Abstract

In the long-baseline laser interferometers that are now under construction to measure gravitational waves, the alignment of the optical components with respect to the incoming laser beam is crucial for maintaining maximum phase sensitivity. We present a basic formalism to calculate the effects of misalignment and beam distortions analytically in an arbitrarily complex optical system, including coupled cavities and Michelson interferometer configurations coupled with cavities. The electromagnetic field is decomposed into a superposition of higher-order Gaussian modes, while misaligned and distorting optical components along with free-space propagators are represented by matrix operators that act on the state vectors in this basis. We show how to deduce useful alignment signals generally, in order to design angular control systems.

© 1997 Optical Society of America

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References

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  1. K. S. Thorne, “Gravitational radiation,” in 300 Years of Gravitation, S. W. Hawking and W. Israel, eds. (Cambridge U. Press, Cambridge, 1987), Chap. 9, pp. 330–458.
  2. A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
    [CrossRef] [PubMed]
  3. M. W. Regehr, F. J. Raab, and S. E. Whitcomb, “Demonstration of a power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Appl. Opt. 20, 1507–1509 (1995).
  4. C. N. Man, D. Shoemaker, M. Pham Tu, and D. Dewey, “External modulation technique for sensitive interferometric detection of displacements,” Phys. Lett. A 148, 8–16 (1990).
    [CrossRef]
  5. A. Schenzle, R. DeVoe, and G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982).
    [CrossRef]
  6. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [CrossRef]
  7. L. Schnupp, Max Planck Institute for Quantum Optics, Garching, Germany (personal communication, 1986).
  8. Y. Hefetz and N. Mavalvala, in “Sensitivity of the LIGO interferometer to mirror misalignment and method for automatic alignment,” in Proceedings of the Seventh Marcel Grossmann Meeting on General Relativity, R. Ruffini, ed. (World Scientific, Singapore, 1996), pp. 1349–1351.
  9. D. Z. Anderson, “Alignment of resonant optical cavities,” Appl. Opt. 23, 2944–2949 (1984).
    [CrossRef] [PubMed]
  10. N. Sampas and D. Z. Anderson, “Stabilization of laser beam alignment to an optical resonator by heterodyne detection of off-axis modes,” Appl. Opt. 29, 394–403 (1990).
    [CrossRef] [PubMed]
  11. E. Morrison, B. J. Meers, D. I. Robertson, and H. Ward, “Experimental demonstration of an automatic alignment system for optical interferometers,” Appl. Opt. 33, 5037–5040 (1994).
    [CrossRef] [PubMed]
  12. E. Morrison, B. J. Meers, D. I. Robertson, and H. Ward, “Automatic alignment of optical interferometers,” Appl. Opt. 33, 5041–5049 (1994).
    [CrossRef] [PubMed]
  13. J. Vinet, P. Hello, C. N. Man, and A. Brillet, “A high accuracy method for the simulation of non-ideal optical cavities,” J. Phys. I (Paris) 2, 1287–1303 (1992).
  14. P. Saha, “Fast estimation of transverse fields in high-finesse optical cavities,” J. Opt. Soc. Am. A (to be published).
  15. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 13, p. 663.
  16. An introduction to group theory can be found, e.g., in H. Georgi, Lie Algebras in Particle Physics (Benjamin Cummings, Menlo Park, Calif., 1982), Chap. 2, p. 8.
  17. Note that the eigenvalues of M(Θx, Θy) can exceed unity, if just first-order terms are included. To obtain numerically stable solutions, the terms of at least second order in Θi must be included, if M is to be used in Eq. (20).

1995

M. W. Regehr, F. J. Raab, and S. E. Whitcomb, “Demonstration of a power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Appl. Opt. 20, 1507–1509 (1995).

1994

1992

J. Vinet, P. Hello, C. N. Man, and A. Brillet, “A high accuracy method for the simulation of non-ideal optical cavities,” J. Phys. I (Paris) 2, 1287–1303 (1992).

