Abstract

The conservation equation for a monochromatic field with arbitrary polarization propagating in an inhomogeneous transparent medium is expressed in terms of amplitude and phase variables. The expressions obtained for linearly polarized fields are compared with the results obtained in the eikonal approximation. The electric field wave equation is written in terms of intensity and phase variables. The transport equations for the irradiance and the phase are shown to be particular cases of these derivations. The conservation equation arising from the second-order differential wave equation is shown to be equivalent to that obtained from Poynting’s theorem.

© 2003 Optical Society of America

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References

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  1. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, Vol. XXVIII, E. Wolf, ed. (Elsevier Science, Amsterdam, 1990), pp. 271–359.
  2. S. Mallick, “Common path interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 95–122.
  3. T. R. O’Meara, D. M. Pepper, J. O. White, “Applications of nonlinear optical phase conjugation,” in Optical Phase Conjugation, R. Fisher, ed. (Academic, New York, 1983), pp. 537–584.
  4. M. D. Iturbe Castillo, D. Sánchez de la Llave, R. Ramos Garcı́a, L. I. Olivos Pérez, L. A. González, M. Rodrı́guez Ortiz, “Real-time self-induced nonlinear optical Zernike-type filter in a bacteriorhodopsin film,” Opt. Eng. 40, 2367–2368 (2001).
    [CrossRef]
  5. A. E. Conrady, Applied Optics and Optical Design, Part II (Dover, New York, 1960), p. 614.
  6. A. Cornejo-Rodrı́guez, A. Cordero-Dávila, “Wavefront slope measurements in optical testing,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 311–337.
  7. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).
  8. A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 9, pp. 349–350.
  9. M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of the phase,” J. Opt. Soc. Am. 72, 1199–1209 (1982).
    [CrossRef]
  10. P. A. Magaña, F. S. Granados Agustı́n, A. Cornejo Rodrı́guez, “Medición de la fase o frente de onda con un banco nodal,” Rev. Mex. Fis. 46, Suppl. 2, 54–58 (2000).
  11. F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990).
    [CrossRef] [PubMed]
  12. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
    [CrossRef] [PubMed]
  13. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  14. M. Campos Garcı́a, “Prueba de Roddier: una revisión,” B.Sc. thesis (Universidad National Autónoma de México, Distrito Federal, México, 1995), pp. 48–49.
  15. M. Fernández Guasti, “El teorema de Poynting para campos complejos,” Rev. Mex. Fis. 47, 105–106 (2001).
  16. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), p. 298.
  17. D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
    [CrossRef]
  18. A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982), pp. 278–293.
  19. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), p. 7.
  20. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), pp. 32, 111.
  21. M. R. Teague, “Image formation in terms of the transport equation,” J. Opt. Soc. Am. A 2, 2019–2026 (1985).
    [CrossRef]
  22. H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. London Sect. A LXVI, 1129–1137 (1953).
    [CrossRef]
  23. M. A. van Dam, R. G. Lane, “Wave-front sensing from defocused images by use of wave-front slopes,” Appl. Opt. 41, 5497–5502 (2002).
    [CrossRef] [PubMed]
  24. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 626. (In this reference, the opposite sign convention is used in the phase spatial dependence).
  25. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
    [CrossRef]
  26. M. Fernández Guasti, A. Gil-Villegas, “Orthogonal functions invariant for the time-dependent harmonic oscillator,” Phys. Lett. A 292, 243–245 (2002).
    [CrossRef]
  27. M. Fernández Guasti, R. Diamant, A. Gil-Villegas, “Ermakov equation arising from electromagnetic fields propagating in 1D inhomogeneous media,” Rev. Mex. Fis. 46, 530–535 (2000).

2002 (2)

M. Fernández Guasti, A. Gil-Villegas, “Orthogonal functions invariant for the time-dependent harmonic oscillator,” Phys. Lett. A 292, 243–245 (2002).
[CrossRef]

M. A. van Dam, R. G. Lane, “Wave-front sensing from defocused images by use of wave-front slopes,” Appl. Opt. 41, 5497–5502 (2002).
[CrossRef] [PubMed]

2001 (2)

M. D. Iturbe Castillo, D. Sánchez de la Llave, R. Ramos Garcı́a, L. I. Olivos Pérez, L. A. González, M. Rodrı́guez Ortiz, “Real-time self-induced nonlinear optical Zernike-type filter in a bacteriorhodopsin film,” Opt. Eng. 40, 2367–2368 (2001).
[CrossRef]

M. Fernández Guasti, “El teorema de Poynting para campos complejos,” Rev. Mex. Fis. 47, 105–106 (2001).

2000 (2)

P. A. Magaña, F. S. Granados Agustı́n, A. Cornejo Rodrı́guez, “Medición de la fase o frente de onda con un banco nodal,” Rev. Mex. Fis. 46, Suppl. 2, 54–58 (2000).

