Abstract

FM-to-AM conversion is an important issue that could prevent fusion ignition with high-power lasers, such as the Laser MegaJoule (LMJ). We first overview the whole problem of FM-to-AM conversion in high-power lasers and we explain why AM spectral content of FM-to-AM conversion is important, although this information was not used in previous studies. We then propose simple analytical models to simulate FM-to-AM conversion in the LMJ frequency conversion system. We succeed in isolating every cause of spectrum distortion and give, for each of them, FM-to-AM predictions that are in very good agreement with simulations of a complex propagation code. Finally, we show how the last grating filters most of the FM-to-AM conversion. We conclude that the FM-to-AM conversion distortion criterion will be, on LMJ, below 40% in the last optics and 10% on the target.

© 2008 Optical Society of America

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References

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  1. www-lmj.cea.fr and www.llnl.gov/nif/project.
  2. J. E. Rothenberg, D. F. Browning, and R. B. Wilcox, “Issue of FM-to-AM conversion on the National Ignition Facility,” Proc. SPIE 3492, 51-61 (1999).
    [CrossRef]
  3. J. D. Lindl, P. Amedt, R. H. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Hann, R. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on National Ignition Facility,” Phys. Plasmas 11, (2004).
    [CrossRef]
  4. D. Penninckx, N. Beck, J.-F. Gleyze, and L. Videau, “Signal propagation over polarization-maintaining fibers: problem and solutions,” J. Lightwave Technol. 24, 4197-4207 (2006).
    [CrossRef]
  5. B. Lyot, “Le filtre monochromatique polarisant,” Annales d'astrophysique (1944).
  6. A. Boscheron, “Etude de nouvelles configurations de conversion de fréquence pour l'optimisation des lasers de haute puissance,” PhD dissertation (Université Paris XI,1996).
  7. E. Hugonnot, G. Deschaseaux, O. Hartmann, and H. Coïc, “Design of PETAL multipetawatt high-energy laser front end based on optical parametric chirped pulse amplification,” Appl. Opt. 46, 8181-8187 (2007).
    [CrossRef]
  8. O. Morice, “Miró: complete modeling and software for pulse amplification and propagation in high-power laser systems,” Opt. Eng. 42, 1530-1541 (2003).
    [CrossRef]
  9. J. R. Murray, J. R. Smith, R. B. Ehrlich, D. T. Karazys, C. E. Thompson, T. L. Weiland, and R. B. Wilcox, “Experimental observation and suppression of transverse stimulated Brillouin scattering in large optical components,” J. Opt. Soc. Am. B 6, 2402-2411 (1989).
  10. J. Garnier, L. Videau, C. Gouédard, and A. Migus, “Statistical analysis for beam smoothing and some applications,” J. Opt. Soc. Am. A 14, 1928-1937 (1997).
    [CrossRef]
  11. R. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2002).
  12. J. R. Carson, “Notes on the theory of modulation,” Proc. IRE 10, 57-64 (1922).
    [CrossRef]
  13. B. Wedding, “New method for optical transmission beyond dispersion limit,” Electron. Lett. 28, 1298-1300 (1992).
    [CrossRef]
  14. D. Penninckx, J.-M. Di Nicola, J.-F. Gleyze, and S. Hocquet, “Paradox in the measurement of FM-to-AM conversion in high-power lasers,” CG-5-WED, presented at CLEO Europe, Munich, Germany, 17-22 June, 2007.
  15. V. D. Volosov, S. G. Karpenko, N. E. Kornienko, and V. L. Strishevkii,“Method for compensating the phase-matching dispersion in nonlinear optics,” Sov. J. Quantum Electron. 4, 1090-1098 (1975).
    [CrossRef]
  16. E. B. Treacy: “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5, 454-458 (1969).
    [CrossRef]
  17. J. Néauport, N. Blanchot, C. Rouyer, and C. Sauteret, “Chromatism compensation of the PETAL multipetawatt high energy laser,” Appl. Opt. 46, 1568-1574 (2007)
  18. J. E. Rothenberg, “Ultrafast picket fence pulse trains to enhance frequency conversion of shaped inertial confinement fusion laser pulses,” Appl. Opt. 39, 6931-6938 (2000).
    [CrossRef]
  19. S. Hocquet, E. Bordenave, J.-P. Goossens, C. Gouedard, L. Videau, and D. Penninckx, “Amplitude modulation filtering of FM-to-AM conversion due to the focusing grating of LMJ,” in IFSA 2007 proceedings.
  20. Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (Dover, 1965).

