Abstract

Several techniques are described for studying the propagation of off-axis polynomial Gaussian beams in media having straight axes and periodic z variations of the quadratic refraction and loss coefficients. For some periodic variations, exact analytical solutions of the paraxial equations are possible, and for sufficiently slow variations, WKB solutions can always be obtained. All results are expressed in conventional beam matrix form.

© 1985 Optical Society of America

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References

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  1. L. W. Casperson, J. L. Kirkwood, “Beam Propagation in Tapered Quadratic Index Media: Numerical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 256 (1985).
    [CrossRef]
  2. L. W. Casperson, “Beam Propagation in Tapered Quadratic Index Media: Analytical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 264 (1985).
    [CrossRef]
  3. G. Goubau, F. Schwering, “On the Guided Propagation of Electromagnetic Wave Beams,” IRE Trans. Antennas Propag. AP-9, 248 (1961).
    [CrossRef]
  4. J. R. Pierce, “Modes in Sequences of Lenses,” Proc. Natl. Acad. Sci. U.S.A. 47, 1808 (1961).
    [CrossRef] [PubMed]
  5. D. W. Berreman, “A Lens or Light Guide Using Convectively Distorted Thermal Gradients in Gases,” Bell Syst. Tech J. 43, 1469 (1964).
  6. E. A. J. Marcatili, “Modes in a Sequence of Thick Astigmatic Lens-Like Focusers,” Bell Syst. Tech. J. 43, 2887 (1964).
  7. P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a Light Beam of Gaussian Field Distribution in Continuous and Periodic Lens-Like Media,” Proc. IEEE 53, 129 (1965).
    [CrossRef]
  8. Y. Suematsu, “Light-Beam Waveguide Using Lens-Like Media with Periodic Hyperbolic Temperature Distribution,” Electron. Commun. Jpn. 49, 107 (1966).
  9. M. S. Sodha, A. K. Ghatak, D. P. S. Malik, “Electromagnetic Wave Propagation in Radially and Axially Nonuniform Media: Geometrical-Optics Approximation,” J. Opt. Soc. Am. 61, 1492 (1971).
    [CrossRef]
  10. M. D. Feit, D. E. Maiden, “Unstable Propagation of a Gaussian Laser Beam in a Plasma Waveguide,” Appl. Phys. Lett. 28, 331 (1976).
    [CrossRef]
  11. L. Ronchi, C. Garbarino, “Gaussian Beams in Periodic Lenslike Media,” Opt. Acta 29, 1171 (1982).
    [CrossRef]
  12. L. W. Casperson, “Beam Modes in Complex Lenslike Media and Resonators,” J. Opt. Soc. Am. 66, 1373 (1976).
    [CrossRef]
  13. H. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including Those With a Loss or Gain Variation,” Appl. Opt. 4, 1562 (1965).
    [CrossRef]
  14. F. B. Hildebrand, Advanced Calculus for Applications (Prentice-Hall, Princeton, N.J., 1965), p. 50.
  15. G. W. Hill, “Mean Motion of the Lunar Perigee,” Acta Math. 8, 1 (1886).
    [CrossRef]
  16. E. Mathieu, “Memoire sur le mouvement vibratoire d’une membrane de forme elliptique,” J. Math. Pures Appl. 13, 137 (1868).
  17. N. W. McLachlan, Theory and Applications of Mathieu Functions (Oxford U. P., London, 1951), Part 2.
  18. J. L. Kirkwood, “Propagation of a Gaussian Beam in Tapered Optical Fibers and Concave Metallic Strip Waveguides,” M.S. Thesis, U. California, Los Angeles (1983).
  19. L. W. Casperson, “Solvable Hill Equation,” Phys. Rev. A 30, 2749 (1984); Phys. Rev. A 31, 2743 (1985).
    [CrossRef] [PubMed]
  20. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 67.
  21. I. S. Gradshteyn, I. W. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1972), p. 156.
  22. S. Yamamoto, T. Makimoto, “Equivalence Relations in a Class of Distributed Optical Systems—Lenslike Media,” Appl. Opt. 10, 1160 (1971).
    [CrossRef] [PubMed]

1985 (2)

L. W. Casperson, J. L. Kirkwood, “Beam Propagation in Tapered Quadratic Index Media: Numerical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 256 (1985).
[CrossRef]

L. W. Casperson, “Beam Propagation in Tapered Quadratic Index Media: Analytical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 264 (1985).
[CrossRef]

