Abstract

The formulas for the reflection and refraction of a narrow Gaussian beam with general astigmatism at a tilted optical surface are derived by ray-tracing techniques. The propagation direction of the reflected and refracted beams is computed by tracing the central ray of the incident beam, and the characteristic parameters of the respective wavefronts are worked out by applying the formulas developed for the generalized ray tracing. Moreover, the Gaussian form of the reflected and refracted amplitude distributions along the transverse coordinates is determined by requiring the matching of the incident, reflected, and refracted light spots on the optical surface. No limiting assumptions are made regarding the form of the optical interface or the orientation of the incident astigmatic wavefront. In the end, to illustrate a simple application of these formulas, the reflection of a Gaussian beam at a conicoid is considered, and a simple property of the conicoidal mirrors is reported.

© 2007 Optical Society of America

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References

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  1. G. A. Massey and A. E. Siegman, "Reflection and refraction of Gaussian light beams at tilted ellipsoidal surfaces," Appl. Opt. 8, 975-978 (1969).
    [CrossRef] [PubMed]
  2. S. Gangopadhyay and S. Sarkar, "ABCD matrix for reflection and refraction of Gaussian light beams at surfaces of hyperboloid of revolution and efficiency computation for laser diode to single-mode fiber coupling by way of a hyperboloic lens on the fiber tip," Appl. Opt. 36, 8582-8586 (1997).
    [CrossRef]
  3. H. Liu, L. Liu, R. Xu, and Z. Luan, "ABCD matrix for reflection and refraction of Gaussian beams at the surface of parabola of revolution," Appl. Opt. 44, 4809-4813 (2005).
    [CrossRef] [PubMed]
  4. G. A. Deschamps, "Ray techniques in electromagnetics," Proc. IEEE 60, 1022-1035 (1972).
    [CrossRef]
  5. A. Rohani, A. A. Shishegar, and S. Safavi-Naeini, "A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces," Opt. Commun. 232, 1-10 (2004).
    [CrossRef]
  6. J. A. Arnaud and H. Kogelnik, "Gaussian light beams with general astigmatism," Appl. Opt. 8, 1687-1693 (1969).
    [CrossRef] [PubMed]
  7. A. Gullstrand, "Die reelle optische Abbildung," K. Sven. Vetenskapsakad. Handl. 41, 1-119 (1906).
  8. J. A. Kneisly II, "Local curvature of wavefronts in an optical system," J. Opt. Soc. Am. 54, 229-235 (1964).
    [CrossRef]
  9. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), pp. 136-179.
  10. O. N. Stavroudis, "Simpler derivation of the formulas for generalized ray tracing," J. Opt. Soc. Am. 66, 1330-1333 (1976).
    [CrossRef]
  11. D. G. Burkhard and D. L. Shealy, "Simplified formula for the illuminance in an optical system," Appl. Opt. 20, 897-909 (1981).
    [CrossRef] [PubMed]
  12. D. G. Burkhard and D. L. Shealy, "A different approach to lighting and imaging: formulas for flux density, exact lens and mirror equations and caustic surfaces in terms of the differential geometry of surfaces," in Proc. SPIE 692, 248-272 (1986).
  13. S. Solimeno, B. Crosignani, and P. Di Porto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, 1986), Chap. 2, pp. 81-89.
  14. N. Lindlein and J. Schwider, "Local wave fronts at diffractive elements," J. Opt. Soc. Am. A 10, 2563-2572 (1993).
    [CrossRef]
  15. J. E. A. Landgrave and J. R. Moya-Cessa, "Generalized Coddington equations in ophthalmic lens design," J. Opt. Soc. Am. A 13, 1637-1644 (1996).
    [CrossRef]
  16. C. E. Campbell, "Generalized Coddington equations found via an operator method," J. Opt. Soc. Am. A 23, 1691-1698 (2006).
    [CrossRef]
  17. For example, M. J. Kidger, Fundamental Optical Design (SPIE, 2002), pp. 45-62.
  18. R. Kingslake, "Who discovered Coddington's equations?" Opt. Photon. News 5, 20-23 (1994).
    [CrossRef]
  19. W. T. Welford, Aberrations of Optical Systems (Hilger, 1986), pp. 185-191.
  20. R. Zeleny, "Post-objective mirrors allow for large fields," Laser Focus World, 89-93 (April 2003).
  21. G. N. Lawrence, "Optical modeling," in Applied Optics and Optical Engineering, R.R.Shannon and J.C.Wyant, eds. (Academic, 1992), Vol. 11, pp. 125-200.
  22. P. Hello and J. Vinet, "Simulation of beam propagation in off-axis optical systems," J. Opt. 27, 265-276 (1996).
    [CrossRef]
  23. Code V Version 9.6 Reference Manual (Optical Research Associates, 2005), Vol. 3, Chap. 19, pp. 193-231.
  24. For example, D. Malacara, "An optical surface and its characteristics," in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, 1992), pp. 743-753.
  25. For example, I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics (Springer, 2004), p. 245.
  26. For example: I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics (Ref. 25), pp. 205-206.
  27. O. N. Stavroudis, "Ray-tracing formulas for uniaxial crystals," J. Opt. Soc. Am. 52, 187-191 (1962).
    [CrossRef]

