Abstract

Transmission and reflection of Gaussian beams from a general anisotropic multilayer structure are investigated. The principal axes of the layers are oriented arbitrarily with respect to each other and with respect to a fixed reference coordinate system. The Gaussian beam is assumed to have an arbitrary angle of incidence and linear polarization orientation. Two numerical examples are presented: a single slab of uniaxial calcite and a multilayer structure of biaxial 4-(N,N-dimethylamino)-3-acetamidonitrobenzene with antireflection coatings on the input and output faces. Results show the distortions of the beam caused by the anisotropy of the structure.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. Chiu Chan, T. Tamir, “Beam propagation phenomena at and near critical incidence upon a dielectric interface,” J. Opt. Soc. Am. A 4, 655–663 (1987).
    [CrossRef]
  2. C. Chiu Chan, T. Tamir, “Angular shift of a Gaussian beam reflected near the Brewster angle,” Opt. Lett. 10, 378–380 (1985).
    [CrossRef]
  3. S. Kozaki, H. Harada, “Beam displacement of a reflected beam at an interface between an inhomogeneous medium and free space,” J. Opt. Soc. Am. 68, 1592–1596 (1978).
    [CrossRef]
  4. S. Kozaki, H. Sakurai, “Characteristics of a Gaussian beam at a dielectric interface,” J. Opt. Soc. Am. 68, 508–514 (1978).
    [CrossRef]
  5. M. McGuirk, C. K. Carniglia, “An angular representation approach to the Goos–Hanchen shift,” J. Opt. Soc. Am. 67, 103–107 (1977).
    [CrossRef]
  6. B. R. Horowitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am 61, 586–594 (1971).
    [CrossRef]
  7. T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam of a uniaxially anisotropic medium,” Electron. Commun. Jpn. Part I 72, 95–103 (1989).
    [CrossRef]
  8. R. P. Riesz, R. Simon, “Reflection of a Gaussian beam from a dielectric slab,” J. Opt. Soc. Am. A 2, 1809–1817 (1985).
    [CrossRef]
  9. C. W. Haue, T. Tamir, “Lateral displacement and distortion of beams incident upon a transmitting-layer configuration,” J. Opt. Soc. Am. A 2, 978–988 (1985).
    [CrossRef]
  10. M. Tanaka, K. Tanaka, O. Fakumitsu, “Transmission and reflection of a Gaussian beam at oblique incidence on a dielectric slab,” J. Opt. Soc. Am. 67, 819–825 (1977).
    [CrossRef]
  11. T. Sonoda, S. Kozaki, “Reflection and transmission of a Gaussian beam from an anisotropic dielectric slab,” Electron. Commun. Jpn. Part I 73, 85–92 (1990).
    [CrossRef]
  12. F. Falco, T. Tamir, “Improved analysis of nonspecular phenomena in beams reflected from stratified media,” J. Opt. Soc. Am. A 7, 185–190 (1990).
    [CrossRef]
  13. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. A 3, 558–565 (1986).
    [CrossRef]
  14. V. Shah, T. Tamir, “Absorption and lateral shift of beams incident upon lossy multilayered media,” J. Opt. Soc. Am. 73, 37–44 (1983).
    [CrossRef]
  15. T. Tamir, H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures,” J. Opt. Soc. Am. 61, 1397–1413 (1971).
    [CrossRef]
  16. D. W. Berreman, “Optics in stratified and anisotropic media,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  17. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  18. P. Yeh, A. Yariv, C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
    [CrossRef]
  19. P. Yeh, A. Yariv, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. 67, 438–447 (1977).
    [CrossRef]
  20. S. Teitler, B. W. Henvis, “Refraction in stratified, anisotropic media,” J. Opt. Soc. Am. 60, 830–834 (1970).
    [CrossRef]
  21. P. J. Lin-Chung, S. Teitler, “4 × 4 matrix formalisms for optics in stratified anisotropic media,” J. Opt. Soc. Am. A 1, 703–705 (1984).
    [CrossRef]
  22. K. Eidner, “Light propagation in stratified anisotropic media: orthogonality and symmetry properties of the 4 × 4 matrix formalisms,” J. Opt. Soc. Am. A 6, 1657–1660 (1989).
    [CrossRef]
  23. H. Wöhler, G. Haas, M. Fritch, D. A. Mlynski, “Faster 4 × 4 matrix method for uniaxial inhomogeneous media,” J. Opt. Soc. Am. A 5, 1554–1557 (1988).
    [CrossRef]
  24. R. S. Weis, T. K. Gaylord, “Electromagnetic transmission and reflection characteristics of anisotropic multilayered structures,” J. Opt. Soc. Am. A 4, 1720–1739 (1987).
    [CrossRef]
  25. J. F. Nicoud, R. J. Twieg, “Organic SHG powder test data,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. Chemla, J. Zyss, eds. (Academic, Orlando, Fla., 1987), Vol. 2, p. 251.
  26. P. Kerkoc, M. Zgonik, K. Sutter, Ch. Bossard, P. Günter, “4-(N, N-dimethylamino)-3-acetamidonitrobenzene single crystals for nonlinear-optical applications,” J. Opt. Soc. Am. B 7, 313–319 (1990).
    [CrossRef]
  27. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass, 1981).

