Abstract

The free-space propagation of initially aberrated optical beams is considered with an emphasis on aero-optical applications. An exact statistical solution of the paraxial wave equation is derived that can be used to obtain statistics of the beam such as beam center, spread, and higher-order statistics as algebraic functions of propagation distance, wavelength, and statistics of the initial wavefront. Correlations between the proposed description and intensity-based statistics, such as the Strehl ratio, are investigated. It is found that the root-mean-square (rms) of the gradient of the wavefront plays an important role in causing coherence degradation and that the rms of the wavefront error is not always an appropriate measure of the degradation. To illustrate the use of this statistical tool, index of refraction data from a numerical simulation of compressible flow over a cylinder are employed to perform an aero-optical analysis.

© 2006 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  2. L. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).
  3. J. C. Ricklin and F. M. Davidson, "Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication," J. Opt. Soc. Am. A 19, 1794-1802 (2002).
    [CrossRef]
  4. G. W. Sutton, "Aero-optical foundations and applications," AIAA J. 23, 1525-1537 (1985).
    [CrossRef]
  5. E. J. Jumper and E. J. Fitzgerald, "Recent advances in aero-optics," Prog. Aerosp. Sci. 37, 299-339 (2001).
    [CrossRef]
  6. E. Tromeur, E. Garnier, P. Sagaut, and C. Basdevant, "Large eddy simulations of aero-optical effects in a turbulent boundary layer," J. Turbul. 4, 1-22 (2003).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  8. A. Maréchal, "Etude des effets combinés de la diffraction et des aberrations géométriques sur l'image d'un point lumineux," Rev. Opt., Theor. Instrum. 26, 257-277 (1947).
  9. V. N. Mahajan, "Strehl ratio for aberration in terms of their aberration variance," J. Opt. Soc. Am. 73, 860-861 (1983).
    [CrossRef]
  10. P. E. Cassady, S. F. Birch, and P. J. Terry, "Aero-optical analysis of compressible flow over an open cavity," AIAA J. 27, 758-762 (1989).
    [CrossRef]
  11. R. C. Aguirre, J. O. Mason, and H. J. Catrakis, "Experimental studies of turbulent interfaces in mixing aero-optics and high-speed flows," Int. Assoc. Mech. Eng. Trans. 2, 50-58 (2005).
  12. J. O. Mason, R. C. Aguirre, and H. J. Catrakis, "Computational aero-optics and electromagnetics: compressible vortices and laser beam propagation," Int. Assoc. Mech. Eng. Trans. 42, 1973-1981 (2005).
  13. N. Sinha, S. Arunajatesan, J. M. Seiner, and L. S. Ukeiley, "Large-eddy simulations of aero-optic flow fields and control application," AIAA Pap.2004-2448 (2004).
  14. E. J. Fitzgerald and E. J. Jumper, "Scaling aerooptical aberrations produced by high-subsonic-Mach shear layers," AIAA J. 40, 1373-1381 (2002).
    [CrossRef]
  15. S. Gordeyev, E. J. Jumper, T. T. Ng, and A. B. Cain, "Aero-optical characteristics of compressible, subsonic turbulent boundary layers," AIAA Pap.2003-3606 (2003).
  16. S. Gordeyev, E. J. Jumper, T. T. Ng, and A. B. Cain, "Optical disturbances caused by transonic separated boundary layer behind a 20-degree ramp: physics and control," in Proceedings of the American Institute of Aeronautics and Astronautics Conference (AAIA 2004) Pap.2004-0472 (AIAA, 2004).
  17. M. I. Jones and E. E. Bender, "CFD-based computer simulation of optical turbulence through aircraft flowfields and wakes," AIAA Pap.2001-2798 (2001).
  18. J. M. Cicchiello and E. J. Jumper, "Far-field optical degradation due to near-field transmission through a turbulent heated jet," Appl. Opt. 36, 6441-6452 (1997).
    [CrossRef]
  19. B. Y. Zeldovich, A. V. Mamaev, and V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, 1995), pp. 1-10.
  20. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), pp. 50-51.
  21. W. Wolf and G. J. Zissis, The Infrared Handbook (U.S. Office of Naval Research, 1978), pp. 16-24.
  22. G. W. Sutton, "Effect of turbulence fluctuations in an optically active fluid medium," AIAA J. 7, 1737-1743 (1969).
    [CrossRef]
  23. A. Mani, M. Wang, and P. Moin, "Computational study of aero-optical distortion by turbulent wake," in Proceedings of the American Institute of Aeronautics and Astronautics Conference (AAIA 2005) Pap. 2005-4655 (AIAA, 2005).

