–1674 (2002).

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Figures (2)

Fig. 1
Fig. 1

Ordinary and extraordinary wavelets and the crystal axis. 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. Also shown is the crystal axis, which is not parallel to the surface normal.

Fig. 2
Fig. 2

Fictive ordinary ray. Shown is a prism of a birefringent medium with a ray incident on its upper surface and the refracted ordinary and extraordinary rays. At the point of incidence on the second surface, the equations for the extraordinary ray are inverted to get a fictive ordinary ray, indicated by a dashed line, which is then refracted through the prism’s exit surface.

Tables (1)

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Table 1 Refraction at a Plane Surfacea

Equations (67)

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Hne2{A×[(Pe-H)×A]}2+no2[(Pe-H)·A]2-no2(So·H)2=0.
Hne2xe-hye-kT+ne2ze2+Nxe-hye-kTαβ+zeγ2-no2hkTξoηo2=0,
N=no2-ne2.
(ne2I+NAm-no2Em)hk=ne2xeye+N(xeα+yeβ+zeγ)αβ,
Am=α2αβαββ2,
An=β2-αβ-αβα2.
AmAn=0,αβTAn=0,
αβTAm=(1-γ2)αβT,
Am2=(1-γ2)Am,
(1-γ2)I-An=Am.
(xeα+yeβ+zeγ)αβ=Amxeye+zeγαβ.
(ne2I+NAm-no2Em)hk=(ne2I+NAm)xeye+Nzeγαβ.
ne2I+NAm-no2Em=ne2+Nα2-no2ξ2Nαβ-no2ξηNαβ-no2ξηne2+Nβ2-no2η2,
Δ2=(ne2+Nα2-no2ξ2)(ne2+Nβ2-no2η2)-(Nαβ-no2ξη)2=ne4+ne2N(α2+β2)-ne2no2(ξo2+ηo2)-no2N(α2ηo2+β2ξo2)+2no2Nαβξoηo=ne4+ne2N(1-γ2)-ne2no2(1-ζo2)-no2N(αηo-βξo)2.
(αηo-βξo)2=(1-γ2)(1-ζo2)-(αξo+βηo)2.
Δ2=ne4+ne2N(1-γ2)-no2ne2(1-ζo2)-no2N[(1-γ2)(1-ζo2)-(αξo+βηo)2]=[ne2+N(1-γ2)][ne2-no2(1-ζo2)]+no2N(αξo+βη)2=Γδo2+no2Na2,
Γ=ne2+N(1-γ2),
a=αξo+βηo.
(ne2I+NAm-no2Em)-1=1Δ2ne2+Nβ2-no2ηo2-Nαβ+no2ξoηo-Nαβ+no2ξoηone2+Nα2-no2ξo2
=1Δ2(ne2I+NAn-no2En),
hk=1Δ2[no2(ne2I+NAn)Em+Δ2I]xeye+Nγze(ne2I-no2En)αβ.
b=ξoxe+ηoye.
xe-hye-kTxe-hye-k=1Δ2no4[ne4(1-ζo2)+N(ne2+Γ)(αηo-βξo)2]b2+2γzeno2N[ne4+no2N(αηo-βξo)2]ab+γ2N2ze2[ne4(1-γ2)-no2(ne2+δo2)×(αηo-βξo)2],
xe-hye-kTαβ+γze=-ne2Δ2(no2ab-γzeδo2),
hkTξoηo=ne2Δ2(Γb+γNzea).