1984 (1)

L. W. Casperson, “Solvable Hill Equation,” Phys. Rev. A 30, 2749 (1984); Phys. Rev. A 31, 2743 (1985).
[CrossRef] [PubMed]

1982 (1)

L. Ronchi, C. Garbarino, “Gaussian Beams in Periodic Lenslike Media,” Opt. Acta 29, 1171 (1982).
[CrossRef]

1976 (2)

L. W. Casperson, “Beam Modes in Complex Lenslike Media and Resonators,” J. Opt. Soc. Am. 66, 1373 (1976).
[CrossRef]

M. D. Feit, D. E. Maiden, “Unstable Propagation of a Gaussian Laser Beam in a Plasma Waveguide,” Appl. Phys. Lett. 28, 331 (1976).
[CrossRef]

1971 (2)

1966 (1)

Y. Suematsu, “Light-Beam Waveguide Using Lens-Like Media with Periodic Hyperbolic Temperature Distribution,” Electron. Commun. Jpn. 49, 107 (1966).

1965 (2)

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a Light Beam of Gaussian Field Distribution in Continuous and Periodic Lens-Like Media,” Proc. IEEE 53, 129 (1965).
[CrossRef]

H. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including Those With a Loss or Gain Variation,” Appl. Opt. 4, 1562 (1965).
[CrossRef]

1964 (2)

D. W. Berreman, “A Lens or Light Guide Using Convectively Distorted Thermal Gradients in Gases,” Bell Syst. Tech J. 43, 1469 (1964).

E. A. J. Marcatili, “Modes in a Sequence of Thick Astigmatic Lens-Like Focusers,” Bell Syst. Tech. J. 43, 2887 (1964).

1961 (2)

G. Goubau, F. Schwering, “On the Guided Propagation of Electromagnetic Wave Beams,” IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

J. R. Pierce, “Modes in Sequences of Lenses,” Proc. Natl. Acad. Sci. U.S.A. 47, 1808 (1961).
[CrossRef] [PubMed]

1886 (1)

G. W. Hill, “Mean Motion of the Lunar Perigee,” Acta Math. 8, 1 (1886).
[CrossRef]

1868 (1)

E. Mathieu, “Memoire sur le mouvement vibratoire d’une membrane de forme elliptique,” J. Math. Pures Appl. 13, 137 (1868).

Berreman, D. W.

D. W. Berreman, “A Lens or Light Guide Using Convectively Distorted Thermal Gradients in Gases,” Bell Syst. Tech J. 43, 1469 (1964).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 67.

Casperson, L. W.

L. W. Casperson, J. L. Kirkwood, “Beam Propagation in Tapered Quadratic Index Media: Numerical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 256 (1985).
[CrossRef]

L. W. Casperson, “Beam Propagation in Tapered Quadratic Index Media: Analytical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 264 (1985).
[CrossRef]

L. W. Casperson, “Solvable Hill Equation,” Phys. Rev. A 30, 2749 (1984); Phys. Rev. A 31, 2743 (1985).
[CrossRef] [PubMed]

L. W. Casperson, “Beam Modes in Complex Lenslike Media and Resonators,” J. Opt. Soc. Am. 66, 1373 (1976).
[CrossRef]

Feit, M. D.

M. D. Feit, D. E. Maiden, “Unstable Propagation of a Gaussian Laser Beam in a Plasma Waveguide,” Appl. Phys. Lett. 28, 331 (1976).
[CrossRef]

Garbarino, C.

L. Ronchi, C. Garbarino, “Gaussian Beams in Periodic Lenslike Media,” Opt. Acta 29, 1171 (1982).
[CrossRef]

Ghatak, A. K.

Gordon, J. P.

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a Light Beam of Gaussian Field Distribution in Continuous and Periodic Lens-Like Media,” Proc. IEEE 53, 129 (1965).
[CrossRef]

Goubau, G.

G. Goubau, F. Schwering, “On the Guided Propagation of Electromagnetic Wave Beams,” IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. W. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1972), p. 156.

Hildebrand, F. B.

F. B. Hildebrand, Advanced Calculus for Applications (Prentice-Hall, Princeton, N.J., 1965), p. 50.

Hill, G. W.

G. W. Hill, “Mean Motion of the Lunar Perigee,” Acta Math. 8, 1 (1886).
[CrossRef]

Kirkwood, J. L.

