Abstract

Simple formulas are derived for the degradation in the beam-quality factor, M2, of an arbitrary laser beam caused by quartic phase distortions such as those that might occur in a spherically aberrated optical component, a thermally aberrated laser output window, or a divergent beam emerging from a high-index dielectric medium as in a wide-stripe, unstable-resonator diode laser. A new formula for the defocus correction that is needed to collimate optimally a beam with quartic phase aberration is also derived. Analytical results and numerical examples are given for both radially aberrated and one-dimensional transversely aberrated cases, and a simple experimental measurement of the beam-quality degradation produced by a thin plano–convex lens is shown to be in good agreement with the theory.

© 1993 Optical Society of America

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References

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  1. H. H. Hopkins, Wave Theory of Aberrations (Oxford University, London, 1950).
  2. W. T. Welford, Aberrations of the Symmetric Optical System (Academic, New York, 1974).
  3. V. N. Mahajan, Aberration Theory Made Simple (Society of Photo-Optical Instrumentation Engineers, Bellingham, WA, 1991).
    [CrossRef]
  4. S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, Sect. E3, pp. 1185–1201.
  5. F. W. Dabby, J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” IEEE J. Quantum Electron. QE-3, 382–383 (1968).
  6. I. Miyamoto, “Analysis of thermally induced optical distortion in lens during focusing high power CO2 laser beam,” in CO2 Lasers and Applications II, H. Opower, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1276, 112–121 (1990).
  7. C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29, 343–350 (1990); C. A. Klein, “Optical distortion coefficients of laser windows: one more time,” in Mirrors and Windows for High Power/High Energy Laser Systems, C. A. Klein, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1047, 58–79 (1989).
    [CrossRef]
  8. C. A. Klein, “Power handling capability of Faraday rotation isolators for CO2 laser radars,” Appl. Opt. 28, 904–914 (1989).
    [CrossRef] [PubMed]
  9. J. R. Whinnery, D. T. Miller, F. Dabby, “Thermal convection and spherical aberration distortion of laser beams in low-loss liquids,” Appl. Phys. Lett. 13, 284–286 (1968).
  10. M. L. Tilton, G. Dente, A. H. Paxton, J. Cser, R. K. DeFreez, C. E. Moeller, D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
    [CrossRef]
  11. R. J. Lang, “Geometric formulation of unstable-resonator design and application to self-collimating unstable-resonator diode lasers,” Opt. Lett. 16, 1319–1321 (1991).
    [CrossRef] [PubMed]
  12. L. Marshall, “Applications a la mode,” Laser Focus 4(4), 26–28 (1971).
  13. M. W. Sasnett, “Propagation of multimode laser beams: the M2 factor,” in The Physics and Technology of Laser Resonators, D. R. Hall, P. E. Jackson, eds. (Hilger, London, 1989), pp. 132–142.
  14. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Soc. Photo-Opt. Instrum. Eng.1224, 2–14 (1990).
  15. M. W. Sasnett, T. F. Johnston, “Beam characterization and measurement of propagation attributes,” in Laser Beam Diagnostics, R. N. Hindy, Y. Kohanzadeh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1414, 21–32 (1991).
  16. A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
    [CrossRef]
  17. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Secs. 16.7 and 18.4.
  18. A. E. Siegman, J. Ruff, “Effects of spherical aberration on laser beam quality,” in Laser Energy Distribution Profiles: Measurement and Applications, J. M. Darchuk, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1834, 100–103 (1992).
  19. R. Martinez-Herrero, P. M. Mejias, G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distortions,” Opt. Lett. 17, 1650–1652 (1992).
    [CrossRef] [PubMed]
  20. R. Martinez-Herrero, P. M. Mejias, “Quality improvements of symmetric-intensity beams propagating through pure phase plates,” Opt. Commun. 95, 1–3 (1993).
    [CrossRef]

1993

R. Martinez-Herrero, P. M. Mejias, “Quality improvements of symmetric-intensity beams propagating through pure phase plates,” Opt. Commun. 95, 1–3 (1993).
[CrossRef]