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

1990

C. N. Man, D. Shoemaker, M. Pham Tu, and D. Dewey, “External modulation technique for sensitive interferometric detection of displacements,” Phys. Lett. A 148, 8–16 (1990).
[CrossRef]

N. Sampas and D. Z. Anderson, “Stabilization of laser beam alignment to an optical resonator by heterodyne detection of off-axis modes,” Appl. Opt. 29, 394–403 (1990).
[CrossRef] [PubMed]

1984

1983

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

1982

A. Schenzle, R. DeVoe, and G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982).
[CrossRef]

Abramovici, A.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Althouse, W. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Anderson, D. Z.

Brewer, G.

A. Schenzle, R. DeVoe, and G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982).
[CrossRef]

Brillet, A.

J. Vinet, P. Hello, C. N. Man, and A. Brillet, “A high accuracy method for the simulation of non-ideal optical cavities,” J. Phys. I (Paris) 2, 1287–1303 (1992).

DeVoe, R.

A. Schenzle, R. DeVoe, and G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982).
[CrossRef]

Dewey, D.

C. N. Man, D. Shoemaker, M. Pham Tu, and D. Dewey, “External modulation technique for sensitive interferometric detection of displacements,” Phys. Lett. A 148, 8–16 (1990).
[CrossRef]

Drever, R. W. P.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Gursel, Y.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Hello, P.

J. Vinet, P. Hello, C. N. Man, and A. Brillet, “A high accuracy method for the simulation of non-ideal optical cavities,” J. Phys. I (Paris) 2, 1287–1303 (1992).

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Kawamura, S.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Man, C. N.

J. Vinet, P. Hello, C. N. Man, and A. Brillet, “A high accuracy method for the simulation of non-ideal optical cavities,” J. Phys. I (Paris) 2, 1287–1303 (1992).

C. N. Man, D. Shoemaker, M. Pham Tu, and D. Dewey, “External modulation technique for sensitive interferometric detection of displacements,” Phys. Lett. A 148, 8–16 (1990).
[CrossRef]

Meers, B. J.

Morrison, E.

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Pham Tu, M.

C. N. Man, D. Shoemaker, M. Pham Tu, and D. Dewey, “External modulation technique for sensitive interferometric detection of displacements,” Phys. Lett. A 148, 8–16 (1990).
[CrossRef]

Raab, F. J.

M. W. Regehr, F. J. Raab, and S. E. Whitcomb, “Demonstration of a power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Appl. Opt. 20, 1507–1509 (1995).

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Regehr, M. W.

M. W. Regehr, F. J. Raab, and S. E. Whitcomb, “Demonstration of a power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Appl. Opt. 20, 1507–1509 (1995).

Robertson, D. I.

Sampas, N.

Schenzle, A.

A. Schenzle, R. DeVoe, and G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982).
[CrossRef]

Shoemaker, D.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

C. N. Man, D. Shoemaker, M. Pham Tu, and D. Dewey, “External modulation technique for sensitive interferometric detection of displacements,” Phys. Lett. A 148, 8–16 (1990).
[CrossRef]

Sievers, L.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Spero, R. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Thorne, K. S.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Vinet, J.

J. Vinet, P. Hello, C. N. Man, and A. Brillet, “A high accuracy method for the simulation of non-ideal optical cavities,” J. Phys. I (Paris) 2, 1287–1303 (1992).

Vogt, R. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Ward, H.

Weiss, R.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Whitcomb, S. E.

M. W. Regehr, F. J. Raab, and S. E. Whitcomb, “Demonstration of a power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Appl. Opt. 20, 1507–1509 (1995).

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Zucker, M. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. B

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

J. Phys. I (Paris)

J. Vinet, P. Hello, C. N. Man, and A. Brillet, “A high accuracy method for the simulation of non-ideal optical cavities,” J. Phys. I (Paris) 2, 1287–1303 (1992).