L. W. Casperson, J. L. Kirkwood, “Beam Propagation in Tapered Quadratic Index Media: Numerical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 256 (1985).
[CrossRef]

J. L. Kirkwood, “Propagation of a Gaussian Beam in Tapered Optical Fibers and Concave Metallic Strip Waveguides,” M.S. Thesis, U. California, Los Angeles (1983).

Kogelnik, H.

Maiden, D. E.

M. D. Feit, D. E. Maiden, “Unstable Propagation of a Gaussian Laser Beam in a Plasma Waveguide,” Appl. Phys. Lett. 28, 331 (1976).
[CrossRef]

Makimoto, T.

Malik, D. P. S.

Marcatili, E. A. J.

E. A. J. Marcatili, “Modes in a Sequence of Thick Astigmatic Lens-Like Focusers,” Bell Syst. Tech. J. 43, 2887 (1964).

Mathieu, E.

E. Mathieu, “Memoire sur le mouvement vibratoire d’une membrane de forme elliptique,” J. Math. Pures Appl. 13, 137 (1868).

McLachlan, N. W.

N. W. McLachlan, Theory and Applications of Mathieu Functions (Oxford U. P., London, 1951), Part 2.

Pierce, J. R.

J. R. Pierce, “Modes in Sequences of Lenses,” Proc. Natl. Acad. Sci. U.S.A. 47, 1808 (1961).
[CrossRef] [PubMed]

Ronchi, L.

L. Ronchi, C. Garbarino, “Gaussian Beams in Periodic Lenslike Media,” Opt. Acta 29, 1171 (1982).
[CrossRef]

Ryzhik, I. W.

I. S. Gradshteyn, I. W. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1972), p. 156.

Schwering, F.

G. Goubau, F. Schwering, “On the Guided Propagation of Electromagnetic Wave Beams,” IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

Sodha, M. S.

Suematsu, Y.

Y. Suematsu, “Light-Beam Waveguide Using Lens-Like Media with Periodic Hyperbolic Temperature Distribution,” Electron. Commun. Jpn. 49, 107 (1966).

Tien, P. K.

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a Light Beam of Gaussian Field Distribution in Continuous and Periodic Lens-Like Media,” Proc. IEEE 53, 129 (1965).
[CrossRef]

Whinnery, J. R.

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a Light Beam of Gaussian Field Distribution in Continuous and Periodic Lens-Like Media,” Proc. IEEE 53, 129 (1965).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 67.

Yamamoto, S.

Acta Math. (1)

G. W. Hill, “Mean Motion of the Lunar Perigee,” Acta Math. 8, 1 (1886).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

M. D. Feit, D. E. Maiden, “Unstable Propagation of a Gaussian Laser Beam in a Plasma Waveguide,” Appl. Phys. Lett. 28, 331 (1976).
[CrossRef]

Bell Syst. Tech J. (1)

D. W. Berreman, “A Lens or Light Guide Using Convectively Distorted Thermal Gradients in Gases,” Bell Syst. Tech J. 43, 1469 (1964).

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, “Modes in a Sequence of Thick Astigmatic Lens-Like Focusers,” Bell Syst. Tech. J. 43, 2887 (1964).

Electron. Commun. Jpn. (1)

Y. Suematsu, “Light-Beam Waveguide Using Lens-Like Media with Periodic Hyperbolic Temperature Distribution,” Electron. Commun. Jpn. 49, 107 (1966).

IEEE/OSA J. Lightwave Technol. (2)

L. W. Casperson, J. L. Kirkwood, “Beam Propagation in Tapered Quadratic Index Media: Numerical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 256 (1985).
[CrossRef]

L. W. Casperson, “Beam Propagation in Tapered Quadratic Index Media: Analytical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 264 (1985).
[CrossRef]

IRE Trans. Antennas Propag. (1)

G. Goubau, F. Schwering, “On the Guided Propagation of Electromagnetic Wave Beams,” IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

J. Math. Pures Appl. (1)

E. Mathieu, “Memoire sur le mouvement vibratoire d’une membrane de forme elliptique,” J. Math. Pures Appl. 13, 137 (1868).