1990 (3)

1989 (2)

K. Eidner, “Light propagation in stratified anisotropic media: orthogonality and symmetry properties of the 4 × 4 matrix formalisms,” J. Opt. Soc. Am. A 6, 1657–1660 (1989).
[CrossRef]

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam of a uniaxially anisotropic medium,” Electron. Commun. Jpn. Part I 72, 95–103 (1989).
[CrossRef]

1988 (1)

1987 (2)

1986 (1)

T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. A 3, 558–565 (1986).
[CrossRef]

1985 (3)

1984 (1)

1983 (1)

1979 (1)

1978 (2)

1977 (4)

1972 (1)

1971 (2)

T. Tamir, H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures,” J. Opt. Soc. Am. 61, 1397–1413 (1971).
[CrossRef]

B. R. Horowitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am 61, 586–594 (1971).
[CrossRef]

1970 (1)

Berreman, D. W.

Bertoni, H. L.

Bossard, Ch.

Carniglia, C. K.

Chiu Chan, C.

Eidner, K.

Fakumitsu, O.

Falco, F.

Fritch, M.

Gaylord, T. K.

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass, 1981).

Günter, P.

Haas, G.

Harada, H.

Haue, C. W.

Henvis, B. W.

Hong, C. S.

Horowitz, B. R.

B. R. Horowitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am 61, 586–594 (1971).
[CrossRef]

Kerkoc, P.

Kozaki, S.

T. Sonoda, S. Kozaki, “Reflection and transmission of a Gaussian beam from an anisotropic dielectric slab,” Electron. Commun. Jpn. Part I 73, 85–92 (1990).
[CrossRef]

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam of a uniaxially anisotropic medium,” Electron. Commun. Jpn. Part I 72, 95–103 (1989).
[CrossRef]

S. Kozaki, H. Sakurai, “Characteristics of a Gaussian beam at a dielectric interface,” J. Opt. Soc. Am. 68, 508–514 (1978).
[CrossRef]

S. Kozaki, H. Harada, “Beam displacement of a reflected beam at an interface between an inhomogeneous medium and free space,” J. Opt. Soc. Am. 68, 1592–1596 (1978).
[CrossRef]

Lin-Chung, P. J.

McGuirk, M.

Mlynski, D. A.

Nicoud, J. F.

J. F. Nicoud, R. J. Twieg, “Organic SHG powder test data,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. Chemla, J. Zyss, eds. (Academic, Orlando, Fla., 1987), Vol. 2, p. 251.

Riesz, R. P.

Sakurai, H.

Shah, V.

Simon, R.

Sonoda, T.

T. Sonoda, S. Kozaki, “Reflection and transmission of a Gaussian beam from an anisotropic dielectric slab,” Electron. Commun. Jpn. Part I 73, 85–92 (1990).
[CrossRef]

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam of a uniaxially anisotropic medium,” Electron. Commun. Jpn. Part I 72, 95–103 (1989).
[CrossRef]

Sutter, K.

Tamir, T.

Tanaka, K.

Tanaka, M.

Teitler, S.

Twieg, R. J.

J. F. Nicoud, R. J. Twieg, “Organic SHG powder test data,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. Chemla, J. Zyss, eds. (Academic, Orlando, Fla., 1987), Vol. 2, p. 251.

Weis, R. S.

Wöhler, H.

Yariv, A.

Yeh, P.

Zgonik, M.