2007

2006

2004

J. D. Lindl, P. Amedt, R. H. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Hann, R. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on National Ignition Facility,” Phys. Plasmas 11, (2004).
[CrossRef]

2003

O. Morice, “Miró: complete modeling and software for pulse amplification and propagation in high-power laser systems,” Opt. Eng. 42, 1530-1541 (2003).
[CrossRef]

2002

R. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2002).

2000

1999

J. E. Rothenberg, D. F. Browning, and R. B. Wilcox, “Issue of FM-to-AM conversion on the National Ignition Facility,” Proc. SPIE 3492, 51-61 (1999).
[CrossRef]

1997

1996

A. Boscheron, “Etude de nouvelles configurations de conversion de fréquence pour l'optimisation des lasers de haute puissance,” PhD dissertation (Université Paris XI,1996).

1992

B. Wedding, “New method for optical transmission beyond dispersion limit,” Electron. Lett. 28, 1298-1300 (1992).
[CrossRef]

1989

1975

V. D. Volosov, S. G. Karpenko, N. E. Kornienko, and V. L. Strishevkii,“Method for compensating the phase-matching dispersion in nonlinear optics,” Sov. J. Quantum Electron. 4, 1090-1098 (1975).
[CrossRef]

1969

E. B. Treacy: “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5, 454-458 (1969).
[CrossRef]

1965

Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (Dover, 1965).

1944

B. Lyot, “Le filtre monochromatique polarisant,” Annales d'astrophysique (1944).

1922

J. R. Carson, “Notes on the theory of modulation,” Proc. IRE 10, 57-64 (1922).
[CrossRef]

Amedt, P.

J. D. Lindl, P. Amedt, R. H. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Hann, R. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on National Ignition Facility,” Phys. Plasmas 11, (2004).
[CrossRef]

Beck, N.

Berger, R. H.

J. D. Lindl, P. Amedt, R. H. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Hann, R. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on National Ignition Facility,” Phys. Plasmas 11, (2004).
[CrossRef]

Blanchot, N.

Bordenave, E.

S. Hocquet, E. Bordenave, J.-P. Goossens, C. Gouedard, L. Videau, and D. Penninckx, “Amplitude modulation filtering of FM-to-AM conversion due to the focusing grating of LMJ,” in IFSA 2007 proceedings.

Boscheron, A.

A. Boscheron, “Etude de nouvelles configurations de conversion de fréquence pour l'optimisation des lasers de haute puissance,” PhD dissertation (Université Paris XI,1996).

Boyd, R.

R. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2002).

Browning, D. F.

J. E. Rothenberg, D. F. Browning, and R. B. Wilcox, “Issue of FM-to-AM conversion on the National Ignition Facility,” Proc. SPIE 3492, 51-61 (1999).
[CrossRef]

Carson, J. R.

J. R. Carson, “Notes on the theory of modulation,” Proc. IRE 10, 57-64 (1922).
[CrossRef]

Coïc, H.

Deschaseaux, G.

Di Nicola, J.-M.

D. Penninckx, J.-M. Di Nicola, J.-F. Gleyze, and S. Hocquet, “Paradox in the measurement of FM-to-AM conversion in high-power lasers,” CG-5-WED, presented at CLEO Europe, Munich, Germany, 17-22 June, 2007.

Ehrlich, R. B.

Garnier, J.

Glendinning, S. G.

J. D. Lindl, P. Amedt, R. H. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Hann, R. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on National Ignition Facility,” Phys. Plasmas 11, (2004).
[CrossRef]

Glenzer, S. H.

