Abstract

The main relationships of wave optics are derived from a combination of the Huygens–Fresnel principle and the Feynman integral over all paths. The stationary-phase approximation of the wave relations gives the correspondent relations from the point of view of geometrical optics.

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References

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  1. “Leonhard Euler,” http://en.wikiquote.org/wiki/Leonhard_Euler .
  2. R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
    [CrossRef]
  3. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).
  4. R. J. Black and A. Ankiewicz, “Fiber-optic analogies with mechanics,” Am. J. Phys. 53, 554–563 (1985).
    [CrossRef]
  5. R. P. Feynman, QED: The Strange Theory of Light and Matter (Princeton University, 1985).
  6. G. Eichmann, “Quasi-geometric optics of media with inhomogeneous index of refraction,” J. Opt. Soc. Am. A 61, 161–168 (1971).
    [CrossRef]
  7. A. V. Gitin, “Using continuous Feynman integrals in the mathematical apparatus of geometrical and wave optics. A complex approach,” J. Opt. Technol. 64, 729–735 (1997).
  8. A. V. Gitin, “Mathematical fundamentals of modern linear optics,” Int. J. Antennas Propag. 2012, 273107 (2012).
    [CrossRef]
  9. “Huygens–Fresnel Principle,” http://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle .
  10. A. Walther, “Systematic approach to the teaching of lens theory,” Am. J. Phys. 35, 808–816 (1967).
    [CrossRef]
  11. A. Walther, “Lenses, wave optics, and eikonal functions,” J. Opt. Soc. Am. 59, 1325–1331 (1969).
    [CrossRef]
  12. A. Walther, The Ray and Wave Theory of Lenses (Cambridge University, 1995).
  13. I. L. Sakin, Engineering Optics (Mashinostroenie, 1976, in Russian).
  14. A. V. Gitin, “Radiometry as a section of optical system theory,” Opt. Spectrosc. 63, 106–109 (1987).
  15. A. V. Gitin, “The dirac bra-ket in radiometry of quasi-homogeneous sources,” Appl. Opt. 50, 6073–6083 (2011).
    [CrossRef]
  16. A. V. Gitin, “Radiometry of optical systems with quasi-homogeneous sources: a linear systems approach,” Optik 122, 1713–1718 (2011).
    [CrossRef]
  17. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  18. V. A. Zverev, Radiooptics (Soviet Radio, 1975, in Russian).
  19. O. N. Litvinenko, Fundaments of Radio Optics (Technics, 1974, in Russian).
  20. V. P. Maslov and M. V. Fedoryuk, Semi-Classical Approximation in Quantum Mechanics (D. Reidel, 1981).
  21. M. V. Fedoryuk, The Method of Steepest Descent (Nauka, 1977, in Russian).
  22. Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, 1990).
  23. Y. A. Kravtsov and Y. I. Orlov, “Boundaries of geometrical optics applicability and related problems,” Radio Sci. 16, 975–978 (1981).
    [CrossRef]
  24. C. R. Giuliano, “Applications of optical phase conjugation,” Phys. Today 34(4), 27–35 (1981).
    [CrossRef]
  25. A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
    [CrossRef]
  26. C. Morette, “On the definition and approximation of Feynman’s path integrals,” Phys. Rev. 81, 848–852 (1951).
    [CrossRef]
  27. A. V. Gitin, “Legendre transformations in Hamiltonian optics,” J. Eur. Opt. Soc. Rapid Publ. 5, 10022 (2010).
    [CrossRef]
  28. R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).
  29. M. Born and E. Wolf, Principles of Optics (Pergamon, 1965).
  30. V. Guillemin and S. Sternberg, Geometric Asymptotics (American Mathematical Society, 1977).

2012 (1)

A. V. Gitin, “Mathematical fundamentals of modern linear optics,” Int. J. Antennas Propag. 2012, 273107 (2012).
[CrossRef]

2011 (2)

A. V. Gitin, “The dirac bra-ket in radiometry of quasi-homogeneous sources,” Appl. Opt. 50, 6073–6083 (2011).
[CrossRef]

A. V. Gitin, “Radiometry of optical systems with quasi-homogeneous sources: a linear systems approach,” Optik 122, 1713–1718 (2011).
[CrossRef]

2010 (1)

A. V. Gitin, “Legendre transformations in Hamiltonian optics,” J. Eur. Opt. Soc. Rapid Publ. 5, 10022 (2010).
[CrossRef]

1997 (1)

A. V. Gitin, “Using continuous Feynman integrals in the mathematical apparatus of geometrical and wave optics. A complex approach,” J. Opt. Technol. 64, 729–735 (1997).