Phys. Lett. A

C. N. Man, D. Shoemaker, M. Pham Tu, and D. Dewey, “External modulation technique for sensitive interferometric detection of displacements,” Phys. Lett. A 148, 8–16 (1990).
[CrossRef]

Phys. Rev. A

A. Schenzle, R. DeVoe, and G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982).
[CrossRef]

Science

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Other

K. S. Thorne, “Gravitational radiation,” in 300 Years of Gravitation, S. W. Hawking and W. Israel, eds. (Cambridge U. Press, Cambridge, 1987), Chap. 9, pp. 330–458.

P. Saha, “Fast estimation of transverse fields in high-finesse optical cavities,” J. Opt. Soc. Am. A (to be published).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 13, p. 663.

An introduction to group theory can be found, e.g., in H. Georgi, Lie Algebras in Particle Physics (Benjamin Cummings, Menlo Park, Calif., 1982), Chap. 2, p. 8.

Note that the eigenvalues of M(Θx, Θy) can exceed unity, if just first-order terms are included. To obtain numerically stable solutions, the terms of at least second order in Θi must be included, if M is to be used in Eq. (20).

L. Schnupp, Max Planck Institute for Quantum Optics, Garching, Germany (personal communication, 1986).

Y. Hefetz and N. Mavalvala, in “Sensitivity of the LIGO interferometer to mirror misalignment and method for automatic alignment,” in Proceedings of the Seventh Marcel Grossmann Meeting on General Relativity, R. Ruffini, ed. (World Scientific, Singapore, 1996), pp. 1349–1351.

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Figures (4)

Fig. 1
Fig. 1

Setup of (a) a Fabry–Perot interferometer, (b) a coupled pair of cavities, (c) a recycled Michelson interferometer, and (d) a recycled Michelson interferometer with arm cavities.

Fig. 2
Fig. 2

Reflection on an imperfect and misaligned mirror surface. Direction of ① incoming laser beam, ② reflected beam, ③ undistorted wave front, ④ ideal mirror surface, ⑤ physical mirror surface, Z(x, y), deviation from ideal mirror surface.

Fig. 3
Fig. 3

Alignment signals from a misaligned Fabry-Perot interferometer: ① incoming laser beam, ② resonant cavity mode, ③ partially transmitting input mirror, ④ tilted rear mirror, ⑤ segmented photodetector, ⑥ reflected sideband (gray curve), ⑦ reflected carrier light (solid curve) with its modal decomposition (dashed curve).

Fig. 4
Fig. 4

Angular error signal (top) and mode decomposition (bottom) as a function of the misalignment angle of the front mirror in a resonant Fabry–Perot interferometer. The calculations were made with one transverse degree of freedom only, with 2 modes (dashed curve), 4 modes (dash-dotted curve), and 22 modes (solid curves), respectively. The cavity fields are given in units of the input field, whereas the error signal is given relative to the input power and the modulation depth calculated for the Guoy phase of the detector, which gives the maximum signal at small angles. The cavity parameters are given in the text.

Equations (54)