2006 (1)

2005 (1)

2004 (1)

A. Rohani, A. A. Shishegar, and S. Safavi-Naeini, "A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces," Opt. Commun. 232, 1-10 (2004).
[CrossRef]

2003 (1)

R. Zeleny, "Post-objective mirrors allow for large fields," Laser Focus World, 89-93 (April 2003).

1997 (1)

1996 (2)

P. Hello and J. Vinet, "Simulation of beam propagation in off-axis optical systems," J. Opt. 27, 265-276 (1996).
[CrossRef]

J. E. A. Landgrave and J. R. Moya-Cessa, "Generalized Coddington equations in ophthalmic lens design," J. Opt. Soc. Am. A 13, 1637-1644 (1996).
[CrossRef]

1994 (1)

R. Kingslake, "Who discovered Coddington's equations?" Opt. Photon. News 5, 20-23 (1994).
[CrossRef]

1993 (1)

1986 (1)

D. G. Burkhard and D. L. Shealy, "A different approach to lighting and imaging: formulas for flux density, exact lens and mirror equations and caustic surfaces in terms of the differential geometry of surfaces," in Proc. SPIE 692, 248-272 (1986).

1981 (1)

1976 (1)

1972 (1)

G. A. Deschamps, "Ray techniques in electromagnetics," Proc. IEEE 60, 1022-1035 (1972).
[CrossRef]

1969 (2)

1964 (1)

1962 (1)

1906 (1)

A. Gullstrand, "Die reelle optische Abbildung," K. Sven. Vetenskapsakad. Handl. 41, 1-119 (1906).

Arnaud, J. A.

Burkhard, D. G.

D. G. Burkhard and D. L. Shealy, "A different approach to lighting and imaging: formulas for flux density, exact lens and mirror equations and caustic surfaces in terms of the differential geometry of surfaces," in Proc. SPIE 692, 248-272 (1986).

D. G. Burkhard and D. L. Shealy, "Simplified formula for the illuminance in an optical system," Appl. Opt. 20, 897-909 (1981).
[CrossRef] [PubMed]

Campbell, C. E.

Crosignani, B.

S. Solimeno, B. Crosignani, and P. Di Porto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, 1986), Chap. 2, pp. 81-89.

Deschamps, G. A.

G. A. Deschamps, "Ray techniques in electromagnetics," Proc. IEEE 60, 1022-1035 (1972).
[CrossRef]

Di Porto, P.

S. Solimeno, B. Crosignani, and P. Di Porto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, 1986), Chap. 2, pp. 81-89.

Gangopadhyay, S.

Gullstrand, A.

A. Gullstrand, "Die reelle optische Abbildung," K. Sven. Vetenskapsakad. Handl. 41, 1-119 (1906).

Hello, P.

P. Hello and J. Vinet, "Simulation of beam propagation in off-axis optical systems," J. Opt. 27, 265-276 (1996).
[CrossRef]

Kingslake, R.

R. Kingslake, "Who discovered Coddington's equations?" Opt. Photon. News 5, 20-23 (1994).
[CrossRef]

Kneisly, J. A.

Kogelnik, H.

Landgrave, J. E. A.

Lawrence, G. N.

G. N. Lawrence, "Optical modeling," in Applied Optics and Optical Engineering, R.R.Shannon and J.C.Wyant, eds. (Academic, 1992), Vol. 11, pp. 125-200.