1987 (1)

A. V. Gitin, “Radiometry as a section of optical system theory,” Opt. Spectrosc. 63, 106–109 (1987).

1985 (1)

R. J. Black and A. Ankiewicz, “Fiber-optic analogies with mechanics,” Am. J. Phys. 53, 554–563 (1985).
[CrossRef]

1981 (2)

Y. A. Kravtsov and Y. I. Orlov, “Boundaries of geometrical optics applicability and related problems,” Radio Sci. 16, 975–978 (1981).
[CrossRef]

C. R. Giuliano, “Applications of optical phase conjugation,” Phys. Today 34(4), 27–35 (1981).
[CrossRef]

1978 (1)

A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[CrossRef]

1971 (1)

G. Eichmann, “Quasi-geometric optics of media with inhomogeneous index of refraction,” J. Opt. Soc. Am. A 61, 161–168 (1971).
[CrossRef]

1969 (1)

1967 (1)

A. Walther, “Systematic approach to the teaching of lens theory,” Am. J. Phys. 35, 808–816 (1967).
[CrossRef]

1951 (1)

C. Morette, “On the definition and approximation of Feynman’s path integrals,” Phys. Rev. 81, 848–852 (1951).
[CrossRef]

1948 (1)

R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
[CrossRef]

Ankiewicz, A.

R. J. Black and A. Ankiewicz, “Fiber-optic analogies with mechanics,” Am. J. Phys. 53, 554–563 (1985).
[CrossRef]

Black, R. J.

R. J. Black and A. Ankiewicz, “Fiber-optic analogies with mechanics,” Am. J. Phys. 53, 554–563 (1985).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1965).

Eichmann, G.

G. Eichmann, “Quasi-geometric optics of media with inhomogeneous index of refraction,” J. Opt. Soc. Am. A 61, 161–168 (1971).
[CrossRef]

Fedoryuk, M. V.

M. V. Fedoryuk, The Method of Steepest Descent (Nauka, 1977, in Russian).

V. P. Maslov and M. V. Fedoryuk, Semi-Classical Approximation in Quantum Mechanics (D. Reidel, 1981).

Feynman, R. P.

R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
[CrossRef]

R. P. Feynman, QED: The Strange Theory of Light and Matter (Princeton University, 1985).

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).

Gitin, A. V.

A. V. Gitin, “Mathematical fundamentals of modern linear optics,” Int. J. Antennas Propag. 2012, 273107 (2012).
[CrossRef]

A. V. Gitin, “The dirac bra-ket in radiometry of quasi-homogeneous sources,” Appl. Opt. 50, 6073–6083 (2011).
[CrossRef]

A. V. Gitin, “Radiometry of optical systems with quasi-homogeneous sources: a linear systems approach,” Optik 122, 1713–1718 (2011).
[CrossRef]

A. V. Gitin, “Legendre transformations in Hamiltonian optics,” J. Eur. Opt. Soc. Rapid Publ. 5, 10022 (2010).
[CrossRef]

A. V. Gitin, “Using continuous Feynman integrals in the mathematical apparatus of geometrical and wave optics. A complex approach,” J. Opt. Technol. 64, 729–735 (1997).

A. V. Gitin, “Radiometry as a section of optical system theory,” Opt. Spectrosc. 63, 106–109 (1987).

Giuliano, C. R.

C. R. Giuliano, “Applications of optical phase conjugation,” Phys. Today 34(4), 27–35 (1981).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Guillemin, V.

V. Guillemin and S. Sternberg, Geometric Asymptotics (American Mathematical Society, 1977).

Hibbs, A. R.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).

Kravtsov, Y. A.

Y. A. Kravtsov and Y. I. Orlov, “Boundaries of geometrical optics applicability and related problems,” Radio Sci. 16, 975–978 (1981).
[CrossRef]

Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, 1990).

Litvinenko, O. N.

O. N. Litvinenko, Fundaments of Radio Optics (Technics, 1974, in Russian).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).

Maslov, V. P.

V. P. Maslov and M. V. Fedoryuk, Semi-Classical Approximation in Quantum Mechanics (D. Reidel, 1981).

Morette, C.