J. D. Lindl, P. Amedt, R. H. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Hann, R. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on National Ignition Facility,” Phys. Plasmas 11, (2004).
[CrossRef]

Gleyze, J.-F.

D. Penninckx, N. Beck, J.-F. Gleyze, and L. Videau, “Signal propagation over polarization-maintaining fibers: problem and solutions,” J. Lightwave Technol. 24, 4197-4207 (2006).
[CrossRef]

D. Penninckx, J.-M. Di Nicola, J.-F. Gleyze, and S. Hocquet, “Paradox in the measurement of FM-to-AM conversion in high-power lasers,” CG-5-WED, presented at CLEO Europe, Munich, Germany, 17-22 June, 2007.

Goossens, J.-P.

S. Hocquet, E. Bordenave, J.-P. Goossens, C. Gouedard, L. Videau, and D. Penninckx, “Amplitude modulation filtering of FM-to-AM conversion due to the focusing grating of LMJ,” in IFSA 2007 proceedings.

Gouedard, C.

S. Hocquet, E. Bordenave, J.-P. Goossens, C. Gouedard, L. Videau, and D. Penninckx, “Amplitude modulation filtering of FM-to-AM conversion due to the focusing grating of LMJ,” in IFSA 2007 proceedings.

Gouédard, C.

Hann, S. W.

J. D. Lindl, P. Amedt, R. H. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Hann, R. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on National Ignition Facility,” Phys. Plasmas 11, (2004).
[CrossRef]

Hartmann, O.

Hocquet, S.

D. Penninckx, J.-M. Di Nicola, J.-F. Gleyze, and S. Hocquet, “Paradox in the measurement of FM-to-AM conversion in high-power lasers,” CG-5-WED, presented at CLEO Europe, Munich, Germany, 17-22 June, 2007.

S. Hocquet, E. Bordenave, J.-P. Goossens, C. Gouedard, L. Videau, and D. Penninckx, “Amplitude modulation filtering of FM-to-AM conversion due to the focusing grating of LMJ,” in IFSA 2007 proceedings.

Hugonnot, E.

Karazys, D. T.

Karpenko, S. G.

V. D. Volosov, S. G. Karpenko, N. E. Kornienko, and V. L. Strishevkii,“Method for compensating the phase-matching dispersion in nonlinear optics,” Sov. J. Quantum Electron. 4, 1090-1098 (1975).
[CrossRef]

Kornienko, N. E.

V. D. Volosov, S. G. Karpenko, N. E. Kornienko, and V. L. Strishevkii,“Method for compensating the phase-matching dispersion in nonlinear optics,” Sov. J. Quantum Electron. 4, 1090-1098 (1975).
[CrossRef]

Landen, R. L.

J. D. Lindl, P. Amedt, R. H. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Hann, R. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on National Ignition Facility,” Phys. Plasmas 11, (2004).
[CrossRef]

Lindl, J. D.

J. D. Lindl, P. Amedt, R. H. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Hann, R. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on National Ignition Facility,” Phys. Plasmas 11, (2004).
[CrossRef]

Lyot, B.

B. Lyot, “Le filtre monochromatique polarisant,” Annales d'astrophysique (1944).

Migus, A.

Morice, O.

O. Morice, “Miró: complete modeling and software for pulse amplification and propagation in high-power laser systems,” Opt. Eng. 42, 1530-1541 (2003).
[CrossRef]

Murray, J. R.

Néauport, J.

Penninckx, D.

D. Penninckx, N. Beck, J.-F. Gleyze, and L. Videau, “Signal propagation over polarization-maintaining fibers: problem and solutions,” J. Lightwave Technol. 24, 4197-4207 (2006).
[CrossRef]

D. Penninckx, J.-M. Di Nicola, J.-F. Gleyze, and S. Hocquet, “Paradox in the measurement of FM-to-AM conversion in high-power lasers,” CG-5-WED, presented at CLEO Europe, Munich, Germany, 17-22 June, 2007.