M. Fernández Guasti, R. Diamant, A. Gil-Villegas, “Ermakov equation arising from electromagnetic fields propagating in 1D inhomogeneous media,” Rev. Mex. Fis. 46, 530–535 (2000).

1998 (1)

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

1996 (1)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

1990 (1)

1985 (1)

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

1983 (1)

1982 (1)

1953 (1)

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. London Sect. A LXVI, 1129–1137 (1953).
[CrossRef]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), pp. 32, 111.

Campos Garci´a, M.

M. Campos Garcı́a, “Prueba de Roddier: una revisión,” B.Sc. thesis (Universidad National Autónoma de México, Distrito Federal, México, 1995), pp. 48–49.

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design, Part II (Dover, New York, 1960), p. 614.

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Cordero-Dávila, A.

A. Cornejo-Rodrı́guez, A. Cordero-Dávila, “Wavefront slope measurements in optical testing,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 311–337.

Cornejo Rodri´guez, A.

P. A. Magaña, F. S. Granados Agustı́n, A. Cornejo Rodrı́guez, “Medición de la fase o frente de onda con un banco nodal,” Rev. Mex. Fis. 46, Suppl. 2, 54–58 (2000).

Cornejo-Rodri´guez, A.

A. Cornejo-Rodrı́guez, A. Cordero-Dávila, “Wavefront slope measurements in optical testing,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 311–337.

Cornejo-Rodriguez, A.

A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 9, pp. 349–350.

Diamant, R.

M. Fernández Guasti, R. Diamant, A. Gil-Villegas, “Ermakov equation arising from electromagnetic fields propagating in 1D inhomogeneous media,” Rev. Mex. Fis. 46, 530–535 (2000).

Fernández Guasti, M.

M. Fernández Guasti, A. Gil-Villegas, “Orthogonal functions invariant for the time-dependent harmonic oscillator,” Phys. Lett. A 292, 243–245 (2002).
[CrossRef]

M. Fernández Guasti, “El teorema de Poynting para campos complejos,” Rev. Mex. Fis. 47, 105–106 (2001).

M. Fernández Guasti, R. Diamant, A. Gil-Villegas, “Ermakov equation arising from electromagnetic fields propagating in 1D inhomogeneous media,” Rev. Mex. Fis. 46, 530–535 (2000).

Gil-Villegas, A.

M. Fernández Guasti, A. Gil-Villegas, “Orthogonal functions invariant for the time-dependent harmonic oscillator,” Phys. Lett. A 292, 243–245 (2002).
[CrossRef]

M. Fernández Guasti, R. Diamant, A. Gil-Villegas, “Ermakov equation arising from electromagnetic fields propagating in 1D inhomogeneous media,” Rev. Mex. Fis. 46, 530–535 (2000).

González, L. A.

M. D. Iturbe Castillo, D. Sánchez de la Llave, R. Ramos Garcı́a, L. I. Olivos Pérez, L. A. González, M. Rodrı́guez Ortiz, “Real-time self-induced nonlinear optical Zernike-type filter in a bacteriorhodopsin film,” Opt. Eng. 40, 2367–2368 (2001).
[CrossRef]

Granados Agusti´n, F. S.

P. A. Magaña, F. S. Granados Agustı́n, A. Cornejo Rodrı́guez, “Medición de la fase o frente de onda con un banco nodal,” Rev. Mex. Fis. 46, Suppl. 2, 54–58 (2000).

Green, H. S.

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. London Sect. A LXVI, 1129–1137 (1953).
[CrossRef]

Gureyev, T. E.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Iturbe Castillo, M. D.