1992

1991

R. J. Lang, “Geometric formulation of unstable-resonator design and application to self-collimating unstable-resonator diode lasers,” Opt. Lett. 16, 1319–1321 (1991).
[CrossRef] [PubMed]

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

M. L. Tilton, G. Dente, A. H. Paxton, J. Cser, R. K. DeFreez, C. E. Moeller, D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

1990

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29, 343–350 (1990); C. A. Klein, “Optical distortion coefficients of laser windows: one more time,” in Mirrors and Windows for High Power/High Energy Laser Systems, C. A. Klein, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1047, 58–79 (1989).
[CrossRef]

1989

1971

L. Marshall, “Applications a la mode,” Laser Focus 4(4), 26–28 (1971).

1968

J. R. Whinnery, D. T. Miller, F. Dabby, “Thermal convection and spherical aberration distortion of laser beams in low-loss liquids,” Appl. Phys. Lett. 13, 284–286 (1968).

F. W. Dabby, J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” IEEE J. Quantum Electron. QE-3, 382–383 (1968).

Akhmanov, S. A.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, Sect. E3, pp. 1185–1201.

Cser, J.

M. L. Tilton, G. Dente, A. H. Paxton, J. Cser, R. K. DeFreez, C. E. Moeller, D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

Dabby, F.

J. R. Whinnery, D. T. Miller, F. Dabby, “Thermal convection and spherical aberration distortion of laser beams in low-loss liquids,” Appl. Phys. Lett. 13, 284–286 (1968).

Dabby, F. W.

F. W. Dabby, J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” IEEE J. Quantum Electron. QE-3, 382–383 (1968).

DeFreez, R. K.

M. L. Tilton, G. Dente, A. H. Paxton, J. Cser, R. K. DeFreez, C. E. Moeller, D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

Dente, G.

M. L. Tilton, G. Dente, A. H. Paxton, J. Cser, R. K. DeFreez, C. E. Moeller, D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

Depatie, D.

M. L. Tilton, G. Dente, A. H. Paxton, J. Cser, R. K. DeFreez, C. E. Moeller, D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Oxford University, London, 1950).

Johnston, T. F.

M. W. Sasnett, T. F. Johnston, “Beam characterization and measurement of propagation attributes,” in Laser Beam Diagnostics, R. N. Hindy, Y. Kohanzadeh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1414, 21–32 (1991).

Khokhlov, R. V.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, Sect. E3, pp. 1185–1201.

Klein, C. A.

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29, 343–350 (1990); C. A. Klein, “Optical distortion coefficients of laser windows: one more time,” in Mirrors and Windows for High Power/High Energy Laser Systems, C. A. Klein, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1047, 58–79 (1989).
[CrossRef]

C. A. Klein, “Power handling capability of Faraday rotation isolators for CO2 laser radars,” Appl. Opt. 28, 904–914 (1989).
[CrossRef] [PubMed]

Lang, R. J.

Mahajan, V. N.

V. N. Mahajan, Aberration Theory Made Simple (Society of Photo-Optical Instrumentation Engineers, Bellingham, WA, 1991).
[CrossRef]

Marshall, L.

L. Marshall, “Applications a la mode,” Laser Focus 4(4), 26–28 (1971).

Martinez-Herrero, R.

R. Martinez-Herrero, P. M. Mejias, “Quality improvements of symmetric-intensity beams propagating through pure phase plates,” Opt. Commun. 95, 1–3 (1993).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distortions,” Opt. Lett. 17, 1650–1652 (1992).
[CrossRef] [PubMed]

Mejias, P. M.

R. Martinez-Herrero, P. M. Mejias, “Quality improvements of symmetric-intensity beams propagating through pure phase plates,” Opt. Commun. 95, 1–3 (1993).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distortions,” Opt. Lett. 17, 1650–1652 (1992).
[CrossRef] [PubMed]

Miller, D. T.

J. R. Whinnery, D. T. Miller, F. Dabby, “Thermal convection and spherical aberration distortion of laser beams in low-loss liquids,” Appl. Phys. Lett. 13, 284–286 (1968).

