Abstract

Fermat's principle and the optical metric are generalized to the case of an anisotropic medium. The metric tensor of a three-dimensional Riemannian manifold is related to the dielectric tensor of the medium. The general eikonal equation in a static anisotropic medium is derived. The expressions for the curvature tensor and the curvature scalar that characterize the geometrical structure of a three-dimensional manifold are given. For an isotropic medium the derived expressions for the curvature tensor and curvature scalar reduce to the previous results.

© 1997 Optical Society of America

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  1. S. Zhu, W. Shen, “General relativistic ponderomotive force in a moving medium,” J. Opt. Soc. Am. B 4, 739–742 (1987).
    [CrossRef]
  2. S. Zhu, W. Shen, X. Deng, Z. Wang, “A general covariant derivation of electron energy gain in a laser accelerator,” Acta Phys. Sin. 38, 559–566 (1989).
  3. S. Zhu, “Frequency matching in beat wave laser accelerator,” Acta Phys. Sin. 38, 1167–1171 (1989).
  4. S. Zhu, W. Shen, P. Ji, F. Lin, “Effect of medium background on the hydrogen spectrum,” in Proceedings of the Topical Meeting on Laser Materials and Laser Spectroscopy (A Satellite Meeting of IQEC'88), Z. Wang, Z. Zhang, eds. (World Scientific, Singapore, 1988), p. 190.
  5. S. Zhu, Q. Guo, W. Shen, S. Wang, “Riemannian geometry of strong-laser plasma,” Int. J. Theor. Phys. 34, 169–178 (1995).
    [CrossRef]
  6. Q. Guo, W. Shen, S. Zhu, “Motion of photons in a strong laser plasma,” Acta Phys. Sin. 44, 396–400 (1995).
  7. Q. Guo, W. Shen, S. Zhu, “Classical behavior of a free electron in a strong laser plasma,” Acta Phys. Sin. 44, 210–215 (1995).
  8. W. Shen, S. Zhu, Q. Guo, “Classical description of the radiation of a charged particle in a strong-laserplasma,” Int. J. Theor. Phys. 34, 2095–2104 (1995).
    [CrossRef]
  9. W. Shen, S. Zhu, “Wave function of a free electron in a laser plasma via Riemannian geometry,” Int. J. Theor. Phys. 34, 2085–2094 (1995).
    [CrossRef]
  10. S. Zhu, W. Shen, Q. Guo, “Wave function of a free electron in a strong laser plasma,” Acta Phys. Sin. 42, 1471–1478 (1993).
  11. S. Zhu, W. Shen, “Fermat's principle in the geometrical optics and null geodesic in the metric optics,” presented at the World Optical Conference, Shanghai, China, 1993.
  12. W. Shen, S. Zhu, X. Deng, “The light tracks in the optical fibers with two types of parabolicrefractive indices,” Chin. J. Lasers B 5, 516–525 (1996).
  13. H. Guo, X. Deng, “Differential geometrical methods in the study of optical transmission(scalar theory). I. Static transmission case,” J. Opt. Soc. Am. A 12, 600–606 (1995).
    [CrossRef]
  14. H. Guo, X. Deng, “Differential geometrical methods in the study of optical transmission(scalar theory). II. Time-dependent transmission theory,” J. Opt. Soc. Am. A 12, 607–610 (1995).
    [CrossRef]
  15. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 14, p. 666.
  16. A. L. Rivera, S. M. Chumakov, K. B. Wolf, “Hamiltonian foundation of geometrical anisotropic optics,” J. Opt. Soc. Am. A 12, 1380–1389 (1995).
    [CrossRef]
  17. J. F. Carinena, J. Nasarre, “On the symplectic structures arising in geometric optics,” Fortschr. Phys. 44, 181–198 (1996).
    [CrossRef]
  18. J. F. Carinena, J. Nasarre, “Prosymplectic geometry and Fermat's principle for anisotropic media,” J. Phys. A 29, 1695–1702 (1996).
    [CrossRef]
  19. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 3, p. 122.
  20. M. Carmeli, Classical Fields: General Relativity and Gauge Theory (Wiley–Interscience, New York, 1982), Chap. 2, p. 67.
  21. C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (Freeman, San Francisco, Calif., 1973), Chap. 12, p. 293.