Electron. Commun. Jpn. Part I (2)

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam of a uniaxially anisotropic medium,” Electron. Commun. Jpn. Part I 72, 95–103 (1989).
[CrossRef]

T. Sonoda, S. Kozaki, “Reflection and transmission of a Gaussian beam from an anisotropic dielectric slab,” Electron. Commun. Jpn. Part I 73, 85–92 (1990).
[CrossRef]

J. Opt. Soc. A (1)

T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. A 3, 558–565 (1986).
[CrossRef]

J. Opt. Soc. Am (1)

B. R. Horowitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am 61, 586–594 (1971).
[CrossRef]

J. Opt. Soc. Am. (11)

M. Tanaka, K. Tanaka, O. Fakumitsu, “Transmission and reflection of a Gaussian beam at oblique incidence on a dielectric slab,” J. Opt. Soc. Am. 67, 819–825 (1977).
[CrossRef]

S. Kozaki, H. Harada, “Beam displacement of a reflected beam at an interface between an inhomogeneous medium and free space,” J. Opt. Soc. Am. 68, 1592–1596 (1978).
[CrossRef]

S. Kozaki, H. Sakurai, “Characteristics of a Gaussian beam at a dielectric interface,” J. Opt. Soc. Am. 68, 508–514 (1978).
[CrossRef]

M. McGuirk, C. K. Carniglia, “An angular representation approach to the Goos–Hanchen shift,” J. Opt. Soc. Am. 67, 103–107 (1977).
[CrossRef]

V. Shah, T. Tamir, “Absorption and lateral shift of beams incident upon lossy multilayered media,” J. Opt. Soc. Am. 73, 37–44 (1983).
[CrossRef]

T. Tamir, H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures,” J. Opt. Soc. Am. 61, 1397–1413 (1971).
[CrossRef]

D. W. Berreman, “Optics in stratified and anisotropic media,” J. Opt. Soc. Am. 62, 502–510 (1972).
[CrossRef]

P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
[CrossRef]

P. Yeh, A. Yariv, C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
[CrossRef]

P. Yeh, A. Yariv, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. 67, 438–447 (1977).
[CrossRef]

S. Teitler, B. W. Henvis, “Refraction in stratified, anisotropic media,” J. Opt. Soc. Am. 60, 830–834 (1970).
[CrossRef]

J. Opt. Soc. Am. A (8)

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

Opt. Lett. (1)

Other (2)

J. F. Nicoud, R. J. Twieg, “Organic SHG powder test data,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. Chemla, J. Zyss, eds. (Academic, Orlando, Fla., 1987), Vol. 2, p. 251.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass, 1981).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Coordinate system definition for a multilayer dielectric structure between two isotropic regions. The laboratory coordinate system is such that the front surface of the multilayer structure is the (x, y) plane, and the plane of incidence is the (x, z) plane. The indices of refraction for the input and output regions are ni and nt, respectively, and y′ || y″ || y‴ || y. The index of refraction for each layer in the multilayer structure is a tensor. The incident, transmitted, and reflected beam coordinate systems are specified by single-, double-, and triple-primed coordinate systems, respectively. The distance in the (x, z) plane from the multilayered structure to the input, transmitted, and reflected plane origins are specified by z1, z2, and z3, respectively. The incident symmetric Gaussian beam is focused in the input (x′, y′) plane and has a linear polarization oriented in the (x′, y′) plane at an angle θi with respect to the y′ axis (not shown).

Fig. 2
Fig. 2

Magnitude of the transmitted TE and TM field distributions for a 6000-λ0-thick calcite slab. The input Gaussian beam is incident at an angle υi = 30°, is TE polarized (θi = 0°) with wavelength λ0 = 633 nm, and has a spot size of ω0 = 500 λ0. The input (x′, y′) and output (x″, y″) and (x‴, y‴) planes are located 10 cm from the calcite slab (z1 = z2 = z3 = 10 cm). (a) TE and (b) TM distributions calculated with the methodology developed in this paper compared with calculations for (c) TE and (d) TM distributions calculated with the method in Ref. 11.

Fig. 3
Fig. 3

Magnitude of the reflected TE and TM field distributions for a 6000-λ0-thick calcite slab. The input Gaussian beam is incident at an angle υi = 30°, TE polarized (θi = 0°) with wavelength λ0 = 633 nm, and has a spot size of ω0 = 500 λ0. The incident and output planes are located 10 cm from the calcite slab (z1 = z2 = z3 = 10 cm). (a) TE and (b) TM distributions calculated with the methodology developed in this paper compared to calculations for (c) TE and (d) TM distributions calculated with a method in Ref. 11.