Miyamoto, I.

I. Miyamoto, “Analysis of thermally induced optical distortion in lens during focusing high power CO2 laser beam,” in CO2 Lasers and Applications II, H. Opower, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1276, 112–121 (1990).

Moeller, C. E.

M. L. Tilton, G. Dente, A. H. Paxton, J. Cser, R. K. DeFreez, C. E. Moeller, D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

Paxton, A. H.

M. L. Tilton, G. Dente, A. H. Paxton, J. Cser, R. K. DeFreez, C. E. Moeller, D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

Piquero, G.

Ruff, J.

A. E. Siegman, J. Ruff, “Effects of spherical aberration on laser beam quality,” in Laser Energy Distribution Profiles: Measurement and Applications, J. M. Darchuk, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1834, 100–103 (1992).

Sasnett, M. W.

M. W. Sasnett, “Propagation of multimode laser beams: the M2 factor,” in The Physics and Technology of Laser Resonators, D. R. Hall, P. E. Jackson, eds. (Hilger, London, 1989), pp. 132–142.

M. W. Sasnett, T. F. Johnston, “Beam characterization and measurement of propagation attributes,” in Laser Beam Diagnostics, R. N. Hindy, Y. Kohanzadeh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1414, 21–32 (1991).

Siegman, A. E.

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Soc. Photo-Opt. Instrum. Eng.1224, 2–14 (1990).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Secs. 16.7 and 18.4.

A. E. Siegman, J. Ruff, “Effects of spherical aberration on laser beam quality,” in Laser Energy Distribution Profiles: Measurement and Applications, J. M. Darchuk, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1834, 100–103 (1992).

Sukhorukov, A. P.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, Sect. E3, pp. 1185–1201.

Tilton, M. L.

M. L. Tilton, G. Dente, A. H. Paxton, J. Cser, R. K. DeFreez, C. E. Moeller, D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of the Symmetric Optical System (Academic, New York, 1974).

Whinnery, J. R.

J. R. Whinnery, D. T. Miller, F. Dabby, “Thermal convection and spherical aberration distortion of laser beams in low-loss liquids,” Appl. Phys. Lett. 13, 284–286 (1968).

F. W. Dabby, J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” IEEE J. Quantum Electron. QE-3, 382–383 (1968).

Appl. Opt.

Appl. Phys. Lett.

J. R. Whinnery, D. T. Miller, F. Dabby, “Thermal convection and spherical aberration distortion of laser beams in low-loss liquids,” Appl. Phys. Lett. 13, 284–286 (1968).

IEEE J. Quantum Electron.

M. L. Tilton, G. Dente, A. H. Paxton, J. Cser, R. K. DeFreez, C. E. Moeller, D. Depatie, “High power, nearly diffraction-limited output from a semiconductor laser with an unstable resonator,” IEEE J. Quantum Electron. 27, 2098–2108 (1991).
[CrossRef]

F. W. Dabby, J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” IEEE J. Quantum Electron. QE-3, 382–383 (1968).

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

Laser Focus

L. Marshall, “Applications a la mode,” Laser Focus 4(4), 26–28 (1971).

Opt. Commun.

R. Martinez-Herrero, P. M. Mejias, “Quality improvements of symmetric-intensity beams propagating through pure phase plates,” Opt. Commun. 95, 1–3 (1993).
[CrossRef]

Opt. Eng.

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29, 343–350 (1990); C. A. Klein, “Optical distortion coefficients of laser windows: one more time,” in Mirrors and Windows for High Power/High Energy Laser Systems, C. A. Klein, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1047, 58–79 (1989).
[CrossRef]

Opt. Lett.

Other

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Secs. 16.7 and 18.4.

A. E. Siegman, J. Ruff, “Effects of spherical aberration on laser beam quality,” in Laser Energy Distribution Profiles: Measurement and Applications, J. M. Darchuk, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1834, 100–103 (1992).