Lindlein, N.

Liu, H.

Liu, L.

Luan, Z.

Massey, G. A.

Moya-Cessa, J. R.

Rohani, A.

A. Rohani, A. A. Shishegar, and S. Safavi-Naeini, "A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces," Opt. Commun. 232, 1-10 (2004).
[CrossRef]

Safavi-Naeini, S.

A. Rohani, A. A. Shishegar, and S. Safavi-Naeini, "A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces," Opt. Commun. 232, 1-10 (2004).
[CrossRef]

Sarkar, S.

Schwider, J.

Shealy, D. L.

D. G. Burkhard and D. L. Shealy, "A different approach to lighting and imaging: formulas for flux density, exact lens and mirror equations and caustic surfaces in terms of the differential geometry of surfaces," in Proc. SPIE 692, 248-272 (1986).

D. G. Burkhard and D. L. Shealy, "Simplified formula for the illuminance in an optical system," Appl. Opt. 20, 897-909 (1981).
[CrossRef] [PubMed]

Shishegar, A. A.

A. Rohani, A. A. Shishegar, and S. Safavi-Naeini, "A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces," Opt. Commun. 232, 1-10 (2004).
[CrossRef]

Siegman, A. E.

Solimeno, S.

S. Solimeno, B. Crosignani, and P. Di Porto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, 1986), Chap. 2, pp. 81-89.

Stavroudis, O. N.

Vinet, J.

P. Hello and J. Vinet, "Simulation of beam propagation in off-axis optical systems," J. Opt. 27, 265-276 (1996).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Hilger, 1986), pp. 185-191.

Xu, R.

Zeleny, R.

R. Zeleny, "Post-objective mirrors allow for large fields," Laser Focus World, 89-93 (April 2003).

Appl. Opt. (5)

J. Opt. (1)

P. Hello and J. Vinet, "Simulation of beam propagation in off-axis optical systems," J. Opt. 27, 265-276 (1996).
[CrossRef]

J. Opt. Soc. Am. (3)

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

K. Sven. Vetenskapsakad. Handl. (1)

A. Gullstrand, "Die reelle optische Abbildung," K. Sven. Vetenskapsakad. Handl. 41, 1-119 (1906).

Opt. Commun. (1)

A. Rohani, A. A. Shishegar, and S. Safavi-Naeini, "A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces," Opt. Commun. 232, 1-10 (2004).
[CrossRef]

Opt. Photon. News (1)

R. Kingslake, "Who discovered Coddington's equations?" Opt. Photon. News 5, 20-23 (1994).
[CrossRef]

Proc. IEEE (1)

G. A. Deschamps, "Ray techniques in electromagnetics," Proc. IEEE 60, 1022-1035 (1972).
[CrossRef]

Proc. SPIE (1)

D. G. Burkhard and D. L. Shealy, "A different approach to lighting and imaging: formulas for flux density, exact lens and mirror equations and caustic surfaces in terms of the differential geometry of surfaces," in Proc. SPIE 692, 248-272 (1986).

Other (10)

S. Solimeno, B. Crosignani, and P. Di Porto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, 1986), Chap. 2, pp. 81-89.

For example, M. J. Kidger, Fundamental Optical Design (SPIE, 2002), pp. 45-62.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), pp. 136-179.

W. T. Welford, Aberrations of Optical Systems (Hilger, 1986), pp. 185-191.

R. Zeleny, "Post-objective mirrors allow for large fields," Laser Focus World, 89-93 (April 2003).

G. N. Lawrence, "Optical modeling," in Applied Optics and Optical Engineering, R.R.Shannon and J.C.Wyant, eds. (Academic, 1992), Vol. 11, pp. 125-200.

Code V Version 9.6 Reference Manual (Optical Research Associates, 2005), Vol. 3, Chap. 19, pp. 193-231.

For example, D. Malacara, "An optical surface and its characteristics," in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, 1992), pp. 743-753.

For example, I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics (Springer, 2004), p. 245.

For example: I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics (Ref. 25), pp. 205-206.

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Figures (2)

Fig. 1
Fig. 1

Parallel projection from plane xy to plane x ¯ y ¯ along unit vector N.