C. Morette, “On the definition and approximation of Feynman’s path integrals,” Phys. Rev. 81, 848–852 (1951).
[CrossRef]

Orlov, Y. I.

Y. A. Kravtsov and Y. I. Orlov, “Boundaries of geometrical optics applicability and related problems,” Radio Sci. 16, 975–978 (1981).
[CrossRef]

Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, 1990).

Sakin, I. L.

I. L. Sakin, Engineering Optics (Mashinostroenie, 1976, in Russian).

Sternberg, S.

V. Guillemin and S. Sternberg, Geometric Asymptotics (American Mathematical Society, 1977).

Walther, A.

A. Walther, “Lenses, wave optics, and eikonal functions,” J. Opt. Soc. Am. 59, 1325–1331 (1969).
[CrossRef]

A. Walther, “Systematic approach to the teaching of lens theory,” Am. J. Phys. 35, 808–816 (1967).
[CrossRef]

A. Walther, The Ray and Wave Theory of Lenses (Cambridge University, 1995).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1965).

Yariv, A.

A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[CrossRef]

Zverev, V. A.

V. A. Zverev, Radiooptics (Soviet Radio, 1975, in Russian).

Am. J. Phys. (2)

R. J. Black and A. Ankiewicz, “Fiber-optic analogies with mechanics,” Am. J. Phys. 53, 554–563 (1985).
[CrossRef]

A. Walther, “Systematic approach to the teaching of lens theory,” Am. J. Phys. 35, 808–816 (1967).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[CrossRef]

Int. J. Antennas Propag. (1)

A. V. Gitin, “Mathematical fundamentals of modern linear optics,” Int. J. Antennas Propag. 2012, 273107 (2012).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ. (1)

A. V. Gitin, “Legendre transformations in Hamiltonian optics,” J. Eur. Opt. Soc. Rapid Publ. 5, 10022 (2010).
[CrossRef]

J. Opt. Soc. Am. (1)

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

G. Eichmann, “Quasi-geometric optics of media with inhomogeneous index of refraction,” J. Opt. Soc. Am. A 61, 161–168 (1971).
[CrossRef]

J. Opt. Technol. (1)

A. V. Gitin, “Using continuous Feynman integrals in the mathematical apparatus of geometrical and wave optics. A complex approach,” J. Opt. Technol. 64, 729–735 (1997).

Opt. Spectrosc. (1)

A. V. Gitin, “Radiometry as a section of optical system theory,” Opt. Spectrosc. 63, 106–109 (1987).

Optik (1)

A. V. Gitin, “Radiometry of optical systems with quasi-homogeneous sources: a linear systems approach,” Optik 122, 1713–1718 (2011).
[CrossRef]

Phys. Rev. (1)

C. Morette, “On the definition and approximation of Feynman’s path integrals,” Phys. Rev. 81, 848–852 (1951).
[CrossRef]

Phys. Today (1)

C. R. Giuliano, “Applications of optical phase conjugation,” Phys. Today 34(4), 27–35 (1981).
[CrossRef]

Radio Sci. (1)

Y. A. Kravtsov and Y. I. Orlov, “Boundaries of geometrical optics applicability and related problems,” Radio Sci. 16, 975–978 (1981).
[CrossRef]

Rev. Mod. Phys. (1)

R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
[CrossRef]

Other (15)

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).

R. P. Feynman, QED: The Strange Theory of Light and Matter (Princeton University, 1985).

“Huygens–Fresnel Principle,” http://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle .

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

V. A. Zverev, Radiooptics (Soviet Radio, 1975, in Russian).

O. N. Litvinenko, Fundaments of Radio Optics (Technics, 1974, in Russian).

V. P. Maslov and M. V. Fedoryuk, Semi-Classical Approximation in Quantum Mechanics (D. Reidel, 1981).

M. V. Fedoryuk, The Method of Steepest Descent (Nauka, 1977, in Russian).

Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, 1990).

“Leonhard Euler,” http://en.wikiquote.org/wiki/Leonhard_Euler .

A. Walther, The Ray and Wave Theory of Lenses (Cambridge University, 1995).

I. L. Sakin, Engineering Optics (Mashinostroenie, 1976, in Russian).

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1965).

V. Guillemin and S. Sternberg, Geometric Asymptotics (American Mathematical Society, 1977).

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Figures (9)

Fig. 1.
Fig. 1.

An optically homogeneous layer as a linear system.

Fig. 2.
Fig. 2.