Fig. 4
Fig. 4

Input plane electric-field distribution of the incident Gaussian beam in the example presented in Section 5. The input beam is incident at an angle υi = 30°, TE polarized (θi = 0°) with wavelength λ0 = 1.064 μm, and a spot size of ω0 = 250 μm. The distance from the input plane to the laboratory coordinate system origin is z1 = 10 cm.

Fig. 5
Fig. 5

Magnitude of (a) TE and (b) TM transmitted field distributions for a 2-mm-thick DAN slab with AR-coated entrance and exit interfaces. The DAN slab has refractive indices of nx = 1.517, ny = 1.636, nz = 1.843; its principal dielectric axes are oriented at Euler angles (ϕ, θ, ψ) = (0°, −35.1°, 90°) with respect to the laboratory coordinate system. The AR layers have a refractive index of 1.290 and a thickness of 206 nm. This output plane is located 10 cm from the multilayer structure. The input Gaussian beam profile is shown in Fig. 4.

Fig. 6
Fig. 6

Magnitude of (a) TE and (b) TM reflected field distributions for the 2-mm-thick DAN slab with AR-coated entrance and exit interfaces of Fig. 5.

Equations (47)

Equations on this page are rendered with MathJax. Learn more.

υ t = sin - 1 [ n i n t sin ( υ i ) ] .
E ( x , y , z = 0 ) = exp ( - x 2 + y 2 2 ω 0 2 ) ,
E ( x , y , z = 0 ) = 1 2 π - - A ( k x , k y ) × exp [ - j ( k x x + k y y ) ] d k x d k y .
A ( k x , k y ) = 1 2 π - - E ( x , y , z = 0 ) × exp [ j ( k x x + k y y ) ] d x d y = ω 0 2 exp ( - k x 2 + k y 2 2 ω 0 - 2 ) .
E ( x , y , z = 0 ) = ω 0 2 2 π - - exp ( - k x 2 + k y 2 2 ω 0 - 2 ) × exp [ - j ( k x x + k y y ) ] d k x d k y .
E ( x , y , z ) = ω 0 2 2 π - - exp ( - k x 2 + k y 2 2 ω 0 - 2 ) × exp [ - j ( k x x + k y y ) ] × exp ( - j α ) d k x d k y ,
α = k z z = ( k 0 2 n i 2 - k x 2 - k y 2 ) 1 / 2 z .
E ( x , y , z = 0 ) = ω 0 2 2 π - - t ( k x , k y ) × exp ( - k x 2 + k y 2 2 ω 0 - 2 ) exp [ - j ( k x x + k y y ) ] exp ( - j β ) d k x d k y .
β = k z z 1 + k z z 2 = ( k 0 2 n i 2 - k x 2 - k y 2 ) 1 / 2 z 1 + ( k 0 2 n t 2 - k x 2 - k y 2 ) 1 / 2 z 2 ,
k x = k x { n i n t sin 2 ( υ i ) + cos ( υ i ) [ 1 - n i 2 n t 2 sin 2 ( υ i ) ] 1 / 2 } + k z { sin ( υ i ) [ 1 - n i 2 n t 2 sin 2 ( υ i ) ] 1 / 2 - n i 2 n t sin ( 2 υ i ) } , k z = k z { n i n t sin 2 ( υ i ) + cos ( υ i ) [ 1 - n i 2 n t 2 sin 2 ( υ i ) ] 1 / 2 } + k x { - sin ( υ i ) [ 1 - n i 2 n t 2 sin 2 ( υ i ) ] 1 / 2 + n i 2 n t sin ( 2 υ i ) } .
k x = cos ( υ i ) k x + sin ( υ i ) ( k 0 2 n i 2 - k x 2 - k y 2 ) 1 / 2 , k y = k y .
E ( x , y , z = 0 ) = ω 0 2 2 π - - r ( k x , k y ) × exp ( - k x 2 + k y 2 2 ω 0 - 2 ) exp [ - j ( k x x + k y y ) ] exp ( - j γ ) d k x d k y ,