J. Opt. Soc. Am. (2)

Opt. Acta (1)

L. Ronchi, C. Garbarino, “Gaussian Beams in Periodic Lenslike Media,” Opt. Acta 29, 1171 (1982).
[CrossRef]

Phys. Rev. A (1)

L. W. Casperson, “Solvable Hill Equation,” Phys. Rev. A 30, 2749 (1984); Phys. Rev. A 31, 2743 (1985).
[CrossRef] [PubMed]

Proc. IEEE (1)

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a Light Beam of Gaussian Field Distribution in Continuous and Periodic Lens-Like Media,” Proc. IEEE 53, 129 (1965).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

J. R. Pierce, “Modes in Sequences of Lenses,” Proc. Natl. Acad. Sci. U.S.A. 47, 1808 (1961).
[CrossRef] [PubMed]

Other (5)

F. B. Hildebrand, Advanced Calculus for Applications (Prentice-Hall, Princeton, N.J., 1965), p. 50.

N. W. McLachlan, Theory and Applications of Mathieu Functions (Oxford U. P., London, 1951), Part 2.

J. L. Kirkwood, “Propagation of a Gaussian Beam in Tapered Optical Fibers and Concave Metallic Strip Waveguides,” M.S. Thesis, U. California, Los Angeles (1983).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 67.

I. S. Gradshteyn, I. W. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1972), p. 156.

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Figures (3)

Fig. 1
Fig. 1

Schematic representation of (a) a continuous periodic lenslike medium and (b) a composite periodic medium consisting of simpler uniform and tapered segments. The radius changes are meant to suggest changes in the loss and refraction profiles.

Fig. 2
Fig. 2

Normalized beamwidth w(z)/ws as a function of distance along a periodically modulated quadratic-index waveguide with the modulation frequency (a) γ = 0.5γ0, (b) γ = 1.0γ0, and (c) γ = 1.5γ0 (after Ref. 18).

Fig. 3
Fig. 3

Displacement of the amplitude center of the beam away from the waveguide axis in micrometers as a function of distance along a periodically modulated quadratic-index waveguide with the modulation frequency (a) γ = 0.5γ0, (b) γ = 1.0γ0, and (c) γ = 1.5γ0 (after Ref. 18).

Equations (72)