2005 (2)

R. C. Aguirre, J. O. Mason, and H. J. Catrakis, "Experimental studies of turbulent interfaces in mixing aero-optics and high-speed flows," Int. Assoc. Mech. Eng. Trans. 2, 50-58 (2005).

J. O. Mason, R. C. Aguirre, and H. J. Catrakis, "Computational aero-optics and electromagnetics: compressible vortices and laser beam propagation," Int. Assoc. Mech. Eng. Trans. 42, 1973-1981 (2005).

2003 (1)

E. Tromeur, E. Garnier, P. Sagaut, and C. Basdevant, "Large eddy simulations of aero-optical effects in a turbulent boundary layer," J. Turbul. 4, 1-22 (2003).
[CrossRef]

2002 (2)

2001 (1)

E. J. Jumper and E. J. Fitzgerald, "Recent advances in aero-optics," Prog. Aerosp. Sci. 37, 299-339 (2001).
[CrossRef]

1997 (1)

1989 (1)

P. E. Cassady, S. F. Birch, and P. J. Terry, "Aero-optical analysis of compressible flow over an open cavity," AIAA J. 27, 758-762 (1989).
[CrossRef]

1985 (1)

G. W. Sutton, "Aero-optical foundations and applications," AIAA J. 23, 1525-1537 (1985).
[CrossRef]

1983 (1)

1969 (1)

G. W. Sutton, "Effect of turbulence fluctuations in an optically active fluid medium," AIAA J. 7, 1737-1743 (1969).
[CrossRef]

1947 (1)

A. Maréchal, "Etude des effets combinés de la diffraction et des aberrations géométriques sur l'image d'un point lumineux," Rev. Opt., Theor. Instrum. 26, 257-277 (1947).

Aguirre, R. C.

R. C. Aguirre, J. O. Mason, and H. J. Catrakis, "Experimental studies of turbulent interfaces in mixing aero-optics and high-speed flows," Int. Assoc. Mech. Eng. Trans. 2, 50-58 (2005).

J. O. Mason, R. C. Aguirre, and H. J. Catrakis, "Computational aero-optics and electromagnetics: compressible vortices and laser beam propagation," Int. Assoc. Mech. Eng. Trans. 42, 1973-1981 (2005).

Andrews, L.

L. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

Arunajatesan, S.

N. Sinha, S. Arunajatesan, J. M. Seiner, and L. S. Ukeiley, "Large-eddy simulations of aero-optic flow fields and control application," AIAA Pap.2004-2448 (2004).

Basdevant, C.

E. Tromeur, E. Garnier, P. Sagaut, and C. Basdevant, "Large eddy simulations of aero-optical effects in a turbulent boundary layer," J. Turbul. 4, 1-22 (2003).
[CrossRef]

Bender, E. E.

M. I. Jones and E. E. Bender, "CFD-based computer simulation of optical turbulence through aircraft flowfields and wakes," AIAA Pap.2001-2798 (2001).

Birch, S. F.

P. E. Cassady, S. F. Birch, and P. J. Terry, "Aero-optical analysis of compressible flow over an open cavity," AIAA J. 27, 758-762 (1989).
[CrossRef]

Cain, A. B.

S. Gordeyev, E. J. Jumper, T. T. Ng, and A. B. Cain, "Aero-optical characteristics of compressible, subsonic turbulent boundary layers," AIAA Pap.2003-3606 (2003).