γ = k z ( z 1 + z 3 ) = ( k 0 2 n i 2 - k x 2 - k y 2 ) 1 / 2 ( z 1 + z 3 ) .
w N k < 2 π k w .
[ 0 0 0 0 - z y 0 0 0 z 0 - x 0 0 0 - y x 0 0 z - y 0 0 0 - z 0 x 0 0 0 y - x 0 0 0 0 ] [ E x E y E z H x H y H z ] = j ω [ D x D y D z B x B y B z ]
R ¯ G = j ω C .
C = M ¯ G ,
M ¯ = [ ɛ x x ɛ x y ɛ x z 0 0 0 ɛ y x ɛ y y ɛ y z 0 0 0 ɛ z x ɛ z y ɛ z z 0 0 0 0 0 0 μ 0 0 0 0 0 0 0 μ 0 0 0 0 0 0 0 μ 0 ] .
R ¯ G = j ω M ¯ G .
G = exp ( - j k x x ) exp ( - j k y y ) Γ ( z ) ,
Γ ( z ) = [ E x ( z )             E y ( z )             E z ( z )             H x ( z )             H y ( z )             H z ( z ) ] T .
z ψ = j k 0 Δ ¯ ψ ,
Δ ¯ = [ k ˜ x ɛ z x ɛ z z k ˜ x 2 ɛ z z - 1 k ˜ x ɛ z y ɛ z z - k ˜ x k ˜ y ɛ z z ɛ x z ɛ z x ɛ z z - ɛ x x k ˜ x ɛ x z ɛ z z ɛ x z ɛ z y ɛ z z - ɛ x y - k ˜ y ɛ x z ɛ z z 0 0 0 1 ɛ y x - ɛ y z ɛ z x ɛ z z + k ˜ x k ˜ y - k ˜ x ɛ y z ɛ z z ɛ y y - ɛ y z ɛ z y ɛ z z - k ˜ x 2 - k ˜ y ɛ y z ɛ z z ] ,
k ˜ x = k x / k 0 ,
k ˜ y = k y / k 0 ,
ψ = ( E x             η 0 H y             E y             η 0 H x ) T .
ψ m ( z ) = exp ( j k z m z ) ψ m ( 0 ) ,
k 0 Δ ¯ ψ m = k z m ψ m ,
P ¯ = Ψ ¯ K ¯ Ψ ¯ - 1 ,
ψ t = Ψ ¯ N K ¯ N Ψ ¯ N - 1 Ψ ¯ 2 K ¯ 2 Ψ ¯ 2 - 1 Ψ ¯ 1 K ¯ 1 Ψ ¯ 1 - 1 ψ i .
ψ α = Ψ ¯ α ϕ α ,
Ψ ¯ α = [ cos ( υ α ) - cos ( υ α ) 0 0 n α n α 0 0 0 0 1 1 0 0 - n α cos ( υ α ) n α cos ( υ α ) ] ,
ϕ i = [ E i TM             E r TM             E i TE             E r TE ] T , ϕ t = [ E t TM             0             E t TE             0 ] T .
ϕ t = Ψ ¯ t - 1 Ψ ¯ N K ¯ N Ψ ¯ N - 1 Ψ ¯ 2 K ¯ 2 Ψ ¯ 2 - 1 Ψ ¯ 1 K ¯ 1 Ψ ¯ 1 - 1 Ψ ¯ i ϕ i = T ¯ ϕ i .
r TE - TE = | E r TE E i TE | E i TM = 0 = T 43 T 11 - T 41 T 13 T 33 T 11 - T 13 T 31 , r TE - TM = | E r TM E i TE | E i TM = 0 = T 23 T 11 - T 21 T 13 T 33 T 11 - T 13 T 31 , r TM - TM = | E r TM E i TM | E i TE = 0 = T 33 T 21 - T 31 T 23 T 33 T 11 - T 13 T 31 , r TM - TE = | E r TE E i TM | E i TE = 0 = T 33 T 41 - T 31 T 43 T 33 T 11 - T 13 T 31 , t TE - TE = | E t TE E i TE | E i TM = 0 = T 11 T 33 T 11 - T 13 T 31 , t TE - TM = | E t TM E i TE | E i TM = 0 = - T 13 T 33 T 11 - T 13 T 31 , t TM - TE = | E t TE E i TM | E i TE = 0 = - T 31 T 33 T 11 - T 13 T 31 , t TM - TM = | E t TM E i TM | E i TE = 0 = T 33 T 33 T 11 - T 13 T 31 ,
t TE = t TE - TE cos ( θ i ) + t TM - TE sin ( θ i ) , t TM = t TE - TM cos ( θ i ) + t TM - TM sin ( θ i ) , r TE = r TE - TE cos ( θ i ) + r TM - TE sin ( θ i ) , r TM = r TE - TM cos ( θ i ) + r TM - TM sin ( θ i ) .
n AR = [ n air ( n x + n y + n z 3 ) ] 1 / 2 = 1.290 ,