M. D. Iturbe Castillo, D. Sánchez de la Llave, R. Ramos Garcı́a, L. I. Olivos Pérez, L. A. González, M. Rodrı́guez Ortiz, “Real-time self-induced nonlinear optical Zernike-type filter in a bacteriorhodopsin film,” Opt. Eng. 40, 2367–2368 (2001).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), p. 298.

Lane, R. G.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).

Magaña, P. A.

P. A. Magaña, F. S. Granados Agustı́n, A. Cornejo Rodrı́guez, “Medición de la fase o frente de onda con un banco nodal,” Rev. Mex. Fis. 46, Suppl. 2, 54–58 (2000).

Mallick, S.

S. Mallick, “Common path interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 95–122.

Marathay, A. S.

A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982), pp. 278–293.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), p. 7.

Nugent, K. A.

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

O’Meara, T. R.

T. R. O’Meara, D. M. Pepper, J. O. White, “Applications of nonlinear optical phase conjugation,” in Optical Phase Conjugation, R. Fisher, ed. (Academic, New York, 1983), pp. 537–584.

Olivos Pérez, L. I.

M. D. Iturbe Castillo, D. Sánchez de la Llave, R. Ramos Garcı́a, L. I. Olivos Pérez, L. A. González, M. Rodrı́guez Ortiz, “Real-time self-induced nonlinear optical Zernike-type filter in a bacteriorhodopsin film,” Opt. Eng. 40, 2367–2368 (2001).
[CrossRef]

Paganin, D.

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Pepper, D. M.

T. R. O’Meara, D. M. Pepper, J. O. White, “Applications of nonlinear optical phase conjugation,” in Optical Phase Conjugation, R. Fisher, ed. (Academic, New York, 1983), pp. 537–584.

Ramos Garci´a, R.

M. D. Iturbe Castillo, D. Sánchez de la Llave, R. Ramos Garcı́a, L. I. Olivos Pérez, L. A. González, M. Rodrı́guez Ortiz, “Real-time self-induced nonlinear optical Zernike-type filter in a bacteriorhodopsin film,” Opt. Eng. 40, 2367–2368 (2001).
[CrossRef]

Roddier, F.

Rodri´guez Ortiz, M.

M. D. Iturbe Castillo, D. Sánchez de la Llave, R. Ramos Garcı́a, L. I. Olivos Pérez, L. A. González, M. Rodrı́guez Ortiz, “Real-time self-induced nonlinear optical Zernike-type filter in a bacteriorhodopsin film,” Opt. Eng. 40, 2367–2368 (2001).
[CrossRef]

Sánchez de la Llave, D.

M. D. Iturbe Castillo, D. Sánchez de la Llave, R. Ramos Garcı́a, L. I. Olivos Pérez, L. A. González, M. Rodrı́guez Ortiz, “Real-time self-induced nonlinear optical Zernike-type filter in a bacteriorhodopsin film,” Opt. Eng. 40, 2367–2368 (2001).
[CrossRef]

Schwider, J.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, Vol. XXVIII, E. Wolf, ed. (Elsevier Science, Amsterdam, 1990), pp. 271–359.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 626. (In this reference, the opposite sign convention is used in the phase spatial dependence).

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Teague, M. R.

van Dam, M. A.

White, J. O.

T. R. O’Meara, D. M. Pepper, J. O. White, “Applications of nonlinear optical phase conjugation,” in Optical Phase Conjugation, R. Fisher, ed. (Academic, New York, 1983), pp. 537–584.

Wolf, E.

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. London Sect. A LXVI, 1129–1137 (1953).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), pp. 32, 111.

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

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

Opt. Commun. (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Opt. Eng. (1)

M. D. Iturbe Castillo, D. Sánchez de la Llave, R. Ramos Garcı́a, L. I. Olivos Pérez, L. A. González, M. Rodrı́guez Ortiz, “Real-time self-induced nonlinear optical Zernike-type filter in a bacteriorhodopsin film,” Opt. Eng. 40, 2367–2368 (2001).
[CrossRef]

Phys. Lett. A (1)

M. Fernández Guasti, A. Gil-Villegas, “Orthogonal functions invariant for the time-dependent harmonic oscillator,” Phys. Lett. A 292, 243–245 (2002).
[CrossRef]

Phys. Rev. Lett. (2)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

Proc. Phys. Soc. London Sect. A (1)

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. London Sect. A LXVI, 1129–1137 (1953).
[CrossRef]

Rev. Mex. Fis. (3)

P. A. Magaña, F. S. Granados Agustı́n, A. Cornejo Rodrı́guez, “Medición de la fase o frente de onda con un banco nodal,” Rev. Mex. Fis. 46, Suppl. 2, 54–58 (2000).