1996 (3)

W. Shen, S. Zhu, X. Deng, “The light tracks in the optical fibers with two types of parabolicrefractive indices,” Chin. J. Lasers B 5, 516–525 (1996).

J. F. Carinena, J. Nasarre, “On the symplectic structures arising in geometric optics,” Fortschr. Phys. 44, 181–198 (1996).
[CrossRef]

J. F. Carinena, J. Nasarre, “Prosymplectic geometry and Fermat's principle for anisotropic media,” J. Phys. A 29, 1695–1702 (1996).
[CrossRef]

1995 (8)

H. Guo, X. Deng, “Differential geometrical methods in the study of optical transmission(scalar theory). I. Static transmission case,” J. Opt. Soc. Am. A 12, 600–606 (1995).
[CrossRef]

H. Guo, X. Deng, “Differential geometrical methods in the study of optical transmission(scalar theory). II. Time-dependent transmission theory,” J. Opt. Soc. Am. A 12, 607–610 (1995).
[CrossRef]

A. L. Rivera, S. M. Chumakov, K. B. Wolf, “Hamiltonian foundation of geometrical anisotropic optics,” J. Opt. Soc. Am. A 12, 1380–1389 (1995).
[CrossRef]

S. Zhu, Q. Guo, W. Shen, S. Wang, “Riemannian geometry of strong-laser plasma,” Int. J. Theor. Phys. 34, 169–178 (1995).
[CrossRef]

Q. Guo, W. Shen, S. Zhu, “Motion of photons in a strong laser plasma,” Acta Phys. Sin. 44, 396–400 (1995).

Q. Guo, W. Shen, S. Zhu, “Classical behavior of a free electron in a strong laser plasma,” Acta Phys. Sin. 44, 210–215 (1995).

W. Shen, S. Zhu, Q. Guo, “Classical description of the radiation of a charged particle in a strong-laserplasma,” Int. J. Theor. Phys. 34, 2095–2104 (1995).
[CrossRef]

W. Shen, S. Zhu, “Wave function of a free electron in a laser plasma via Riemannian geometry,” Int. J. Theor. Phys. 34, 2085–2094 (1995).
[CrossRef]

1993 (1)

S. Zhu, W. Shen, Q. Guo, “Wave function of a free electron in a strong laser plasma,” Acta Phys. Sin. 42, 1471–1478 (1993).

1989 (2)

S. Zhu, W. Shen, X. Deng, Z. Wang, “A general covariant derivation of electron energy gain in a laser accelerator,” Acta Phys. Sin. 38, 559–566 (1989).

S. Zhu, “Frequency matching in beat wave laser accelerator,” Acta Phys. Sin. 38, 1167–1171 (1989).

1987 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 3, p. 122.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 14, p. 666.

Carinena, J. F.

J. F. Carinena, J. Nasarre, “On the symplectic structures arising in geometric optics,” Fortschr. Phys. 44, 181–198 (1996).
[CrossRef]

J. F. Carinena, J. Nasarre, “Prosymplectic geometry and Fermat's principle for anisotropic media,” J. Phys. A 29, 1695–1702 (1996).
[CrossRef]

Carmeli, M.

M. Carmeli, Classical Fields: General Relativity and Gauge Theory (Wiley–Interscience, New York, 1982), Chap. 2, p. 67.

Chumakov, S. M.

Deng, X.

W. Shen, S. Zhu, X. Deng, “The light tracks in the optical fibers with two types of parabolicrefractive indices,” Chin. J. Lasers B 5, 516–525 (1996).