S. Gordeyev, E. J. Jumper, T. T. Ng, and A. B. Cain, "Optical disturbances caused by transonic separated boundary layer behind a 20-degree ramp: physics and control," in Proceedings of the American Institute of Aeronautics and Astronautics Conference (AAIA 2004) Pap.2004-0472 (AIAA, 2004).

Cassady, P. E.

P. E. Cassady, S. F. Birch, and P. J. Terry, "Aero-optical analysis of compressible flow over an open cavity," AIAA J. 27, 758-762 (1989).
[CrossRef]

Catrakis, H. J.

J. O. Mason, R. C. Aguirre, and H. J. Catrakis, "Computational aero-optics and electromagnetics: compressible vortices and laser beam propagation," Int. Assoc. Mech. Eng. Trans. 42, 1973-1981 (2005).

R. C. Aguirre, J. O. Mason, and H. J. Catrakis, "Experimental studies of turbulent interfaces in mixing aero-optics and high-speed flows," Int. Assoc. Mech. Eng. Trans. 2, 50-58 (2005).

Cicchiello, J. M.

Davidson, F. M.

Fitzgerald, E. J.

E. J. Fitzgerald and E. J. Jumper, "Scaling aerooptical aberrations produced by high-subsonic-Mach shear layers," AIAA J. 40, 1373-1381 (2002).
[CrossRef]

E. J. Jumper and E. J. Fitzgerald, "Recent advances in aero-optics," Prog. Aerosp. Sci. 37, 299-339 (2001).
[CrossRef]

Garnier, E.

E. Tromeur, E. Garnier, P. Sagaut, and C. Basdevant, "Large eddy simulations of aero-optical effects in a turbulent boundary layer," J. Turbul. 4, 1-22 (2003).
[CrossRef]

Goodman, J. W.

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

Gordeyev, S.

S. Gordeyev, E. J. Jumper, T. T. Ng, and A. B. Cain, "Aero-optical characteristics of compressible, subsonic turbulent boundary layers," AIAA Pap.2003-3606 (2003).

S. Gordeyev, E. J. Jumper, T. T. Ng, and A. B. Cain, "Optical disturbances caused by transonic separated boundary layer behind a 20-degree ramp: physics and control," in Proceedings of the American Institute of Aeronautics and Astronautics Conference (AAIA 2004) Pap.2004-0472 (AIAA, 2004).

Jones, M. I.

M. I. Jones and E. E. Bender, "CFD-based computer simulation of optical turbulence through aircraft flowfields and wakes," AIAA Pap.2001-2798 (2001).

Jumper, E. J.

E. J. Fitzgerald and E. J. Jumper, "Scaling aerooptical aberrations produced by high-subsonic-Mach shear layers," AIAA J. 40, 1373-1381 (2002).
[CrossRef]

E. J. Jumper and E. J. Fitzgerald, "Recent advances in aero-optics," Prog. Aerosp. Sci. 37, 299-339 (2001).
[CrossRef]

J. M. Cicchiello and E. J. Jumper, "Far-field optical degradation due to near-field transmission through a turbulent heated jet," Appl. Opt. 36, 6441-6452 (1997).
[CrossRef]

S. Gordeyev, E. J. Jumper, T. T. Ng, and A. B. Cain, "Aero-optical characteristics of compressible, subsonic turbulent boundary layers," AIAA Pap.2003-3606 (2003).

S. Gordeyev, E. J. Jumper, T. T. Ng, and A. B. Cain, "Optical disturbances caused by transonic separated boundary layer behind a 20-degree ramp: physics and control," in Proceedings of the American Institute of Aeronautics and Astronautics Conference (AAIA 2004) Pap.2004-0472 (AIAA, 2004).

Mahajan, V. N.

Mamaev, A. V.

B. Y. Zeldovich, A. V. Mamaev, and V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, 1995), pp. 1-10.

Mani, A.

A. Mani, M. Wang, and P. Moin, "Computational study of aero-optical distortion by turbulent wake," in Proceedings of the American Institute of Aeronautics and Astronautics Conference (AAIA 2005) Pap. 2005-4655 (AIAA, 2005).