The direction of the wave vector can be characterized by the angles (θ,φ) of the spherical coordinate system (φ(0,2π),θ(0,π/2)), or by the direction cosines (p,q).

Fig. 3.
Fig. 3.

An axis-symmetrical optical system {Rj,Zj,Dj}, j{0,1,2,,N+1}.

Fig. 4.
Fig. 4.

Physical light ray γu¯(z) and its Fresnel volume UF.

Fig. 5.
Fig. 5.

(a) A wave passes from a point source through an optical system in the forward direction. (b) The complex conjugate wave passes through this an optical system in the reverse direction and focuses at the point target located at the point source. (c) The system consists of the optical system a phase conjugate mirror. (d) The unfolded scheme of this system.

Fig. 6.
Fig. 6.

The optical system.

Fig. 7.
Fig. 7.

Squares. (a) The propagator, between their two relevant independent variables, is written on each side of the square while the arrows symbolize the k-Fourier transformations. (b) The eikonal, between their two relevant independent variables, is written on each side of the square while the arrows symbolize the Legendre transformations.

Fig. 8.
Fig. 8.

(a) Geometrical interpretation of the point, point-angle, angle-point, and angle eikonals: optical path length xx, xe, ex, and ee, respectively. (b) Wave interpretation of the angle eikonal.

Fig. 9.
Fig. 9.

Compound system.

Equations (152)