Fig. 2
Fig. 2

Coordinate systems associated with a conicoidal mirror and to the incident and reflected beams.

Equations (96)

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S ( X , Y ) = X i + Y j + Z ( X , Y ) k .
Z ( X , Y ) = { 1 ( 1 + K ) C { 1 [ 1 ( 1 + K ) C 2 ( X 2 + Y 2 ) ] 1 / 2 } for   K 1 C 2 ( X 2 + Y 2 ) for   K = 1 .
N ( X , Y ) = S X ( X , Y ) × S Y ( X , Y ) | S X ( X , Y ) × S Y ( X , Y ) | = Z X ( X , Y ) i Z Y ( X , Y ) j + k [ Z X 2 ( X , Y ) + Z Y 2 ( X , Y ) + 1 ] 1 / 2 .
T i ( A ) = i + η i j + [ Z X ( A ) + η i Z Y ( A ) ] k { 1 + Z X 2 ( A ) + 2 η i Z X ( A ) Z Y ( A ) + [ 1 + Z Y 2 ( A ) ] η i 2 } 1 / 2 ,
C i = Z X X ( A ) + η i Z X Y ( A ) [ 1 + Z X 2 ( A ) + Z Y 2 ( A ) ] 1 / 2 [ 1 + Z X 2 ( A ) + η i Z X ( A ) Z Y ( A ) ] ,
0 = { Z X ( A ) Z Y ( A ) Z Y Y ( A ) [ 1 + Z Y 2 ( A ) ] Z X Y ( A ) } η 2 + { [ 1 + Z X 2 ( A ) ] Z Y Y ( A ) [ 1 + Z Y 2 ( A ) ] Z X X ( A ) } η + [ 1 + Z X 2 ( A ) ] Z X Y ( A ) Z X ( A ) Z Y ( A ) Z X X ( A ) .
C P = C 1 cos 2 θ + C 2 sin 2 θ ,
C Q = C 1 sin 2 θ + C 2 cos 2 θ ,
C P Q = 1 2 ( C 1 C 2 ) sin ( 2 θ ) ,
θ = 1 2 tan 1 [ 2 C P Q C P C Q ] ,
C 1 = C P cos 2 θ + C P Q sin ( 2 θ ) + C Q sin 2 θ ,
C 2 = C P sin 2 θ C P Q sin ( 2 θ ) + C Q cos 2 θ .
ψ ( x , y , z ) = [ q 1 ( z ) q 2 ( z ) ] 1 / 2 exp { j 2 π n λ 0 [ Q x ( z ) 2 x 2 + Q y ( z ) 2 y 2 + Q x y ( z ) 2 x y + z ] }
Q x ( z ) = cos 2 φ q 1 ( z ) + sin 2 φ q 2 ( z ) ,
Q y ( z ) = sin 2 φ q 1 ( z ) + cos 2 φ q 2 ( z ) ,
Q x y ( z ) = sin ( 2 φ ) [ 1 q 2 ( z ) 1 q 1 ( z ) ] ,
q 1 ( z ) = q 1 0 + z , q 2 ( z ) = q 2 0 + z ,
π n λ 0 Q x I ( z ¯ ) x 2 + π n λ 0 Q y I ( z ¯ ) y 2 + π n λ 0 Q x y I ( z ¯ ) x y = 1 .
W ( x , y ) = x i + y j + z ( x , y ) k ,
z ( x , y ) = z ¯ 1 2 Q x R ( z ¯ ) x 2 1 2 Q y R ( z ¯ ) y 2 1 2 Q x y R ( z ¯ ) x y .
d = π n λ 0 Q x I ( z ¯ ) ,     e = π n λ 0 Q y I ( z ¯ ) ,     f = π n λ 0 Q x y I ( z ¯ ) ,
m 1 = D ,     a 1 2 = 2 E ,     a 2 2 = 2 F ,
D = e d + [ ( e d ) 2 + f 2 ] 1 / 2 f ,