H. Guo, X. Deng, “Differential geometrical methods in the study of optical transmission(scalar theory). II. Time-dependent transmission theory,” J. Opt. Soc. Am. A 12, 607–610 (1995).
[CrossRef]

H. Guo, X. Deng, “Differential geometrical methods in the study of optical transmission(scalar theory). I. Static transmission case,” J. Opt. Soc. Am. A 12, 600–606 (1995).
[CrossRef]

S. Zhu, W. Shen, X. Deng, Z. Wang, “A general covariant derivation of electron energy gain in a laser accelerator,” Acta Phys. Sin. 38, 559–566 (1989).

Guo, H.

Guo, Q.

S. Zhu, Q. Guo, W. Shen, S. Wang, “Riemannian geometry of strong-laser plasma,” Int. J. Theor. Phys. 34, 169–178 (1995).
[CrossRef]

Q. Guo, W. Shen, S. Zhu, “Motion of photons in a strong laser plasma,” Acta Phys. Sin. 44, 396–400 (1995).

Q. Guo, W. Shen, S. Zhu, “Classical behavior of a free electron in a strong laser plasma,” Acta Phys. Sin. 44, 210–215 (1995).

W. Shen, S. Zhu, Q. Guo, “Classical description of the radiation of a charged particle in a strong-laserplasma,” Int. J. Theor. Phys. 34, 2095–2104 (1995).
[CrossRef]

S. Zhu, W. Shen, Q. Guo, “Wave function of a free electron in a strong laser plasma,” Acta Phys. Sin. 42, 1471–1478 (1993).

Ji, P.

S. Zhu, W. Shen, P. Ji, F. Lin, “Effect of medium background on the hydrogen spectrum,” in Proceedings of the Topical Meeting on Laser Materials and Laser Spectroscopy (A Satellite Meeting of IQEC'88), Z. Wang, Z. Zhang, eds. (World Scientific, Singapore, 1988), p. 190.

Lin, F.

S. Zhu, W. Shen, P. Ji, F. Lin, “Effect of medium background on the hydrogen spectrum,” in Proceedings of the Topical Meeting on Laser Materials and Laser Spectroscopy (A Satellite Meeting of IQEC'88), Z. Wang, Z. Zhang, eds. (World Scientific, Singapore, 1988), p. 190.

Misner, C. W.

C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (Freeman, San Francisco, Calif., 1973), Chap. 12, p. 293.

Nasarre, J.

J. F. Carinena, J. Nasarre, “Prosymplectic geometry and Fermat's principle for anisotropic media,” J. Phys. A 29, 1695–1702 (1996).
[CrossRef]

J. F. Carinena, J. Nasarre, “On the symplectic structures arising in geometric optics,” Fortschr. Phys. 44, 181–198 (1996).
[CrossRef]

Rivera, A. L.

Shen, W.

W. Shen, S. Zhu, X. Deng, “The light tracks in the optical fibers with two types of parabolicrefractive indices,” Chin. J. Lasers B 5, 516–525 (1996).

W. Shen, S. Zhu, “Wave function of a free electron in a laser plasma via Riemannian geometry,” Int. J. Theor. Phys. 34, 2085–2094 (1995).
[CrossRef]

Q. Guo, W. Shen, S. Zhu, “Classical behavior of a free electron in a strong laser plasma,” Acta Phys. Sin. 44, 210–215 (1995).

W. Shen, S. Zhu, Q. Guo, “Classical description of the radiation of a charged particle in a strong-laserplasma,” Int. J. Theor. Phys. 34, 2095–2104 (1995).
[CrossRef]

Q. Guo, W. Shen, S. Zhu, “Motion of photons in a strong laser plasma,” Acta Phys. Sin. 44, 396–400 (1995).

S. Zhu, Q. Guo, W. Shen, S. Wang, “Riemannian geometry of strong-laser plasma,” Int. J. Theor. Phys. 34, 169–178 (1995).
[CrossRef]

S. Zhu, W. Shen, Q. Guo, “Wave function of a free electron in a strong laser plasma,” Acta Phys. Sin. 42, 1471–1478 (1993).