S. Hocquet, E. Bordenave, J.-P. Goossens, C. Gouedard, L. Videau, and D. Penninckx, “Amplitude modulation filtering of FM-to-AM conversion due to the focusing grating of LMJ,” in IFSA 2007 proceedings.

Rothenberg, J. E.

J. E. Rothenberg, “Ultrafast picket fence pulse trains to enhance frequency conversion of shaped inertial confinement fusion laser pulses,” Appl. Opt. 39, 6931-6938 (2000).
[CrossRef]

J. E. Rothenberg, D. F. Browning, and R. B. Wilcox, “Issue of FM-to-AM conversion on the National Ignition Facility,” Proc. SPIE 3492, 51-61 (1999).
[CrossRef]

Rouyer, C.

Sauteret, C.

Smith, J. R.

Strishevkii, V. L.

V. D. Volosov, S. G. Karpenko, N. E. Kornienko, and V. L. Strishevkii,“Method for compensating the phase-matching dispersion in nonlinear optics,” Sov. J. Quantum Electron. 4, 1090-1098 (1975).
[CrossRef]

Suter, L. J.

J. D. Lindl, P. Amedt, R. H. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Hann, R. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on National Ignition Facility,” Phys. Plasmas 11, (2004).
[CrossRef]

Thompson, C. E.

Treacy, E. B.

E. B. Treacy: “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5, 454-458 (1969).
[CrossRef]

Videau, L.

Volosov, V. D.

V. D. Volosov, S. G. Karpenko, N. E. Kornienko, and V. L. Strishevkii,“Method for compensating the phase-matching dispersion in nonlinear optics,” Sov. J. Quantum Electron. 4, 1090-1098 (1975).
[CrossRef]

Wedding, B.

B. Wedding, “New method for optical transmission beyond dispersion limit,” Electron. Lett. 28, 1298-1300 (1992).
[CrossRef]

Weiland, T. L.

Wilcox, R. B.

Annales d'astrophysique

B. Lyot, “Le filtre monochromatique polarisant,” Annales d'astrophysique (1944).

Appl. Opt.

Electron. Lett.

B. Wedding, “New method for optical transmission beyond dispersion limit,” Electron. Lett. 28, 1298-1300 (1992).
[CrossRef]

IEEE J. Quantum Electron.

E. B. Treacy: “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5, 454-458 (1969).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Eng.

O. Morice, “Miró: complete modeling and software for pulse amplification and propagation in high-power laser systems,” Opt. Eng. 42, 1530-1541 (2003).
[CrossRef]

Phys. Plasmas

J. D. Lindl, P. Amedt, R. H. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Hann, R. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on National Ignition Facility,” Phys. Plasmas 11, (2004).
[CrossRef]

Proc. IRE

J. R. Carson, “Notes on the theory of modulation,” Proc. IRE 10, 57-64 (1922).
[CrossRef]

Proc. SPIE

J. E. Rothenberg, D. F. Browning, and R. B. Wilcox, “Issue of FM-to-AM conversion on the National Ignition Facility,” Proc. SPIE 3492, 51-61 (1999).
[CrossRef]

Sov. J. Quantum Electron.

V. D. Volosov, S. G. Karpenko, N. E. Kornienko, and V. L. Strishevkii,“Method for compensating the phase-matching dispersion in nonlinear optics,” Sov. J. Quantum Electron. 4, 1090-1098 (1975).
[CrossRef]

Other

D. Penninckx, J.-M. Di Nicola, J.-F. Gleyze, and S. Hocquet, “Paradox in the measurement of FM-to-AM conversion in high-power lasers,” CG-5-WED, presented at CLEO Europe, Munich, Germany, 17-22 June, 2007.

R. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2002).

www-lmj.cea.fr and www.llnl.gov/nif/project.

A. Boscheron, “Etude de nouvelles configurations de conversion de fréquence pour l'optimisation des lasers de haute puissance,” PhD dissertation (Université Paris XI,1996).

S. Hocquet, E. Bordenave, J.-P. Goossens, C. Gouedard, L. Videau, and D. Penninckx, “Amplitude modulation filtering of FM-to-AM conversion due to the focusing grating of LMJ,” in IFSA 2007 proceedings.

Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (Dover, 1965).

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Figures (13)

Fig. 1
Fig. 1

Temporal measurement of the power on the LIL, prototype of the LMJ (black curve and red curve are from two different detectors). The beam was initially spectrally shaped with a phase modulation at 2 GHz , m = 7 rad . Because of FM-to-AM conversion, the initial flat-top pulse shape is modulated.

Fig. 2
Fig. 2

Functional diagram of an LMJ beam. The laser pulse is shaped temporally and spectrally in the “source.” Then the pulse is amplified in the MPA and in the amplification chain. The transport section is composed of dielectric mirrors. Its function is to propagate the beam to the SCF that converts the beam into UV light. The SCF also focuses the beam to the target.

Fig. 3
Fig. 3

(a) Optical spectrum for the 2 GHz sinusoidal phase modulation with an index m of 7 (also called anti-Brillouin modulation). (b) Optical spectrum for both modulations used on the LMJ: anti-Brillouin modulation and smoothing modulation at the frequency of 14.25 GHz and a modulation index of 5.

Fig. 4
Fig. 4

Theoretical transfer function H (in blue) applied on a phase modulation (in green with f m = 14.25 GHz , m = 5 ). Top line: Gaussian bandpass filter with a 170 GHz FWHM (see Section 3B for another similar filter). Bottom line: Chromatic dispersion. For example, dispersion in a 200 m long fiber at 1053 nm (dispersion parameter of 11 ps / nm ) (see Section 3C for another similar filter) The two AM spectra are totally different, although the resulting values of distortion criterion α are about the same.

Fig. 5
Fig. 5

(a) Anti-Brillouin modulation ( f m = 2 GHz , m = 7 ) is filtered by a Hermitian transfer function. (b) Resulting AM measured (insets: AM spectrum associated) with a 5 GHz photodiode and a 6 GHz bandwidth oscilloscope. There is no AM at 2 GHz ; because of the filter, only even harmonics of frequency modulation can exist (here, 4 GHz ).

Fig. 6
Fig. 6

Experimental results of FM-to-AM conversion on 23 consecutive shots on LIL. The measured values of α are among the lowest when anti-Brillouin (solid squares) and smoothing (empty squares) phase modulations are both activated, although FM-to-AM is supposed to be higher for a larger spectrum.

Fig. 7
Fig. 7

AM is reduced at 2 GHz when the spectrum is broadened with the second phase modulation at 14.25 GHz .

Fig. 8
Fig. 8

Simple description of the SCF on LMJ. Crystals convert IR into UV. The plane grating improves efficiency conversion by dispersion; phase mismatch over the spectrum is considerably lowered. The focusing grating separates UV light from the first and second harmonics and it focuses the beam to the target.

Fig. 9
Fig. 9

Miró simulations for FM-to-AM conversion in THG (case of a perfect phase matching overall the spectrum). Amplification of AM is always verified, but the value of amplification decreases with the intensity. Curves are predictions with the model α 3 ω = β α 1 ω where β is defined by the fact that I 3 ω I 1 ω β . Differences come from the fact that β instantaneously depends on the intensity.

Fig. 10
Fig. 10

(a) Spectral acceptance for the frequency conversion doubler–tripler system at 2 GW / cm 2 . x: case without plane grating and +, case with plane grating (LMJ). (b) AM resulting from spectral acceptance transfer function. Red curve corresponds to analytical formula results according to Eq. (12).

Fig. 11
Fig. 11

FM-to-AM conversion due to propagation after 1 ω plane grating: Miró simulations (blue triangles) versus analytical model according to Eq. (16) (red curve) for a phase modulation m = 5 rad at 14.25 GHz . Distortion criterion is proportional to the distance propagation with a coefficient of 13 % m 1 . Green squares consider also the effects of crystals on FM-to-AM conversion. Green dashed curve is the analytical model according to Eq. (18) considering the addition of different effects: for long propagation distance, the crystals’ effect is negligible.