I. Miyamoto, “Analysis of thermally induced optical distortion in lens during focusing high power CO2 laser beam,” in CO2 Lasers and Applications II, H. Opower, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1276, 112–121 (1990).

M. W. Sasnett, “Propagation of multimode laser beams: the M2 factor,” in The Physics and Technology of Laser Resonators, D. R. Hall, P. E. Jackson, eds. (Hilger, London, 1989), pp. 132–142.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Soc. Photo-Opt. Instrum. Eng.1224, 2–14 (1990).

M. W. Sasnett, T. F. Johnston, “Beam characterization and measurement of propagation attributes,” in Laser Beam Diagnostics, R. N. Hindy, Y. Kohanzadeh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1414, 21–32 (1991).

H. H. Hopkins, Wave Theory of Aberrations (Oxford University, London, 1950).

W. T. Welford, Aberrations of the Symmetric Optical System (Academic, New York, 1974).

V. N. Mahajan, Aberration Theory Made Simple (Society of Photo-Optical Instrumentation Engineers, Bellingham, WA, 1991).
[CrossRef]

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, Sect. E3, pp. 1185–1201.

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Figures (7)

Fig. 1
Fig. 1

Analytical model for evaluating the effect of quartic phase aberrations on the beam-quality factor, M2, of an arbitrary laser beam profile ũ(x, y).

Fig. 2
Fig. 2

(a) Collimating output mirror as used in many laser devices. (b) Degradation in beam-quality factor for a TEM00 Gaussian beam passing such a mirror as a function of the Gaussian spot size, w, for various values of the beam (and mirror) radius of curvature R.

Fig. 3
Fig. 3

(a) Plano–convex thin lens of focal length f. (b) Increase in the beam-quality factor, M2, versus the Gaussian spot size, w, at the lens for a Gaussian laser beam being collimated or focused by this lens used the wrong way.

Fig. 4
Fig. 4

Dimensionless aberration coefficient C4f for a thin lens as a function of the lens-shape factor, q, and the imaging parameter, p.

Fig. 5
Fig. 5

Laser beam quality factor versus stripe width D for an unstable resonator mode with internal radius of curvature R exiting through the planar-cleaved facet of a diode laser assuming refractive index n = 3.6 and wavelength λ = 800 nm.

Fig. 6
Fig. 6

Stripe width Dq, above which the beam-quality factor deteriorates as (D/Dq)4, and also the critical width, Dc, for total internal reflection, versus internal radius of curvature R in a diode laser with planar output face, using the same assumptions as in Fig. 5.

Fig. 7
Fig. 7

Experimental results for the beam-quality degradation produced by spherical aberration in a thin plano–convex lens as a function of the Gaussian beam spot size of the beam passing through the lens. The solid curves are obtained from the theoretical expressions of this paper.

Tables (1)

Tables Icon

Table 1 Quartic Aberration Analysis: Values of βr and βx

Equations (34)