S. Zhu, W. Shen, X. Deng, Z. Wang, “A general covariant derivation of electron energy gain in a laser accelerator,” Acta Phys. Sin. 38, 559–566 (1989).

S. Zhu, W. Shen, “General relativistic ponderomotive force in a moving medium,” J. Opt. Soc. Am. B 4, 739–742 (1987).
[CrossRef]

S. Zhu, W. Shen, P. Ji, F. Lin, “Effect of medium background on the hydrogen spectrum,” in Proceedings of the Topical Meeting on Laser Materials and Laser Spectroscopy (A Satellite Meeting of IQEC'88), Z. Wang, Z. Zhang, eds. (World Scientific, Singapore, 1988), p. 190.

S. Zhu, W. Shen, “Fermat's principle in the geometrical optics and null geodesic in the metric optics,” presented at the World Optical Conference, Shanghai, China, 1993.

Thorne, K. S.

C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (Freeman, San Francisco, Calif., 1973), Chap. 12, p. 293.

Wang, S.

S. Zhu, Q. Guo, W. Shen, S. Wang, “Riemannian geometry of strong-laser plasma,” Int. J. Theor. Phys. 34, 169–178 (1995).
[CrossRef]

Wang, Z.

S. Zhu, W. Shen, X. Deng, Z. Wang, “A general covariant derivation of electron energy gain in a laser accelerator,” Acta Phys. Sin. 38, 559–566 (1989).

Wheeler, J. A.

C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (Freeman, San Francisco, Calif., 1973), Chap. 12, p. 293.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 14, p. 666.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 3, p. 122.

Wolf, K. B.

Zhu, S.

W. Shen, S. Zhu, X. Deng, “The light tracks in the optical fibers with two types of parabolicrefractive indices,” Chin. J. Lasers B 5, 516–525 (1996).

W. Shen, S. Zhu, “Wave function of a free electron in a laser plasma via Riemannian geometry,” Int. J. Theor. Phys. 34, 2085–2094 (1995).
[CrossRef]

S. Zhu, Q. Guo, W. Shen, S. Wang, “Riemannian geometry of strong-laser plasma,” Int. J. Theor. Phys. 34, 169–178 (1995).
[CrossRef]

Q. Guo, W. Shen, S. Zhu, “Motion of photons in a strong laser plasma,” Acta Phys. Sin. 44, 396–400 (1995).

W. Shen, S. Zhu, Q. Guo, “Classical description of the radiation of a charged particle in a strong-laserplasma,” Int. J. Theor. Phys. 34, 2095–2104 (1995).
[CrossRef]

Q. Guo, W. Shen, S. Zhu, “Classical behavior of a free electron in a strong laser plasma,” Acta Phys. Sin. 44, 210–215 (1995).

S. Zhu, W. Shen, Q. Guo, “Wave function of a free electron in a strong laser plasma,” Acta Phys. Sin. 42, 1471–1478 (1993).

S. Zhu, W. Shen, X. Deng, Z. Wang, “A general covariant derivation of electron energy gain in a laser accelerator,” Acta Phys. Sin. 38, 559–566 (1989).

S. Zhu, “Frequency matching in beat wave laser accelerator,” Acta Phys. Sin. 38, 1167–1171 (1989).

S. Zhu, W. Shen, “General relativistic ponderomotive force in a moving medium,” J. Opt. Soc. Am. B 4, 739–742 (1987).
[CrossRef]

S. Zhu, W. Shen, P. Ji, F. Lin, “Effect of medium background on the hydrogen spectrum,” in Proceedings of the Topical Meeting on Laser Materials and Laser Spectroscopy (A Satellite Meeting of IQEC'88), Z. Wang, Z. Zhang, eds. (World Scientific, Singapore, 1988), p. 190.