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

expik(xp+yq+ζn2p2q2),
Uout(xz,yz)=R2U˜in(p,q)·expik(xp+yq+Zn2p2q2)dpdq.
U˜(p,q)=Fxpyqk{U(x,y)}(k2πi)R2U(x,y)·exp[ik(x·p+y·q)]dxdy.
U(x,y)=Fkpxqy1{U˜(p,q)}(k2πi)R2U˜(p,q)·exp[ik(x·p+y·q)]dpdq,
Uin(x,y)=Fkpxqy1{U˜in(p,q)},ifζ=0;
Uout(xz,yz)=Fkpxzqyz1{U˜in(p,q)·expik·Z·n2p2q2},ifζ=Z.
U˜in(p,q)=Fxpyqk{Uin(x,y)}
U˜out(p,q)=Fxzpyzqk{Uout(xz,yz)}
U˜out(p,q)=U˜in(p,q)·K˜Z(p,q),
K˜Z(p,q)=expik·Z·n2p2q2
Uout(xz,yz)=R2Uin(x,y)·Kz(xzx,yzy)dxdy,
Kz(xzx,yzy)=Fkp(xzx)q(yzy)1{expik·Z·n2p2q2}
Kz(xzx,yzy)=12πddZ(exp[inkZ2+(xzx)2+(yzy)2]Z2+(xzx)2+(yzy)2)=(11inkZ2+(xzx)2+(yzy)2)×(nkZ2πi)exp[inkZ2+(xzx)2+(yzy)2]Z2+(xzx)2+(yzy)2.
Kz(xzx,yzy)(k2πi)nZZ2+(xzx)2+(yzy)2×exp[inkZ2+(xzx)2+(yzy)2].
xj=Rjsinθj·sinφj,
yj=Rjsinθj·cosφj,
zj=ZjRjcosθj,
U(x,y)=R2U(x,y)·KS(x,y;x,y)dxdy.
S0(x,y;θ1,φ1)=n·s0(x,y;θ1,φ1),
Sj(θj,φj;θj+1,φj+1)=nj,j+1·sj(θj,φj;θj+1,φj+1),
SN(θ1,φ1;x,y)=n·sN(θ1,φ1;x,y).
au(x,y;x,y)·exp{ik·Su(x,y;x,y)}.
Su(x,y;x,y)=S0(x,y;θ1,φ1)+j=1N1Sj(θj,φj;θj+1,φj+1)+SN(θN,φN;x,y)
KS(x,y;x,y)=R2Nau(x,y;x,y)exp{ikSu(x,y;x,y)}du.
dSu¯(x,y;x,y)Su(x,y;x,y)u|u=u¯=0.
KS(x,y;x,y)aS(x,y;x,y)·exp[ik·S(x,y;x,y)],
UF{u:|Su(x,y;x,y)S(x,y;x,y)|<λ2}.
KS1(x,y;x,y)=KS*(x,y;x,y)=aS(x,y;x,y)·exp{ikS(x,y;x,y)}.
U(x,y)=R2U(x,y)·KS(x,y;x,y)dxdy,
U^(x^,y^)=R2U(x,y)·KS*(x^,y^;x,y)dxdy.
U^(x^,y^)=R2[R2U(x,y)·KS(x,y;x,y)dxdy]·KS*(x^,y^;x,y)dxdy=R2U(x,y)·K(x,y;x^,y^)dxdy,
K(x,y;x^,y^)R2KS(x,y;x,y)KS*(x^,y^;x,y)dxdy
K(x,y;x^,y^)=δ(xx^;yy^).
R2KS(x,y;x,y)·KS*(x^,y^;x,y)dxdy=δ(xx^;yy^).
R2|a(x,y;x,y)|2·exp{ik[S(x,y;x,y)S(x^,y^;x,y)]}dxdy=δ(xx^;yy^).
S(x,y;x,y)S(x^,y^;x,y)Sx·(xx^)+Sy·(yy^).
R2|a(x,y;x,y)|2·exp{ik[Sx·(xx^)+Sy·(yy^)]}dxdy=δ(xx^;yy^).
pSx,
qSy,
R2|a(x,y;x,y)|2·exp{ik[p(xx^)+q(yy^)]}|det(2Sxx2Sxy2Syx2Syy)|1dpdq=δ(xx^;yy^),
|det(2Sxx2Sxy2Syx2Syy)|1
R2exp{ik[p(xx^)+q(yy^)]}d(kp)d(kq)=(2π)2·δ(xx^;yy^).
a(x,y;x,y)=c·k2π|det(2Sxx2Sxy2Syx2Syy)|1/2,
S(x,y;x,y)=nZ2+(xx)2+(yy)2,
a(x,y;x,y)=c·k2π|det(nZ2+(yy)2(Z2+(xx)2+(yy)2)3/2n(xx)(yy)(Z2+(xx)2+(yy)2)3/2n(xx)(yy)(Z2+(xx)2+(yy)2)3/2nZ2+(xx)2(Z2+(xx)2+(yy)2)3/2)|1/2=c·k2π|nZZ2+(xx)2+(yy)2|.
c=i.
KS(x,y;x,y)=k2πi|det(2Sxx2Sxy2Syx2Syy)|1/2·exp{ikS(x,y;x,y)}.