E = e + d + [ ( e d ) 2 + f 2 ] 1 / 2 ,
F = e + d [ ( e d ) 2 + f 2 ] 1 / 2 ,
D = 0 , E = 2 d , F = 2 e .
d = 1 2 Q x R ( z ¯ ) ,     e = 1 2 Q y R ( z ¯ ) ,     f = 1 2 Q x y R ( z ¯ ) ,
M 1 = D ,     C 1 = E ,     C 2 = F ,
d = D 2 F + E 2 ( D 2 + 1 ) , e = D 2 E + F 2 ( D 2 + 1 ) , f = D ( E F ) D 2 + 1 .
φ = 1 2 tan 1 [ Q x y ( z ¯ ) Q y ( z ¯ ) Q x ( z ¯ ) ] , π 4 < φ R < π 4 ,
2 q 1 ( z ¯ ) = Q x ( z ¯ ) + Q y ( z ¯ ) + Q x ( z ¯ ) Q y ( z ¯ ) cos ( 2 φ ) ,
2 q 2 ( z ¯ ) = Q x ( z ¯ ) + Q y ( z ¯ ) Q x ( z ¯ ) Q y ( z ¯ ) cos ( 2 φ ) ,
φ = π 4 ,
2 q 1 ( z ¯ ) = 2 Q x ( z ¯ ) Q x y ( z ¯ ) ,
2 q 2 ( z ¯ ) = 2 Q x ( z ¯ ) + Q x y ( z ¯ ) ,
φ = 0 , 1 q 1 ( z ¯ ) = 1 q 2 ( z ¯ ) = Q x ( z ¯ ) .
x ¯ = x , y ¯ = y N N ¯ .
a x 2 + b y 2 + c x y = 1
a ¯ x ¯ 2 + b ¯ y ¯ 2 + c ¯ x ¯ y ¯ = 1 ,
a ¯ = a , b ¯ = ( N N ¯ ) 2 b , c ¯ = ( N N ¯ ) c .
N = μ N + γ N ¯ , N = N 2 ( N N ¯ ) N ¯ ,
μ = n n ,
γ = { 1 μ 2 [ 1 ( N N ¯ ) 2 ] } 1 / 2 μ ( N N ¯ ) .
P = N × N ¯ | N × N ¯ | .
P = N × N ¯ | N × N ¯ | = N × N ¯ | N × N ¯ | ,
Q = N × P , Q ¯ = N ¯ × P ,
    Q = N × P ,     Q = N × P .
C P = μ C P + γ C ¯ P ,
( N N ¯ ) C P Q = μ ( N N ¯ ) C P Q + γ C ¯ P Q ,
( N N ¯ ) 2 C Q = μ ( N N ¯ ) 2 C Q + γ C ¯ Q ,
C P = C P 2 ( N N ¯ ) C ¯ P ,
C P Q = C P Q + 2 C ¯ P Q ,
C Q = C Q 2 N N ¯ C ¯ Q .
π n λ 0 Q x I ( A ) x 2 + π n λ 0 Q y I ( A ) y 2 + π n λ 0 Q x y I ( A ) x y = 1 ,
π n λ 0 Q x I ( A ) x 2 + π n λ 0 Q y I ( A ) y 2 + π n λ 0 Q x y I ( A ) x y = 1 ,
π n λ 0 Q x I ( A ) x 2 + π n λ 0 Q y I ( A ) y 2 + π n λ 0 Q x y I ( A ) x y = 1 .
Q x I ( A ) = μ Q x I ( A ) ,
( N N ¯ ) 2 Q y I ( A ) = μ ( N N ¯ ) 2 Q y I ( A ) ,
( N N ¯ ) Q x y I ( A ) = μ ( N N ¯ ) Q x y I ( A ) ,
Q x I ( A ) = Q x I ( A ) ,
Q y I ( A ) = Q y I ( A ) ,
Q x y I ( A ) = Q x y I ( A ) .
N ¯ ( A ) = C Y A j + [ 1 ( 1 + K ) C 2 Y A 2 ] 1 / 2 k [ 1 K C 2 Y A 2 ] 1 / 2 ,
T ¯ 1 ( A ) = i ,
C ¯ 1 ( A ) = C ( 1 K C 2 Y A 2 ) 1 / 2 ,
T ¯ 2 ( A ) = [ 1 ( 1 + K ) C 2 Y A 2 ] 1 / 2 j + C Y A k ( 1 K C 2 Y A 2 ) 1 / 2 ,