S. Zhu, W. Shen, “Fermat's principle in the geometrical optics and null geodesic in the metric optics,” presented at the World Optical Conference, Shanghai, China, 1993.

Acta Phys. Sin. (5)

S. Zhu, W. Shen, X. Deng, Z. Wang, “A general covariant derivation of electron energy gain in a laser accelerator,” Acta Phys. Sin. 38, 559–566 (1989).

S. Zhu, “Frequency matching in beat wave laser accelerator,” Acta Phys. Sin. 38, 1167–1171 (1989).

Q. Guo, W. Shen, S. Zhu, “Motion of photons in a strong laser plasma,” Acta Phys. Sin. 44, 396–400 (1995).

Q. Guo, W. Shen, S. Zhu, “Classical behavior of a free electron in a strong laser plasma,” Acta Phys. Sin. 44, 210–215 (1995).

S. Zhu, W. Shen, Q. Guo, “Wave function of a free electron in a strong laser plasma,” Acta Phys. Sin. 42, 1471–1478 (1993).

Chin. J. Lasers B (1)

W. Shen, S. Zhu, X. Deng, “The light tracks in the optical fibers with two types of parabolicrefractive indices,” Chin. J. Lasers B 5, 516–525 (1996).

Fortschr. Phys. (1)

J. F. Carinena, J. Nasarre, “On the symplectic structures arising in geometric optics,” Fortschr. Phys. 44, 181–198 (1996).
[CrossRef]

Int. J. Theor. Phys. (3)

W. Shen, S. Zhu, Q. Guo, “Classical description of the radiation of a charged particle in a strong-laserplasma,” Int. J. Theor. Phys. 34, 2095–2104 (1995).
[CrossRef]

W. Shen, S. Zhu, “Wave function of a free electron in a laser plasma via Riemannian geometry,” Int. J. Theor. Phys. 34, 2085–2094 (1995).
[CrossRef]

S. Zhu, Q. Guo, W. Shen, S. Wang, “Riemannian geometry of strong-laser plasma,” Int. J. Theor. Phys. 34, 169–178 (1995).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

J. Phys. A (1)

J. F. Carinena, J. Nasarre, “Prosymplectic geometry and Fermat's principle for anisotropic media,” J. Phys. A 29, 1695–1702 (1996).
[CrossRef]

Other (6)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 3, p. 122.

M. Carmeli, Classical Fields: General Relativity and Gauge Theory (Wiley–Interscience, New York, 1982), Chap. 2, p. 67.

C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (Freeman, San Francisco, Calif., 1973), Chap. 12, p. 293.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 14, p. 666.

S. Zhu, W. Shen, P. Ji, F. Lin, “Effect of medium background on the hydrogen spectrum,” in Proceedings of the Topical Meeting on Laser Materials and Laser Spectroscopy (A Satellite Meeting of IQEC'88), Z. Wang, Z. Zhang, eds. (World Scientific, Singapore, 1988), p. 190.

S. Zhu, W. Shen, “Fermat's principle in the geometrical optics and null geodesic in the metric optics,” presented at the World Optical Conference, Shanghai, China, 1993.

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Equations (43)