M. Fernández Guasti, R. Diamant, A. Gil-Villegas, “Ermakov equation arising from electromagnetic fields propagating in 1D inhomogeneous media,” Rev. Mex. Fis. 46, 530–535 (2000).

M. Fernández Guasti, “El teorema de Poynting para campos complejos,” Rev. Mex. Fis. 47, 105–106 (2001).

Other (13)

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), p. 298.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, Vol. XXVIII, E. Wolf, ed. (Elsevier Science, Amsterdam, 1990), pp. 271–359.

S. Mallick, “Common path interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 95–122.

T. R. O’Meara, D. M. Pepper, J. O. White, “Applications of nonlinear optical phase conjugation,” in Optical Phase Conjugation, R. Fisher, ed. (Academic, New York, 1983), pp. 537–584.

A. E. Conrady, Applied Optics and Optical Design, Part II (Dover, New York, 1960), p. 614.

A. Cornejo-Rodrı́guez, A. Cordero-Dávila, “Wavefront slope measurements in optical testing,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 311–337.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).

A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 9, pp. 349–350.

M. Campos Garcı́a, “Prueba de Roddier: una revisión,” B.Sc. thesis (Universidad National Autónoma de México, Distrito Federal, México, 1995), pp. 48–49.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 626. (In this reference, the opposite sign convention is used in the phase spatial dependence).

A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982), pp. 278–293.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), p. 7.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), pp. 32, 111.

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Equations (40)

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12(E×H*+E*×H)+t [12(B  H*+E  D*)]
=-12(J*E+JE*),
S=12(E×H*+E*×H),
E(r, t)=a˜(r)exp{i[ϕ(r)-ωt]}.
H=-1ωμ (a˜×ϕ+i×a˜)exp{i[ϕ(r)-ωt]},
S=-12ωμ [a˜×(a˜*×ϕ)-ia˜×(×a˜*)+c.c.],
S=1ωμ (a˜a˜*)ϕ+12ωμ [-(a˜ϕ)a˜*+ia˜×(×a˜*)+c.c.].
u=12(B  H*+E  D*)
1ωμ (a˜  a˜*)ϕ+12ωμ [-(a˜ϕ)a˜*
+ia˜×(×a˜*)+c.c.]=0.
a+iaϕ=-a ln ,
S=1μ (a  a) ϕω.
1μωa  aϕ=0.
S=Iμc W,
I2W+WI-IW ln μ=0.
S=0μ0ra02nˆ,
T(ITϕ)+zI ϕz=0,
ϕϕ=k2=TϕTϕ+(ϕ/z)2.
ϕz2TϕTϕ,
ϕz=k-12k TϕTϕ+ .
T(ITϕ)+k Iz+I kz=0.
IT2ϕ+TITϕ+k Iz=0.
2E-μ 2Et2=-(E ln )- ln μ(×E).
2a-(ϕϕ)a+μω2a=FE,
FE=-(a ln )- ln μ××a=-[a( ln μ)]+(a) ln μ+( ln μ)a.
a2a-aa(ϕϕ)+μω2aa=FEa.
am2am=12 2am2-141am2 am2am2,
12 2I-14m=131Im ImIm-I(ϕϕ)
+μrrk02I=FEa,
12 2I-141I II-I(ϕϕ)+k02I=0.
Iϕz2=12 2I-141I II-ITϕTϕ+k02I.
ϕzk01+12k02I212 IT2I-14 TITI-I2TϕTϕ.
a2ϕ+2(ϕ)a=-(a ln )ϕ+ ln μ×(a×ϕ).
1am (am2ϕ)=-(a ln μ) ϕrm+( ln μϕ)am,
(aaϕ)=-(a ln μ)ϕa+( ln μϕ)aa.
1μaaϕ=1μ (aaϕ)+ 1μϕaa;
1μaaϕ=0.
Q=aaϕz.
2az2-aϕz2+μ0ω2(z)a=0.
2az2-aQ(aa)2+μ0ω2(z)a=0,

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