Maréchal, A.

A. Maréchal, "Etude des effets combinés de la diffraction et des aberrations géométriques sur l'image d'un point lumineux," Rev. Opt., Theor. Instrum. 26, 257-277 (1947).

Mason, J. O.

J. O. Mason, R. C. Aguirre, and H. J. Catrakis, "Computational aero-optics and electromagnetics: compressible vortices and laser beam propagation," Int. Assoc. Mech. Eng. Trans. 42, 1973-1981 (2005).

R. C. Aguirre, J. O. Mason, and H. J. Catrakis, "Experimental studies of turbulent interfaces in mixing aero-optics and high-speed flows," Int. Assoc. Mech. Eng. Trans. 2, 50-58 (2005).

Moin, P.

A. Mani, M. Wang, and P. Moin, "Computational study of aero-optical distortion by turbulent wake," in Proceedings of the American Institute of Aeronautics and Astronautics Conference (AAIA 2005) Pap. 2005-4655 (AIAA, 2005).

Ng, T. T.

S. Gordeyev, E. J. Jumper, T. T. Ng, and A. B. Cain, "Optical disturbances caused by transonic separated boundary layer behind a 20-degree ramp: physics and control," in Proceedings of the American Institute of Aeronautics and Astronautics Conference (AAIA 2004) Pap.2004-0472 (AIAA, 2004).

S. Gordeyev, E. J. Jumper, T. T. Ng, and A. B. Cain, "Aero-optical characteristics of compressible, subsonic turbulent boundary layers," AIAA Pap.2003-3606 (2003).

Phillips, R. L.

L. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

Ricklin, J. C.

Sagaut, P.

E. Tromeur, E. Garnier, P. Sagaut, and C. Basdevant, "Large eddy simulations of aero-optical effects in a turbulent boundary layer," J. Turbul. 4, 1-22 (2003).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), pp. 50-51.

Seiner, J. M.

N. Sinha, S. Arunajatesan, J. M. Seiner, and L. S. Ukeiley, "Large-eddy simulations of aero-optic flow fields and control application," AIAA Pap.2004-2448 (2004).

Shkunov, V. V.

B. Y. Zeldovich, A. V. Mamaev, and V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, 1995), pp. 1-10.

Sinha, N.

N. Sinha, S. Arunajatesan, J. M. Seiner, and L. S. Ukeiley, "Large-eddy simulations of aero-optic flow fields and control application," AIAA Pap.2004-2448 (2004).

Sutton, G. W.

G. W. Sutton, "Aero-optical foundations and applications," AIAA J. 23, 1525-1537 (1985).
[CrossRef]

G. W. Sutton, "Effect of turbulence fluctuations in an optically active fluid medium," AIAA J. 7, 1737-1743 (1969).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), pp. 50-51.

Terry, P. J.

P. E. Cassady, S. F. Birch, and P. J. Terry, "Aero-optical analysis of compressible flow over an open cavity," AIAA J. 27, 758-762 (1989).
[CrossRef]

Tromeur, E.

E. Tromeur, E. Garnier, P. Sagaut, and C. Basdevant, "Large eddy simulations of aero-optical effects in a turbulent boundary layer," J. Turbul. 4, 1-22 (2003).
[CrossRef]

Ukeiley, L. S.

N. Sinha, S. Arunajatesan, J. M. Seiner, and L. S. Ukeiley, "Large-eddy simulations of aero-optic flow fields and control application," AIAA Pap.2004-2448 (2004).

Wang, M.

A. Mani, M. Wang, and P. Moin, "Computational study of aero-optical distortion by turbulent wake," in Proceedings of the American Institute of Aeronautics and Astronautics Conference (AAIA 2005) Pap. 2005-4655 (AIAA, 2005).

Wolf, W.

W. Wolf and G. J. Zissis, The Infrared Handbook (U.S. Office of Naval Research, 1978), pp. 16-24.

Zeldovich, B. Y.