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δds=δ  ndl=0,
ds2=n2dl2=gij(x)dxidxj.
gij=n2δij=n2(i=j) 0(ij)(i, j=1, 2, 3),
δij=1(i=j)0(ij).
D=EE,
E=ε11ε12ε13ε21ε22ε23ε31ε32ε33.
E=ε1 0 ε2 0 ε3=n12 0 n22 0 n32,
δds=δ (n12dx2+n22dy2+n32dz2)1/2=0.
ds2=n12dx2+n22dy2+n32dz2=gij(x)dxidxj.
gij=ni2δij=ni2(i=j) 0(ij)(i, j=1, 2, 3),
gij=n120 n22 0 n32.
g=det(gij)=n12n22n32,
gij=ni-2(i=j)  0(ij)(i, j=1, 2, 3),
gij=n1-2 0 n2-2 0 n3-2.
δABds=δAbn(x, x˙)dl=0,
n(x, x˙)=(n12x˙12+n22x˙22+n32x˙32)1/2.
δzAzBdzL[q(z), z; v(z)]=0,
L(q, z; v)=(1+v2)1/2n(q, z; v).
L=1+n1n32x˙12+n2n32x˙221/2n3(x)=(1+x˙12+x˙22)1/2n3(x1, x2, x3)×1+[(n1/n3)2-1]x˙12+[(n2/n3)2-1]x˙221+x˙12+x˙221/2.
L=(1+v2)1/21+[(n0/n3)2-1]v21+v21/2n3(q, z)=(1+v2)1/2n(q, z; v2)=L(q, z; v2),
n(q, z; v2)=n3(q, z)1+[(n0/n3)2-1]v21+v21/2=n3(q, z)1+(n02-n32)n32 sin2 θ1/2,
δds=δ(gijdxidxj)1/2=0,
d2xids2+Γjki dxids dxkds=0,
Γjki=12 gilgljxk+glkxj-gjkxl.
Γjki=Γkji=ni-1ni,k(i=j)-ni-2njnj,i(ij=k)0(ijki),
dds n12 dxds=12 n1,x2dxds2+n2,x2dyds2+n3,x2dzds2,
dds n22 dyds=12 n1,y2dxds2+n2,y2dyds2+n3,y2dzds2,
dds n32 dzds=12 n1,z2dxds2+n2,z2dyds2+n3,z2dzds2,
dds (ijk)n12 0 n22 0 n32 dx/dsdy/dsdz/ds
=12 dxds dyds dzdsn12 0 n22 0 n32 dx/dsdy/dsdz/ds.
i x+j y+k z.
N2=n12 0 n22 0 n32=E;
dds eN2dxdsT=12 dxds N2dxdsT,
ddl n dxdl=n.
Rjkli=Γjl,ki-Γjk,li+ΓjlmΓmki-ΓjkmΓmli,
Γjk,li=Γkj,li=ni-1ni,kl-ni-2ni,kni,l(i=j)2ni-2njnj,ini,l -ni-2(nj,lnj,i+njnj,il)(ij=k)0(ijki) ,
Rjkli=-Rjlki=14ni-2{2(ni,jj2+nj,ii2)+np-2ni,p2nj,p2-ni-2[(ni,j2)2+ni,i2nj,i2]-nj-2[(nj,i2)2-nj,j2ni,j2](ij, i=l, j=k)14ni-2[2ni,jk2-(ni-2ni,j2ni,k2+nj-2ni,j2nj,k2+nk-2ni,k2nk,j2)](ijki, i=l)-14ni-2[2nj,ik2-(ni-2nj,i2ni,k2+nj-2nj,i2nj,k2+nk-2nj,k2nk,i2)](ijki, j=l),
R=Rii=gikRik=gikRikll.
R=14ni-2nj-2{2(ni,jj2+nj,ii2)-2ni-2[ni,i2nj,i2+(ni,j2)2]+nk-2ni,k2nj,k2},
d2Nids2+Rjkli dxjds Nk dxlds=0,
Rjkli=-Rjlki=12n-2{(njj2+nii2)+n-2[12(np2)2-(ni2)2-(nj2)2]}(ij, i=l, j=k)12n-2(njk2-32n-2nj2nk2)(ijki, i=l)-12n-2(nik2-32n-2ni2nk2)(ijki, j=l)0otherwise,
R=2n-4[(2n2)-34n-2(n2)2],
2=2xx+2yy+2zz=2x1x1+2x2x2+2x3x3.

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