U(x,y)=R2KS(x,y;xy)·U(x,y)dxdy,
U˜(p,q)=R2KV(x,y;pq)·U(x,y)dxdy,
U(x,y)=R2KV(p,q;xy)·U˜(p,q)dpdq,
U˜(p,q)=R2KT(p,q;pq)·U˜(p,q)dpdq.
KV(x,y;p,q)=Fxpyqk{KS(x,y;xy)},
KV(p,q;x,y)=Fkxpyq1{KS(x,y;xy)},
KT(p,q;p,p)=Fkxpyq1Fxpyqk{KS(x,y;xy)}.
KV(x,y;p,q)=Fxpyqk{KS(x,y;xy)}(k2πi)2·R2|det(2Sxx2Sxy2Syx2Syy)|1/2·exp{ik[S(x,y;x,y)xpyq]}dxdy.
[S(x,y;x,y)xpyp]x|x=x¯=0p=Sx,
[S(x,y;x,y)xpyp]y|y=y¯=0q=Sy,
KV(x,y;p,q)k2πi|det(2Sxx2Sxy2Syx2Syy)det(2(Spxqy)xx2(Spxqy)xy2(Spxqy)yx2(Spxqy)yy)|x=x¯y=y¯1/2×exp{ikV(x,y;p,q)}.
V(x,y;p,q)=S(x,y;x,y)pxqy,
pSx,qSy
S(x,y;x,y)=V(x,y;p,q)+px+qy,
xVp,yVq.
(xy)(Spxqy)=0.
[(δx,δy)(xy)+(δx,δy)(xy)](xy)(Spxqy)=0.
(δx,δy)(2Sxx2Sxy2Syx2Syy)+(δx,δy)(2(Spxqy)xx2(Spxqy)xy2(Spxqy)yx2(Spxqy)yy)=0.
(x,y)(x,y)=|det(2Sxx2Sxy2Syx2Syy)det(2(Spxqy)xx2(Spxqy)xy2(Spxqy)yx2(Spxqy)yy)|,
det(2Vxp2Vxp2Vyp2Vyp)=det(xxyyxxyy)=(x,y)(x,y).
|det(2Sxx2Sxy2Syx2Syy)det(2(Spxqy)xx2(Spxqy)xy2(Spxqy)yx2(Spxqy)yy)|=|det(2Vxp2Vxp2Vyp2Vyp)|.
KV(x,y;p,q)k2πi|det(2Vxp2Vxq2Vyp2Vyq)|1/2·exp{ik·V(x,y;p,q)}.
KV(p,q;x,y)k2πi|det(2Vpx2Vpy2Vqx2Vqy)|1/2·exp{ik·V(p,q;x,y)},
V(p,q;x,y)=S(x,y;x,y)+px+qy,
KT(p,q;p,q)k2πi|det(2Tpp2Tpq2Tqp2Tqq)|1/2·exp{ik·T(p,q;p,q)},
T(p,q;p,p)=S(x,y;x,y)+px+qypxqy,
V(x;p)=S(x;x)px,wherepSx,
V(p;x)=S(x;x)+px,wherepSx,
T(p;p)=S(x;x)+xpxp,wherepSx,pSx,
U(x,y)=R2U(x,y)KS1(x,y;xy)dxdy,
U(x,y)=R2U(x,y)KS2(x,y;x,y)dxdy.
U(x,y)=R2[R2U(x,y)KS1(x,y;xy)dxdy]KS2(x,y;x,y)dxdy=R2U(x,y)KS12(x,y;x,y)dxdy,
KS12(x,y;x,y)=R2KS1(x,y;x,y)KS2(x,y;x,y)dxdy
KS12(x,y;x,y)=(k2πi)2R2|det(2S1xx2S1xy2S1yx2S1yy)·(2S2xx2S2xy2S2yx2S2yy)|××exp{ik[S1(x,y;x,y)+S2(x,y;x,y)]}dxdy.
[S1(x,y;x,y)+S2(x,y;x,y)]x|x=x¯=0,[S1(x,y;x,y)+S2(x,y;x,y)]y|y=y¯=0.
KS12(x,y;x,y)k2πi|det(2S1xx2S1xy2S1yx2S1yy)·det(2S2xx2S2xy2S2yx2S2yy)det(2(S1+S2)xx2(S1+S2)xy2(S1+S2)yx2(S1+S2)yy)||x=x¯y=y¯×exp{ik[S12(x,y;x,y)]},
S12(x,y;x,y)S1(x,y;x¯,y¯)+S2(x¯,y¯;x,y).
(xx)(S1+S2)=0.
[(δx,δy)(xy)+(δx,δy)(xy)](xy)(S1+S2)=0.
(δx,δy)(2(S1+S2)xx2(S1+S2)xy2(S1+S2)yx2(S1+S2)yy)+(δx,δy)(2S2xx2S2xy2S2yx2S2yy)=0.
(x,y)(x,y)|x=x¯y=y¯=|det(2S2xx2S2xy2S2yx2S2yy)det(2(S1+S2)xx2(S1+S2)xy2(S1+S2)yx2(S1+S2)yy)|.
dpdq=|det(2S1xx2S1xy2S1yx2S1yy)|dxdy=|det(2Sxx2Sxy2Syx2Syy)|dxdy,