B. Y. Zeldovich, A. V. Mamaev, and V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, 1995), pp. 1-10.

Zissis, G. J.

W. Wolf and G. J. Zissis, The Infrared Handbook (U.S. Office of Naval Research, 1978), pp. 16-24.

AIAA J. (4)

G. W. Sutton, "Aero-optical foundations and applications," AIAA J. 23, 1525-1537 (1985).
[CrossRef]

P. E. Cassady, S. F. Birch, and P. J. Terry, "Aero-optical analysis of compressible flow over an open cavity," AIAA J. 27, 758-762 (1989).
[CrossRef]

E. J. Fitzgerald and E. J. Jumper, "Scaling aerooptical aberrations produced by high-subsonic-Mach shear layers," AIAA J. 40, 1373-1381 (2002).
[CrossRef]

G. W. Sutton, "Effect of turbulence fluctuations in an optically active fluid medium," AIAA J. 7, 1737-1743 (1969).
[CrossRef]

Appl. Opt. (1)

Int. Assoc. Mech. Eng. Trans. (2)

R. C. Aguirre, J. O. Mason, and H. J. Catrakis, "Experimental studies of turbulent interfaces in mixing aero-optics and high-speed flows," Int. Assoc. Mech. Eng. Trans. 2, 50-58 (2005).

J. O. Mason, R. C. Aguirre, and H. J. Catrakis, "Computational aero-optics and electromagnetics: compressible vortices and laser beam propagation," Int. Assoc. Mech. Eng. Trans. 42, 1973-1981 (2005).

J. Opt. Soc. Am. (1)

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

J. Turbul. (1)

E. Tromeur, E. Garnier, P. Sagaut, and C. Basdevant, "Large eddy simulations of aero-optical effects in a turbulent boundary layer," J. Turbul. 4, 1-22 (2003).
[CrossRef]

Prog. Aerosp. Sci. (1)

E. J. Jumper and E. J. Fitzgerald, "Recent advances in aero-optics," Prog. Aerosp. Sci. 37, 299-339 (2001).
[CrossRef]

Rev. Opt., Theor. Instrum. (1)

A. Maréchal, "Etude des effets combinés de la diffraction et des aberrations géométriques sur l'image d'un point lumineux," Rev. Opt., Theor. Instrum. 26, 257-277 (1947).

Other (11)

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

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

L. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

N. Sinha, S. Arunajatesan, J. M. Seiner, and L. S. Ukeiley, "Large-eddy simulations of aero-optic flow fields and control application," AIAA Pap.2004-2448 (2004).

S. Gordeyev, E. J. Jumper, T. T. Ng, and A. B. Cain, "Aero-optical characteristics of compressible, subsonic turbulent boundary layers," AIAA Pap.2003-3606 (2003).

S. Gordeyev, E. J. Jumper, T. T. Ng, and A. B. Cain, "Optical disturbances caused by transonic separated boundary layer behind a 20-degree ramp: physics and control," in Proceedings of the American Institute of Aeronautics and Astronautics Conference (AAIA 2004) Pap.2004-0472 (AIAA, 2004).

M. I. Jones and E. E. Bender, "CFD-based computer simulation of optical turbulence through aircraft flowfields and wakes," AIAA Pap.2001-2798 (2001).

A. Mani, M. Wang, and P. Moin, "Computational study of aero-optical distortion by turbulent wake," in Proceedings of the American Institute of Aeronautics and Astronautics Conference (AAIA 2005) Pap. 2005-4655 (AIAA, 2005).

B. Y. Zeldovich, A. V. Mamaev, and V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, 1995), pp. 1-10.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), pp. 50-51.

W. Wolf and G. J. Zissis, The Infrared Handbook (U.S. Office of Naval Research, 1978), pp. 16-24.

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Figures (5)

Fig. 1
Fig. 1

Schematics of the problem. Initial distortion of an optical beam causes beam spread in the far field.