[(δo2+Na2)ze-b(Naγ+Δ)]×[(δo2+Na2)ze-b(Naγ-Δ)]=0,
(δo2+Na2)ze-(ξoxe+ηoye)(Naγ-Δ)=0.
Ng=1Γg(ξo(Naγ-Δ),ηo(Naγ-Δ),-(δo2+Na2)),
Γg2=(1-ζo2)(Naγ-Δ)2+(δo2+Na2)2.
hk=1ζo2(I-En)xoyo=(ne2I+NAm-no2Em)-1×(ne2I+NAm)xeye+Nzeγαβ.
(ne2I+NAm)xeye=1ζo2(ne2I+NAm-no2Em)(I-En)×xoyo-Nzeγαβ.
xeye=(ne2I+NAm)-11ζo2[(ne2I+NAm)-no2Em]×(I-En)xoyo-Nzeγαβ.
|ne2I+NAm|=ne2+Nα2NαβNαβne2+Nβ2=(ne2+Nα2)(ne2+Nβ2)-N2α2β2=ne2[ne2+N(1-γ2)]=ne2Γ,
(ne2I+NAm)-1=1ne2Γ(ne2I+NAn).
(ne2I+NAm)-1[(ne2I+NAm)-no2Em]=I-no2ne2Γ(ne2I+NAn)Em.
ξoηoT(ne2I+NAm)-1[(ne2I+NAm)-no2Em]
=ξoηoT-no21ne2Γne2ξoηoT+NξoηoTAnEm
=ξoηoT-no2ne2Γne2(1-ζo2)ξoηoT+N(ξoβ-ηoα)2ξoηoT
=1ne2Γ{ne2Γ-no2ne2(1-ζo2)-no2N[(1-γ2)(1-ζo2)-a2]}ξoηoT
=1ne2Γ{ne2Γ-no2(1-ζo2)[ne2+N(1-γ2)]+no2Na2}ξoηoT
=1ne2Γ{Γ[ne2-no2(1-ζo2)]+no2Na2}ξoηoT
=1ne2Γ(Γδo2+no2Na2)ξoηoT
=Δ2ne2ΓξoηoT.
ξoηoT(I-En)xoyo=ξoηoTxoyo=xoξo+yoηo=-ζozo.
ξoηoTαβ=αξo+βηo=a.
xeξo+yeηo=ξoηoTxeye=-1ne2Γζo(Δ2zo+ne2Nzeγζoa).
zezo=Δne2ζo.
(ne2I+NAm-no2Em)hk=(ne2I+NAm)hk+zeζeξeηe+Nzeγαβ,
-no2Emhk=zeζe(ne2I+NAm)ξeηe+Nzeγαβ.
-no2Emhk=-no2(hξo+kηo)ξoηo=no2zoζoξoηo,
(ne2I+NAm)ξeηe=ζeno2zozeζoξoηo-Nzeγαβ.
ξeηe=ζezene2Γ(ne2I+NAn)no2zoζoξoηo-Nzeγαβ=ζene2Γno2zoζoze(ne2I+NAn)ξoηo-Nne2γαβ=ζeΓno2Δ(ne2I+NAn)ξoηo-γNαβ.
1-ζe2=ξeηeTξeηe=ζe2Γ2no2ΔξoηoT(ne2I+NAn)-γNαβT×(ne2I+NAn)ξoηo-γNαβ,
(1-ζe2)Γ2Δ2=ζe2(no4{ne4(1-ζo2)+N(ne2+Γ)×[(1-γ2)(1-ζo2)-a2]}-2Δno2ne2γNa+Δ2γ2N2(1-γ2)).
ζe=ΓΔne[no2Γ2-N(γΔ+no2a)2]1/2.
ξeηeζe=1δegno2(ne2+Nβ2)-no2Nαβ0-no2Nαβno2(ne2+Nα2)000ΓΔ/ζo×ξoηoζo-γNΔ100010000αβγ,
δeg2=ne2[no2Γ2-N(γΔ+no2a)2].
βξe-αηe=ζene2Γno2zoζoze[ne2+N(1-γ2)](βξo-αηo),
ne2ze(βξe-αηe)ζe=no2zo(βξo-αηo)ζo.
ne2ze2ζe2(1-ζe2)+ne2ze2+Nzeζe(αξe+βηe+γζe)2=no2zoζo2,
zoze=ζoμnoζe,
μ2=ne2+N(αξe+βηe+γζe)2.
ξoηo=1noμ(ne2I+NAm)ξeηe+ζeNγαβ.
1-ζo2=ξoηoTξoηo=1no2μ2ξeηeT(ne2I+NAm)+ζeγNαβT×(ne2I+NAm)ξeηe+ζeγNαβ,
ζo=Δenoμ,
Δe2=ne2(N+ne2ζe2)+N(Se·A)×[(Nγ2-ne2)(Se·A)+2ne2γζe].
ξoηoζo=1noμne2+Nα2Nαβ0Nαβne2+Nβ2000Δe/ζeξeηeζe+ζeγN100010000αβγ.

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