C ¯ 2 ( A ) = C ( 1 K C 2 Y A 2 ) 3 / 2 .
N ( A ) N ¯ ( A ) = C Y A N Y + [ 1 ( 1 + K ) C 2 Y A 2 ] 1 / 2 N Z [ 1 K C 2 Y A 2 ] 1 / 2 ,
N ( A ) = [ 1 ( K + 2 ) C 2 Y A 2 ] N Y + 2 [ 1 ( 1 + K ) C 2 Y A 2 ] 1 / 2 C Y A N Z 1 K C 2 Y A 2 j + [ 1 ( K + 2 ) C 2 Y A 2 ] N Z + 2 [ 1 ( 1 + K ) C 2 Y A 2 ] 1 / 2 C Y A N Y 1 K C 2 Y A 2 k .
P ( A ) = s i g n ( Γ ) i , Q ( A ) = s i g n ( Γ ) ( N Z j N Y k ) , Q ¯ ( A ) = s i g n ( Γ ) [ 1 ( 1 + K ) C 2 Y A 2 ] 1 / 2 j + C Y A k ( 1 K C 2 Y A 2 ) 1 / 2 , Q ( A ) = s i g n ( Γ ) { [ 1 ( K + 2 ) C 2 Y A 2 ] N Z + 2 [ 1 ( 1 + K ) C 2 Y A 2 ] 1 / 2 C Y A N Y 1 K C 2 Y A 2 j [ 1 ( K + 2 ) C 2 Y A 2 ] N Y + 2 [ 1 ( 1 + K ) C 2 Y A 2 ] 1 / 2 C Y A N Z 1 K C 2 Y A 2 k } ,
Γ = [ 1 ( 1 + K ) C 2 Y A 2 ] 1 / 2 N Y + C Y A N Z .
   C P = C 1 cos 2 Θ + C 2 sin 2 Θ ,
C Q = C 1 sin 2 Θ + C 2 cos 2 Θ ,
C P Q = 1 2 ( C 1 C 2 ) sin ( 2 Θ ) ,
C ¯ P = C ¯ 1 = C ( 1 K C 2 Y A 2 ) 1 / 2 ,
C ¯ Q = C ¯ 2 = C ( 1 K C 2 Y A 2 ) 3 / 2 ,
C ¯ P Q = 0 .
C P = C 1 cos 2 Θ + C 2 sin 2 Θ 2 C Ω 1 K C 2 Y A 2 ,
C P Q = 1 2 ( C 1 C 2 ) sin ( 2 Θ ) ,
C Q = C 1 sin 2 Θ + C 2 cos 2 Θ 2 C ( 1 K C 2 Y A 2 ) Ω ,
Ω = C Y A N Y + [ 1 ( 1 + K ) C 2 Y A 2 ] 1 / 2 N Z .
tan ( 2 Θ ) = ( C 1 C 2 ) sin ( 2 Θ ) ( C 1 C 2 ) cos ( 2 Θ ) 2 C 1 K C 2 Y A 2 [ Ω 1 Ω ] ,
C 1 = C P cos 2 Θ + C P Q sin ( 2 Θ ) + C Q sin 2 Θ ,
C 2 = C P sin 2 Θ C P Q sin ( 2 Θ ) + C Q cos 2 Θ .
Q x I = λ 0 π n [ cos 2 θ a 1 2 + sin 2 θ a 2 2 ] ,
Q y I = λ 0 π n [ sin 2 θ a 1 2 + cos 2 θ a 2 2 ] ,
Q x y I = λ 0 π n [ 1 a 1 2 1 a 2 2 ] sin ( 2 θ ) .
Q x I = λ 0 π n [ cos 2 θ a 1 2 + sin 2 θ a 2 2 ] ,
Q y I = λ 0 π n [ sin 2 θ a 1 2 + cos 2 θ a 2 2 ] ,
Q x y I = λ 0 π n [ 1 a 1 2 1 a 2 2 ] sin ( 2 θ ) .
tan θ = tan θ , a 1 = a 1 , a 2 = a 2 .
Θ = θ = 0 ,
C 1 = C 1 2 C Ω 1 K C 2 Y A 2 = C 1 2 C ¯ 1 N ( A ) N ¯ ( A ) ,
C 2 = C 2 2 C ( 1 K C 2 Y A 2 ) Ω = C 2 2 C ¯ 2 N ( A ) N ¯ ( A ) ,
a 1 = a 1 ,
a 2 = a 2 .

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