Fig. 2
Fig. 2

Instantaneous vorticity contours of turbulent flow over a circular cylinder at R e = 3900 and M = 0.4 , obtained using a large-eddy simulation. Flow is from left to right, and free-stream velocity and cylinder diameter are used as reference scales to make the contour level nondimensional. The optical beam, which is schematically shown, is shot from the surface of the cylinder.

Fig. 3
Fig. 3

Instantaneous far-field intensity patterns for an aberrated beam (top) and a nonaberrated beam (bottom) at different distances of propagation. The intensity levels are normalized by peak intensity at the aperture, which is assumed to have a Gaussian profile. The coordinates are dimensionless with cylinder diameter as reference length. The dimensionless wavelength is 2.5 × 10 6 .

Fig. 4
Fig. 4

SR and FR as a function of nondimensional time for a distance of propagation equal to 2000 D . The FR is calculated through Eq. (46). The SR is calculated by standard Fourier optics methods and not from expression (6).

Fig. 5
Fig. 5

Same as Fig. 4, except that the SR and FR are plotted in the very far field where the Fraunhofer approximation is valid.

Tables (3)

Tables Icon

Table 1 Primary Statistics of the Example Problem

Tables Icon

Table 2 Very Far Secondary Statistics of the Example Problem

Tables Icon

Table 3 Comparison of Different Measures at Fraunhofer Limit

Equations (51)