|det(2S1xx2S1xy2S1yx2S1yy)|(x,y)(x,y)|x=x¯y=y¯=|det(2Sxx2Sxy2Syx2Syy)|.
KS12(x,y;x,y)k2πi|det(2S12xx2S12xy2S12yx2S12yy)|1/2×exp{ik[S12(x,y;x,y)]},
U˜(p,q)=R2KV1(x,y;pq)·U(x,y)dxdy
U(x,y)=R2KV2(p,q;xy)·U˜(p,q)dpdq,
U˜(p,q)=R2[R2KV1(x,y;pq)·U(x,y)dxdy]KV2(p,q;xy)dpdq==R2KS12(x,y;xy)·U(x,y)dxdy.
KS12(x,y;x,y)=R2KV1(x,y;pq)KV2(p,q;x,y)dpdq.
KS12(x,y;x,y)=(k2πi)2R2|det(2V1xp2V1xq2V1yp2V1yq)·det(2V2px2V2py2V2qx2V2qy)|××exp{ik[V1(x,y;p,q)+V2(p,q;x,y)]}dpdq.
[V1(x,y;p,q)+V2(p,q;x,y)]p|p=p¯=0,[V1(x,y;p,q)+V2(p,q;x,y)]q|q=q¯=0,
KS12(x,y;x,y)k2πi|det(2V1xp2V1xq2V1yp2V1yq)·det(2V2px2V2py2V2qx2V2qy)det(2(V1+V2)pp2(V1+V2)pq2(V1+V2)qp2(V1+V2)qq)||p=p¯q=q¯×exp{ikS12(x,y;xy)},
S12(x,y;xy)V1(x,y;p¯,q¯)+V2(p¯,q¯;x,y).
(pq)(V1+V2)=0.
[(δp,δq)(pq)+(δx,δx)(xx)](pq)(V1+V2)=0.
(δp,δq)(2(V1+V2)pp2(V1+V2)pq2(V1+V2)qp2(V1+V2)qq)+(δx,δy)(2V2px2V2px2V2qx2V2qx)=0.
(p,q)(x,y)|p=p¯q=q¯=|det(2V2px2V2py2V2qx2V2qy)det(2(V1+V2)pp2(V1+V2)pq2(V1+V2)qp2(V1+V2)qq)|,
dpdq=|det(2V1xp2V1xp2V1yp2V1yp)|dpdq=|det(2S12xx2S12xy2S12xy2S12xy)|dxdy,
|det(2V1xp2V1xq2V1yp2V1yq)|(p,q)(p,q)|=|det(2T12pp2T12pq2T12qp2T12qq)|.
|det(2V1xp2V1xq2V1yp2V1yq)·det(2V2px2V2py2V2qx2V2qy)det(2(V1+V2)pp2(V1+V2)pq2(V1+V2)qp2(V1+V2)qq)||p=p¯q=q¯=|det(2S12xy2S12xy2S12xy2S12xy)|.
U˜(p,q)=R2KT1(p,q;pq)·U˜(p,q)dpdq,
U˜(p,q)=R2KT2(p,q;pq)·U˜(p,q)dpdq.
U˜(p,q)=R2[R2KT1(p,q;pq)·U˜(p,q)dpdq]KT2(p,q;pq)·U˜(p,q)dpdq==R2KT12(p,q;pq)·U˜(p,q)dpdq,
KT12(p,q;pq)=R2KT1(p,q;pq)·KT2(p,q;pq)dpdq.
KT12(p,q;pq)=(k2πi)2R2|det(2T1pp2T1pq2T1qp2T1qq)·det(2T2pp2T2pq2T2qp2T2qq)|×.×exp{ik[T1(p,q;p,q)+T2(p,q;p,q)]}dpdq.
[T1(p,q;p,q)+T2(p,q;p,q)]p|p=p¯=0,[T1(p,q;p,q)+T2(p,q;p,q)]q|q=q¯=0,
KT12(p,q;pq)k2πi|det(2T1pp2T1pq2T1qp2T1qq)·det(2T2pp2T2pq2T2qp2T2qq)det(2(T1+T2)pp2(T1+T2)pq2(T1+T2)qp2(S1+S2)qq)||p=p¯q=q¯×exp{ik[T12(p,q;p,q)]},
T12(p,q;p,q)T1(p,q;p¯,q¯)+T2(p¯,q¯;p,q).
(pq)(T1+T2)=0.
[(δp,δq)(pq)+(δp,δq)(pq)](pq)(T1+T2)=0.
(δp,δq)(2(T1+T2)pp2(T1+T2)pq2(T1+T2)qp2(T1+T2)qq)+(δp,δq)(2T2pp2T2pq2T2qp2T2qq)=0.
(p,q)(p,q)|p=p¯q=q¯=|det(2T2pp2T2pq2T2qp2T2qq)det(2(T1+T2)pp2(T1+T2)pq2(T1+T2)qp2(T1+T2)qq)|,
dxdy=|det(2T1pp2T1pq2T1qp2T1qq)|dpdq=|det(2T12pp2T12pq2T12qp2T12qq)|dpdq,