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2 E ¯ + k 2 E ¯ = 0 ,
k 2 ( x , z ) = k 0 ( z ) [ k 0 ( z ) k 2 ( z ) x 2 ] ,
E x = A ( x , z ) exp [ i 0 z k 0 ( z ) d z ] .
2 A x 2 2 i k 0 A z i d k 0 d z A k 0 k 2 x 2 A = 0 ,
A ( x , z ) = exp [ i ( Q x 2 / 2 + S x + P ) ] .
Q 2 + k 0 d Q d z + k 0 k 2 = 0 ,
Q S + k 0 d S d z = 0 ,
d P d z = i Q 2 k 0 S 2 2 k 0 .
Q k 0 = 1 R 2 i k 0 w 2 .
d a = S i / Q i ,
d p = S r / Q r ,
Q = k 0 r d r d z .
d d z [ k 0 ( z ) d r d z ] + k 2 ( z ) r = 0 ,
d 2 r d z 2 + k 2 ( z ) k 0 r = 0 .
d 2 y d x 2 + f ( x ) y = 0 ,
d 2 y d x 2 + ( p 2 q cos 2 x ) y = 0 .
k 2 ( z ) k 0 = p 2 q cos 2 z ,
r ( z ) = a u ( z ) + b υ ( z ) ,
r ( z ) = a u ( z ) + b υ ( z ) .
[ r ( z ) r ( z ) ] = [ A ( z ) B ( z ) C ( z ) D ( z ) ] [ r 1 r 1 ] ,
A ( z ) = u ( z 1 ) υ ( z ) υ ( z 1 ) u ( z ) υ ( z 1 ) u ( z 1 ) u ( z 1 ) υ ( z 1 ) ,
B ( z ) = υ ( z 1 ) u ( z ) u ( z 1 ) υ ( z ) υ ( z 1 ) u ( z 1 ) u ( z 1 ) υ ( z 1 ) ,
C ( z ) = u ( z 1 ) υ ( z ) υ ( z 1 u ( z ) υ ( z 1 ) u ( z 1 ) u ( z 1 ) υ ( z 1 ) ,
D ( z ) = υ ( z 1 ) u ( z ) u ( z 1 ) υ ( z ) υ ( z 1 ) u ( z 1 ) u ( z 1 ) υ ( z 1 ) .
Q ( z ) k 0 = C + D Q ( z 1 ) / k 0 A + B Q ( z 1 ) / k 0 .
d ln S d z = Q k 0 = C + D Q ( z 1 ) / k 0 A + B Q ( z 1 ) / k 0 ,
d ln S d z = d d z ln [ A + B Q ( z 1 ) k 0 ] .
S ( z ) = S ( z 1 ) [ A + B Q ( z 1 ) / k 0 ] 1 .
P ( z ) P ( z 1 ) = i 2 ln [ A + B Q ( z 1 ) k 0 ] S 2 ( z 1 ) 2 k 0 B A + B Q ( z 1 ) / k 0 .
d 2 r d z 2 + F ( 1 + 1 2 cos γ z ) r = 0
w s = [ λ π n 0 ( n 0 n 2 ) 1 / 2 ] 1 / 2 .
γ 0 = ( n 2 / n 0 ) 1 / 2 ,
d 2 r d z 2 + [ F ( 1 + G cos γ z ) 4 + γ 2 G cos γ z 1 + G cos γ z ] r = 0 ,
r ( z ) = a ( 1 + G cos γ z 1 + G ) cos ( F 1 / 2 γ ( 1 G 2 ) { G sin γ z 1 + G cos γ z 2 ( 1 G 2 ) 1 / 2 tan 1 [ ( 1 G 2 ) 1 / 2 1 + G tan ( γ z 2 ) ] } ) + b ( 1 + G cos γ z 1 + G ) sin ( F 1 / 2 γ ( 1 G 2 ) { G sin γ z 1 + G cos γ z 2 ( 1 G 2 ) 1 / 2 tan 1 [ ( 1 G 2 ) 1 / 2 1 + G tan ( γ z 2 ) ] } ) ,
u ( z ) = ( 1 + G cos γ z 1 + G ) cos ( F 1 / 2 γ ( 1 G 2 ) { G sin γ z 1 + G cos γ z 2 ( 1 G 2 ) 1 / 2 × tan 1 [ ( 1 G 2 ) 1 / 2 1 + G tan ( γ z 2 ) ] } ) ,
υ ( z ) = ( 1 + G cos γ z 1 + G ) sin ( F 1 / 2 γ ( 1 G 2 ) { G sin γ z 1 + G cos γ z 2 ( 1 G 2 ) 1 / 2 × tan 1 [ ( 1 G 2 ) 1 / 2 1 + G tan ( γ z 2 ) ] } ) .
u ( z ) = ( 1 + G cos γ z 1 + G ) cos [ F 1 / 2 0 z d z ( 1 + G cos γ z ) 2 ] .
υ ( z ) = ( 1 + G cos γ z 1 + G ) sin [ F 1 / 2 0 z d z ( 1 + G cos γ z ) 2 ] .
u ( z ) = γ G sin γ z 1 + G cos [ ] F 1 / 2 ( 1 + G ) ( 1 + G cos γ z ) 2 sin [ ] ,
υ ( z ) = γ G sin γ z 1 + G sin [ ] + F 1 / 2 ( 1 + G ) ( 1 + G cos γ z ) 2 cos [ ] ,
d 2 r d z 2 + [ F + ( γ 2 4 F ) G cos γ z ] r = 0 .
d 2 r d z 2 + [ F + 4 ( 1 F ) G cos 2 z ] r = 0 ,