d AR = λ 0 4 n AR = 206 nm .
[ x p y p z p ] = N ¯ [ x y z ] .
N ¯ = [ cos ( ψ ) sin ( ψ ) 0 - sin ( ψ ) cos ( ψ ) 0 0 0 1 ] [ 1 0 0 0 cos ( θ ) sin ( θ ) 0 - sin ( θ ) cos ( θ ) ] × [ cos ( ϕ ) sin ( ϕ ) 0 - sin ( ϕ ) cos ( ϕ ) 0 0 0 1 ] .
N ¯ = [ N 11 N 12 N 13 N 21 N 22 N 23 N 31 N 32 N 33 ] ,
N 11 = cos ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) sin ( ψ ) , N 12 = cos ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) sin ( ψ ) , N 13 = sin ( ψ ) sin ( θ ) , N 21 = - sin ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) cos ( ψ ) , N 22 = - sin ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) cos ( ψ ) , N 23 = cos ( ψ ) sin ( θ ) ,             N 31 = sin ( θ ) sin ( ϕ ) , N 32 = - sin ( θ ) cos ( ϕ ) ,             N 33 = cos ( θ ) .
ψ = tan - 1 ( N 13 N 23 ) , θ = sin - 1 [ sin ( ψ ) N 13 + cos ( ψ ) N 23 ] , ϕ = tan - 1 ( - N 31 N 32 ) .
[ D x p D y p D z p ] = [ x p 0 0 0 y p 0 0 0 z p ] [ E x p E y p E z p ] ,
x p = 0 ɛ x p = 0 n x p 2 , y p = 0 ɛ y p = 0 n y p 2 , z p = 0 ɛ z p = 0 n z p 2 .
ɛ ¯ ɛ ¯ laboratory = N ¯ - 1 ɛ ¯ principal N ¯ .
ɛ x x = ɛ x p [ cos ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) sin ( ψ ) ] 2 + ɛ y p [ - sin ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) cos ( ψ ) ] 2 + ɛ z p [ sin ( θ ) sin ( ϕ ) ] 2 , ɛ x y = ɛ x p [ cos ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) sin ( ψ ) ] × [ cos ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) sin ( ψ ) ] + ɛ y p [ - sin ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) cos ( ψ ) ] × [ - sin ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) cos ( ψ ) ] + ɛ z p [ sin ( θ ) sin ( ϕ ) ] [ - sin ( θ ) cos ( ϕ ) ] , ɛ x z = ɛ x p [ cos ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) sin ( ψ ) ] × [ sin ( ψ ) sin ( θ ) ] + ɛ y p [ - sin ( ψ ) cos ( ϕ ) - cos ( θ ) sin ( ϕ ) cos ( ψ ) ] [ cos ( ψ ) sin ( θ ) ] + ɛ z p [ sin ( θ ) sin ( ϕ ) ] [ cos ( θ ) ] , ɛ y x = ɛ x y , ɛ y y = ɛ x p [ cos ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) sin ( ψ ) ] 2 + ɛ y p [ - sin ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) cos ( ψ ) ] 2 + ɛ z [ - sin ( θ ) cos ( ϕ ) ] 2 , ɛ y z = ɛ x p [ cos ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) sin ( ψ ) ] × [ sin ( ψ ) sin ( θ ) ] + ɛ y p [ - sin ( ψ ) sin ( ϕ ) + cos ( θ ) cos ( ϕ ) cos ( ψ ) ] [ cos ( ψ ) sin ( θ ) ] + ɛ z p [ - sin ( θ ) cos ( ϕ ) ] [ cos ( θ ) ] , ɛ z x = ɛ x z , ɛ z y = ɛ y z , ɛ z z = ɛ x p [ sin ( ψ ) sin ( θ ) ] 2 + ɛ y p [ cos ( ψ ) sin ( θ ) ] 2 + ɛ z p [ cos ( θ ) ] 2 .

Metrics