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u ˜ ( r , θ ) = u ˜ 0 ( r , θ ) × exp [ j 2 π λ ( r 2 2 F - C 4 r 4 ) ] .
P ˜ ( s x , x y ) = - - u ˜ ( x , y ) exp [ + j 2 π ( s x x + s y y ) ] d x d y ,
s x 2 ¯ = - - s x 2 P ( s x , s y ) 2 d s x d s y = ( 1 2 π ) 2 - - | u ˜ ( x , y ) x | 2 d x d y
p 2 ¯ = ( 1 2 π ) 2 0 2 π 0 [ | u ˜ ( r , θ ) r | 2 + 1 r 2 | u ˜ ( r , θ ) θ | 2 ] r d r d θ ,
p 2 ¯ = p 0 2 ¯ - ( 2 r 2 ¯ R 0 λ 2 + 8 C 4 r 4 ¯ λ 2 ) ( 1 F ) + r 2 ¯ λ 2 ( 1 F ) 2 + 4 r 3 ϕ 0 ( r ) ¯ π λ C 4 + 16 r 6 ¯ λ 2 C 4 2 .
1 R 0 - j λ 4 π r 2 ¯ 0 2 π 0 ( r u ˜ 0 u ˜ 0 * r - r u ˜ 0 * u ˜ 0 r ) r d r d θ .
1 R 0 = λ 2 π r 2 ¯ 0 2 π 0 [ r ϕ 0 ( r , θ ) r ] u ˜ 0 ( r , θ ) 2 r d r d θ .
r 3 ϕ 0 ( r ) ¯ 0 2 π 0 r 3 ϕ 0 ( r , θ ) r u ˜ 0 ( r , θ ) 2 r d r d θ .
1 F | collimated = 1 R 0 + 4 r 4 ¯ r 2 ¯ C 4 .
p 2 ¯ = p 0 2 ¯ - r 2 R 0 2 λ 2 + [ 4 r 3 ϕ 0 ( r ) ¯ π λ - 8 r 4 ¯ R 0 λ 2 ] C 4 + 16 λ 2 ( r 2 ¯ r 6 ¯ - r 4 ¯ r 2 ) C 4 2 .
p 2 ¯ = ( p 0 2 ¯ - r 2 ¯ R 0 2 λ 2 ) + 16 r 2 ¯ λ 2 ( r 2 ¯ r 6 ¯ - r 4 ¯ 2 r 4 ¯ 2 ) ( C 4 r 4 ) .
β r 2 ( r 2 ¯ r 6 ¯ - r 4 ¯ ) 2 r 4 ¯ 2 .
M r 2 = [ ( M r 0 2 ) 2 + ( M r q 2 ) 2 ] 1 / 2 ,
M r q 2 = 8 π β r λ C 4 r 4 ¯ ,
u ˜ ( x ) = u ˜ 0 ( x ) exp [ j ( π x 2 F λ - 2 π C 4 x 4 λ ) ] .
s x 2 ¯ = - s x 2 P ( s x ) 2 d s x = ( 1 / 2 π ) 2 - u ˜ ( x ) / x 2 d x .
1 F | collimated = 1 R 0 x + 4 x 4 ¯ x 2 C 4 ,
s x 2 ¯ = ( s x 0 2 ¯ - x 2 ¯ R 0 2 λ 2 ) + 16 λ 2 β x 2 C 4 2 ,
β x 2 ( x 2 x 6 ¯ - x 4 ¯ ) 2 ( x 4 ¯ ) 2 .
M x 2 = [ ( M x 0 2 ) 2 + ( M x q 2 ) 2 ] 1 / 2 ,
M x q 2 = 16 π β x λ C 4 x 4 ¯ .
C 4 = n 1 ( n 1 2 - n 2 2 ) 8 n 2 2 ( 1 R - 1 R s ) 2 ( 1 R - n 1 n 1 + n 2 1 R s ) = n 8 ( n - 1 ) 2 R 3 .
M r q 2 = π n w 4 2 3 / 2 ( n - 1 ) 2 R 3 λ = ( w w q ) 4 ,
w q = [ 2 3 / 2 ( n - 1 ) 2 R 3 λ π n ] 1 / 4 .
C 4 = n 3 + ( 3 n + 2 ) ( n - 1 ) 2 p 2 + ( n + 2 ) q 2 + 4 ( n 2 - 1 ) p q 32 n ( n - 1 ) 2 f 3 = C 4 f f 3 ,
M r q 2 = 2 3 / 2 π C 4 f w 4 f 3 λ = ( w w q ) 4 .
w q ( f 3 λ 2 3 / 2 π C 4 f ) 1 / 4 .
M r q 2 = 0.1 C 4 f ( f # ) 3 × D λ .
M x q 2 = 16 π β x C 4 x x 4 ¯ λ ,
M x q 2 = π n ( n 2 - 1 ) D 4 20 ( 21 ) 1 / 2 R 3 λ = ( D D q ) 4 .
M c n + 1 n - 1 1.8.
R = M - M c M - 1 L ,
R s = 2 ( M - M c ) ( M - 1 ) ( M c + 1 ) L ,
C 4 = ( M c 2 - 1 ) ( M - 1 ) 3 32 ( M c - M ) L 3 .

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