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

2 U ( x , y , z ) + k 2 n 2 ( x , y , z ) U ( x , y , z ) = 0 ,
U ( x , y , z 1 ) = U ( x , y , z 0 ) exp ( j k L ( x , y ) ) ,
L ( x , y ) = z 0 z 1 n ( x , y , z ) d z ,
SR max { I } max { I i d e a l }
SR 1 ( k L r m s ) 2 ,
SR exp [ ( k L r m s ) 2 ] .
U ( x , y , z ) = A ( x , y , z ) exp ( j k z ) .
j 2 k A z = A x x + A y y ,
j 2 k A z * = A x x * A y y * .
j 2 k z A A * d x d y = ( A * A x x A A x x * + A * A y y A A y y * ) d x d y .
j 2 k z I d x d y = 0 ,
z I = 0 .
x ¯ ( z ) x I ( x , y , z ) d x d y I , y ¯ ( z ) y I ( x , y , z ) d x d y I .
x ¯ z = 1 I ( x I ) z d x d y = 1 I ( x A * A z + x A A z * ) d x d y .
x ¯ z = 1 j 2 k I ( x A * A x x x A A x x * + x A * A y y x A A y y * ) d x d y .
x ¯ z = 1 j 2 k I ( A A x * A * A x ) d x d y .
z ( A A x * A * A x ) d x d y = ( A A z x * + A x * A z A * A z x A x A z * ) d x d y .
z ( A A x * A * A x ) d x d y = 1 j 2 k ( A A x x x * A * A x x x + A x A x x * + A x * A x x A A y y x * A * A y y x + A x A y y * + A x * A y y ) d x d y .
z ( A A x * A * A x ) d x d y = 0 .
A ( x , y , z 1 ) = A ( x , y ) exp ( j k L ( x , y ) ) ,
( ( A A x * A * A x ) d x d y ) z 1 = j 2 k ( L x A 2 d x d y ) = j 2 k I L x ¯ z 1 ,
x ¯ = L x ¯ z 1 ( z z 1 ) + x ¯ z 1 .
y ¯ = L y ¯ z 1 ( z z 1 ) + y ¯ z 1 .
z x 2 ¯ = z ( 1 I x 2 I d x d y ) = 1 I ( x 2 A * A z + x 2 A A z * ) d x d y .
z x 2 ¯ = 1 j k I ( x A A x * x A * A x ) d x d y .
z ( x A A x * x A * A x ) d x d y = ( x A A z x * + x A x * A z x A * A z x x A x A z * ) d x d y .
z ( x A A x * x A * A x ) d x d y = 1 j 2 k ( x A A x x x * x A * A x x x + x A x A x x * + x A x * A x x x A A y y x * x A * A y y x + x A x A y y * + x A x * A y y ) d x d y .
z ( x A A x * x A * A x ) d x d y = 1 j 2 k ( 4 A x A x * ) d x d y .
z ( A x A x * ) d x d y = ( A x A z x * + A x * A z x ) d x d y = 1 j 2 k ( A x * A x x x A x A x x x * + A x * A y y x A x A y y x * ) d x d y .
z A x A x * d x d y = 0 .
x 2 ¯ = 1 2 α ( z z 1 ) 2 + β ( z z 1 ) + x 2 ¯ z 1 ,
α = 2 z 2 x 2 ¯ = 2 k 2 I ( A x A x * d x d y ) z 1 ,
β = z x 2 ¯ z 1 = 1 j k I ( ( x A A x * x A * A x ) d x d y ) z 1 .
α = 2 k 2 I ( ( k 2 L x 2 I + A x 2 ) d x d y ) z 1 = 2 ( L x 2 ¯ + 1 k 2 ( A x A ) 2 ¯ ) z 1 ,
β = 1 j k I ( j 2 k x L x I d x d y ) z 1 = 2 x L x ¯ z 1 .
x 2 ¯ = ( L x 2 ¯ + 1 k 2 ( A x A ) 2 ¯ ) z 1 ( z z 1 ) 2 + 2 x L x ¯ z 1 ( z z 1 ) + x 2 ¯ z 1 .
y 2 ¯ = ( L y 2 ¯ + 1 k 2 ( A y A ) 2 ¯ ) z 1 ( z z 1 ) 2 + 2 y L y ¯ z 1 ( z z 1 ) + y 2 ¯ z 1 ,
x y ¯ = ( L y L x ¯ + 1 k 2 ( A y A x A 2 ) ¯ ) z 1 ( z z 1 ) 2 + ( y L x ¯ + x L y ¯ ) z 1 ( z z 1 ) + x y ¯ z 1 .
x 2 ¯ = x 2 ¯ x ¯ 2 = ( L x 2 ¯ + 1 k 2 ( A x A ) 2 ¯ ) z 1 ( z z 1 ) 2 + 2 x L x ¯ z 1 ( z z 1 ) + x 2 ¯ z 1 ,
y 2 ¯ = ( L y 2 ¯ + 1 k 2 ( A y A ) 2 ¯ ) z 1 ( z z 1 ) 2 + 2 y L y ¯ z 1 ( z z 1 ) + y 2 ¯ z 1 ,
x y ¯ = ( L x L y ¯ + 1 k 2 ( A y A x A 2 ) ¯ ) z 1 ( z z 1 ) 2 + ( y L x ¯ + x L y ¯ ) z 1 ( z z 1 ) + x y ¯ z 1 .
R 2 r 2 ¯ = x 2 ¯ + y 2 ¯ = ( ( L ) 2 ¯ + 1 k 2 ( A A ) 2 ¯ ) z 1 ( z z 1 ) 2 + 2 X ( L ) ¯ z 1 ( z z 1 ) + R 2 z 1 ,
S ( x 2 ¯ y 2 ¯ x y ¯ 2 ) 1 2 .
Δ R 2 R 2 R z 1 2 = Δ R g 2 + Δ R d 2 ,
Δ R g 2 = ( ( L ) 2 ¯ ) z 1 ( z z 1 ) 2 + 2 X ( L ) ¯ z 1 ( z z 1 ) ,
Δ R d 2 = 1 k 2 ( A A ) 2 ¯ z 1 ( z z 1 ) 2 .
k s a t = ( ( A A ) 2 ¯ ( L ) 2 ¯ ) z 1 1 2 .
Δ z f ( c a ) 1 2 = R z 1 ( ( L ) 2 ¯ + 1 k 2 ( A A ) 2 ¯ ) z 1 1 2 .
FR ( z ) S ( L = 0 ) S .
FR λ 2 4 π 2 ( A x A ) 2 ¯ + x 2 ¯ z 2 L x 2 ¯ + λ 2 4 π 2 ( A x A ) 2 ¯ + x 2 ¯ z 2 ,
n = 1 + G ( λ ) ρ ,

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