|det(2T1pp2T1pq2T1qp2T1qq)|(p,q)(p,q)|=|det(2T12pp2T12pq2T12qp2T12qq)|.
|det(2T1pp2T1pq2T1qp2T1qq)·det(2T2pp2T2pq2T2qp2T2qq)det(2(T1+T2)pp2(T1+T2)pq2(T1+T2)qp2(T1+T2)qq)||p=p¯q=q¯=|det(2T12pp2T12pq2T12qp2T12qq)|.
KT12(p,q;pq)k2πi|det(2T12pp2T12pq2T12qp2T12qq)|1/2·exp{ik[T12(p,q;p,q)]}.
U(x,y)=R2KV1(p,q;x,y)·U˜(p,q)dpdq,
U˜(p,q)=R2KV2(x,y;p,q)·U(x,y)dxdy.
U˜(p,q)=R2[R2KV1(p,q;x,y)·U˜(p,q)dpdq]KV2(x,y;p,q)dxdy=R2KT12(p,q;pq)·U˜(p,q)dpdq.
KT12(p,q;p,q)=R2KV1(p,q;x,y)KV2(x,y;p,q)dxdy.
KT12(p,q;p,q)=(k2πi)2R2|det(2V1px2V1py2V1qx2V1qy)·det(2V2xp2V2xq2V2yp2V2yq)|×exp{ik[V1(p,q;x,y)+V2(x,y;p,q)]}dxdy.
[V1(p,q;x,y)+V2(x,y;p,q)]x|x=x¯=0,[V1(p,q;x,y)+V2(x,y;p,q)]y|y=y¯=0,
KT12(p,q;p,q)=k2πi|det(2V1px2V1py2V1qx2V1qy)·det(2V2xp2V2xq2V2yp2V2yq)det(2(V1+V2)xx2(V1+V2)xy2(V1+V2)yx2(V1+V2)yy)||x=x¯y=y¯×exp{ikT12(p,q;p,q)},
T12(p,q;p,q)V1(p,q;x¯,y¯)+V2(x¯,y¯;p,q).
(xy)(V1+V2)=0.
[(δx,δy)(xy)+(δp,δq)(pq)](xy)(V1+V2)=0.
(δx,δy)(2(V1+V2)xx2(V1+V2)xy2(V1+V2)yx2(V1+V2)yy)+(δp,δq)(2V2xp2V2xq2V2yp2V2yq)=0.
(x,y)(p,q)|p=p¯q=q¯=|det(2V2xp2V2xq2V2yp2V2yq)det(2(V1+V2)xx2(V1+V2)xy2(V1+V2)yx2(V1+V2)yy)|,
dpdq=det(2V1px2V1py2V1qx2V1qy)dxdy=det(2T12pp2T12pq2T12qp2T12qq)dpdq,
|det(2V1px2V1py2V1qx2V1qy)|(x,y)(p,q)|=|det(2T12pp2T12pq2T12qp2T12qq)|.
|det(2V1px2V1py2V1qx2V1qy)·det(2V2xp2V2xq2V2yp2V2yq)det(2(V1+V2)xx2(V1+V2)xy2(V1+V2)yx2(V1+V2)yy)||x=x¯y=y¯=|det(2T12pp2T12pq2T12qp2T12qq)|.
I=R2g(x,y)exp{ikS(x,y)}dxdy,
S(x,y)x|x=x¯y=y¯=0andS(x,y)y|x=x¯y=y¯=0.
S(x,y)=S(x¯,y¯)+(xx¯yy¯)t·(2Sxx2Sxy2Syx2Syy)|x=x¯y=y¯·(xx¯yy¯),
(2Sxx2Sxy2Syx2Syy)|x=x¯y=y¯
det(2Sxx2Sxy2Syx2Syy)|x=x¯y=y¯0.
(vw)=(a11a12a21a22)(xx¯yy¯),
(a11a12a21a22)=(vxvywxwy)(x,y)(v,w),
((x,y)(v,w))t·(2Sxx2Sxy2Syx2Syy)|x=x¯y=y¯·((x,y)(v,w))=(1001).
Ig(x¯,y¯)·exp{ikS(x¯,y¯)}×R2exp{ik12(vw)t(1001)(vw)}|det(x,y)(v,w)|dvdw.
|det((x,y)(v,w))t||det·(2Sxx2Sxy2Syx2Syy)|x=x¯y=y¯|·|det((x,y)(v,w))|=1.
|det((x,y)(v,w))|=|det·(2Sxx2Sxy2Syx2Syy)|x=x¯y=y¯|1/2.
I|det·(2Sxx2Sxy2Syx2Syy)|x=x¯y=y¯|1/2·g(x¯,y¯)·exp{ikS(x¯,y¯)}××Rexp(ikv22)dv·Rexp(ikw22)dw.
Rexp(ikv22)dv=2πkexp[iπ4]
Rexp(ikw22)dw=2πkexp[iπ4],
I2πik·|det·(2Sxx2Sxy2Syx2Syy)|x=x¯y=y¯|1/2g(x¯,y¯)·exp{ikS(x¯,y¯)}.

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