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E(x, y, z)=amnUmn(x, y, z),
E(x, y, z2)=M(x, y, z2, z1)E(x, y, z1),
Mmn,kl(z2, z1)=-Umn(x, y, z2)M(x, y,z2, z1)×Ukl(x, y, z1)dxdy,
Pmn,kl(η)=δmkδnl exp[-ik(z2-z1)]×exp[i(m+n+1)η],
Mmn,kl=mn|M(x, y)|kl
M(x, y)=exp[-2ikZ(x, y)].
M¯(x, y)=exp[2ikZ(x, y)]=M(x, y).
Mmn,kl=mn|exp[-2ikZ(x, y)]|kl=mn|exp-2ikop,qr|opZop,qrqr||kl.
-2kZ(x, y)=i,jcijHi(x)Hj(y),
-2kZop,qr2i,jcijTop,qrij=i,jcijop|Hi(x)Hj(y)|qr|z=0.
-2kZ(x, y)=-8Θx 2xw(z)=-2ΘxH1(x).
Top,qr10=12δpr(qδo,q-1+oδo,q+1).
M(x, y)=exp[-i(n-1)kd(x, y)],
Δ(x)θx1-1nd
Eout=ECR+ESB++ESB-,
ECR=O(ωCR)Einp,
ESB±=i Γ2O(ωCR±Δω)Einp,
S¯=0c2[P(η)Eout]DΩ[P(η)Eout],
S=0c2(ECR)P(η)DΩP(η)ESB-+0c2(ESB+)P(η)DΩP(η)ECR.
S=s=0ds cos(sη)+s=0es sin(sη),
Prt=(-r1)(-r2)M1PM2P,
Eins=PrtEins+t1E0=t1(I-Prt)-1E0,
Erefl=r1M1E0-t1r2PM2PEins=r1M1I-r12+t12r12Prt(I-Prt)-1E0ME0,
E=a00a10a01.
P(z2, z1)=exp[-ik(z2-z1)]eiη000e2iη000e2iη,
M(Θx, Θy)1-2iΘx-2iΘy-2iΘx10-2iΘy01.
S=S0Γ 0cE022[Θ1x cos(η0+η¯)+Θ2x cos(η0+η¯+η)],
Erefl=r3M3I+r32+t32r3M3P1MP1×(I+r3M3P1MP1)-1E0,
Erefl=r3M3I+r32+t32r3M3P0MP0×(I+r3M3P0MP0)-1E0,
M=-tBS2r1P1M1P1-rBS2r2P2M2P2,
Eantisym=t3rBStBS(-r1P1M1P1+r2P2M2P2)×(I+r3M3P0MP0)-1E0,
Um(x, z)=2π1/412mm!w(z)1/2Hm2xw(z)×exp-x21w(z)2+ik2R(z)×expim+12η(z),
η(z)=tan-1zz0,w(z)=w01+zz021/2,
R(z)=z+z02z,
-Um(x, z)Un(x, z)dx=δmn,
2xHm(x)=Hm+1(x)+2mHm-1(x),
ddxHm(x)=2mHm-1(x),
-Um(x, 0) Hi(2x/w0)Hk(2x/w0)Uk(x, 0)dx=2ii!2kk!1/2δmi.
Umn=Um(x, z)Un(y, z)exp(-ikz),
Hi(x)Hj(y)Hq(x)Hr(y)=sthst,qrijHs(x)Ht(y).
Top,qrij=hop,qrij22oo!2pp!2qq!2rr!1/2.
E(x+Δx, y+Δy, z)=O(Δx, Δy)E(x, y, z).
O(Δx, Δy)=expiΔx 1iddx+Δy 1iddy.
Omn,kl(Δx, Δy)=mn|expiop,qr|op2Δxw(z)Top,qrx+2Δyw(z)Top,qryqr||kl,
Top,qrx=δpr-dxUo(x, z) w(z)i2ddxUq(x, z),
Hr2xw(z) w(z)i2ddxUr(x, z)
=12Hr+12xw(z)t+rHr-12xw(z)t*Ur(x, z)
t=i-w(z)2k2R(z)=i-zz0.
Top,qrx=12δpr(otδo,q+1+qt*δo,q-1).
S=0c2Ωdxdyp(x, y)×[P(η, x, y)ECR(x, y)][P(η, x, y)ESB-(x, y)]+0c2 Ωdxdyp(x, y)×[P(η, x, y)ESB+(x, y)][P(η, x, y)ECR(x, y)],
0c2mn,kl(EmnCR) exp[i(m+n+1)η]×Ωdxdyp(x, y)Umn(x, y, z)Ukl(x, y, z)×exp[i(k+l+1)η]EklSB-.
Dmn,klΩ=Ωdxdyp(x, y)Umn(x, y, z)Ukl(x, y, z).
Dmn,klfull-plane=δmkδnl.
Ωdxdyp(x, y)-dy0dx--0dx.

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