( A B C D ) = [ cos ( k 2 b / k 0 ) 1 / 2 d b ( k 2 b / k 0 ) 1 / 2 sin ( k 2 b / k 0 ) 1 / 2 d b ( k 0 / k 2 b ) 1 / 2 sin ( k 2 b / k 0 ) 1 / 2 d b cos ( k 2 b / k 0 ) 1 / 2 d b ] × [ cos ( k 2 a / k 0 ) 1 / 2 d a ( k 2 a / k 0 ) 1 / 2 sin ( k 2 a / k 0 ) 1 / 2 d a ( k 0 / k 2 a ) 1 / 2 sin ( k 2 a / k 0 ) 1 / 2 d a cos ( k 2 a / k 0 ) 1 / 2 d a ] ,
( A B C D ) s = 1 sin ϑ { A sin ( s ϑ ) sin [ ( s 1 ) ϑ ] C sin ( s ϑ ) B sin ( s ϑ ) D sin ( s ϑ ) sin [ ( s 1 ) ϑ ] } ,
y ( x ) [ f ( x ) ] 1 / 4 { a cos 0 x [ f ( x ) ] 1 / 2 d x + b sin 0 x [ f ( x ) ] 1 / 2 d x } ,
A = [ k 2 ( z ) k 2 ( z 1 ) ] 1 / 4 cos z 1 z [ k 2 ( z ) k 0 ] 1 / 2 d z ,
B = [ k 2 ( z ) k 2 ( z 1 ) k 0 2 ] 1 / 4 sin z 1 z [ k 2 ( z ) k 0 ] 1 / 2 d z ,
C = [ k 2 ( z ) k 2 ( z 1 ) k 0 2 ] 1 / 4 sin z 1 z [ k 2 ( z ) k 0 ] 1 / 2 d z ,
D = [ k 2 ( z ) k 2 ( z 1 ) ] 1 / 4 cos z 1 z [ k 2 ( z ) k 0 ] 1 / 2 d z .
A = ( p 2 q cos 2 z p 2 q cos 2 z 1 ) 1 / 4 cos z 1 z ( p 2 q cos 2 z ) 1 / 2 d z ,
d d z [ k 0 ( z ) d r ( z ) d z ] + k 2 ( z ) r ( z ) = 0 .
d d z [ k 0 ( z ) d r ( z ) d z ] + k 2 ( z ) r ( z ) = 0 ,
k 0 ( z ) = k 0 [ z ( z ) ] d z d z ( z ) ,
k 2 ( z ) = k 2 [ z ( z ) ] d z ( z ) d z ,
r ( z ) = r [ z ( z ) ] ,
d d z = d z d z d d z .
d z d z = k 0 k 0 ( z ) .
z = k 0 0 z d z k 0 ( z ) .
k 2 ( z ) = k 2 ( z ) k 0 ( z ) / k 0 .
d 2 r ( z ) d z 2 + k 2 ( z ) k 0 r ( z ) = 0 .
det M = | A B C D | = AD BC = 1 .
d A d z = C ,
d B d z = D .
d C d z = k 2 ( z ) k 0 A ,
d D d z = k 2 ( z ) k 0 B .
d d z ( AD BC ) = D d A d z + A d D d z B d C d z C d B d z = DC k 2 ( z ) k 0 AB + k 2 ( z ) k 0 AB CD = 0 .
y ( z ) = a ( 1 + G cos γ z 1 + G ) cos ( F 1 / 2 γ ( 1 G 2 ) { G sin γ z 1 + G cos γ z 2 ( G 2 1 ) 1 / 2 × tanh 1 [ ( G 2 1 ) 1 / 2 1 + G tan ( γ z 2 ) ] } ) + b ( 1 + G cos γ z 1 + G ) × sin ( F 1 / 2 γ ( 1 G 2 ) { G sin γ z 1 + G cos γ z 2 ( G 2 1 ) 1 / 2 × tanh 1 [ ( G 2 1 ) 1 / 2 1 + G tan ( γ z 2 ) ] } ) .
y ( z ) = a ( 1 + cos γ z 2 ) cos { F 1 / 2 2 γ [ tan ( γ z 2 ) + 1 3 tan 3 ( γ z 2 ) ] } + b ( 1 + cos γ z 2 ) sin { F 1 / 2 2 γ [ tan ( γ z 2 ) + 1 3 tan 3 ( γ z 2 ) ] } .
y ( z ) = a ( 1 + G cos γ z 1 + G ) × cosh ( ( F ) 1 / 2 γ ( 1 G 2 ) { G sin γ z 1 + G cos γ z 2 ( 1 G 2 ) 1 / 2 × tan 1 [ ( 1 + G 2 ) 1 / 2 1 + G tan ( γ z 2 ) ] } ) + b ( 1 + G cos γ z 1 + G ) × sinh ( ( F ) 1 / 2 γ ( 1 G 2 ) { G sin γ z 1 + G cos γ z 2 ( 1 G 2 ) 1 / 2 × tan 1 [ ( 1 G 2 ) 1 / 2 1 + G tan ( γ z 2 ) ] } ) .
y ( z ) = a ( 1 + G cos γ z 1 + G ) + b ( 1 + G cos γ z 1 + G ) [ 1 γ ( 1 G 2 ) ] { G sin γ z 1 + G cos γ z 2 ( 1 G 2 ) 1 / 2 × tan 1 [ ( 1 G 2 ) 1 / 2 1 + G tan ( γ z 2 ) ] } ,
y ( z ) = a cos [ F 1 / 2 z ( 1 + G ) 2 ] + b sin [ F 1 / 2 z ( 1 + G ) 2 ] ,
y ( z ) = a cos ( F 1 / 2 z ) + b sin ( F 1 / 2 z ) ,

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