Fig. 12
Fig. 12

Simple schemes of (a) TSSD and (b) LSSD (LMJ configuration). In both cases, integration of intensity is due to time delay Δ T , so AM is lowered in the focal plane.

Fig. 13
Fig. 13

(a) AM spectrum transfer function: comparison between Miró simulations and numerical model according to Eq. (19) and a low-pass Gaussian filter with a 7 GHz bandwidth. (b) Details of temporal shapes before and after the focusing grating in the LMJ configuration (with two sine phase modulations at 2 and 14.25 GHz ). It illustrates the AM filtering (no reference level).

Tables (2)

Tables Icon

Table 1 Examples of Analytical Values of Distortion Criterion α Due to Simple Filters for a Sinusoidal Phase Modulation at Frequency f m with Modulation Index m a

Tables Icon

Table 2 Distortion Criterion for Frequency Conversion Systems: Miró Results for Different Configurations a

Equations (27)

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

a 0 ( t ) = exp [ i m sin ( 2 π f m t ) ] .
a ˜ 0 ( f ) = n = + J n ( m ) δ ( f n f m ) ,
α = 2 I max I min I max + I min ,
I ˜ out ( k f m ) = n = + H * ( n f m ) H [ ( k n ) f m ] J n ( m ) J k n ( m ) .
I ˜ out ( k f m ) = ( 1 ) n = ( k 1 ) 2 + { H * ( n f m ) H [ ( n + k ) f m ] H * ( n f m ) H [ ( n + k ) f m ] } J n ( m ) J n + k ( m ) .
n ( k + 1 ) 2 , H * ( n f m ) H [ ( n + k ) f m ] H * ( n f m ) H [ ( n + k ) f m ] = 0.
x R , H * ( x ) = ± H ( x ) .
H ( f ) = 1 + A exp [ i ( 2 π f Δ τ + ψ ) ] ,
α 3 ω = β α 1 ω .
A 3 ( Δ k ) = A 3 max L e i Δ k L 1 i Δ k = A 3 max e i Δ k L 2 Sinc ( Δ k L 2 ) ,
H c ( f ) = Sinc [ γ ( f f c ) ] 1 γ 2 6 ( f f c ) 2 ,
γ = L 2 ( Δ k f ) 3 ω .
α c = γ 2 3 ( f c + 3 m f m ) 2 1 γ 2 6 ( f c + 3 m f m ) 2 .
φ ( f ) = φ 0 + φ 1 f + 1 2 φ 2 f 2 + ο [ f 2 ] .
H d ( f ) exp [ i 2 φ 2 f 2 ] 1 + i 2 φ 2 f 2 .
φ 2 = 2 π c N 2 x f 0 3 cos 2 ( θ 0 ) .
α d ( D ) = 2 | φ 2 ( x ) | m f m 2 = 4 π c N 2 f 0 3 cos 2 ( θ 0 ) m f m 2 x .
α sum ( x ) = α c 2 + α d 2 ( x ) .
α N F ( x ) = [ β α d ( l ) ] 2 + α c 2 + α d 2 ( x l ) .
P FF ( t ) = Near Field I NF [ t δ t ( x , y ) ] d x d y ,
P FF ( t ) = S [ 1 + Sinc ( π f AM Δ T ) sin ( 2 π f AM ( t Δ T 2 ) ] ,
a ˜ out ( f ) = n = + H ( n f m ) J n ( m ) δ ( f n f m ) .
I ˜ out ( f ) = F T [ I out ( t ) ] = F T [ | a out ( t ) | 2 ] = F T [ a out ( t ) ] * F T [ a out * ( t ) ] .
I ˜ out ( f ) = [ k = + H ( k f m ) J k ( m ) δ ( f k f m ) ] * [ n = + H * ( n f m ) J n ( m ) δ ( f + n f m ) ] ,
I ˜ out ( f ) = k = + { n = + H * ( n f m ) H [ ( k n ) f m ] J n ( m ) J k n ( m ) } δ ( f k f m ) .
I ˜ out ( k f m ) = n = + H * ( n f m ) H [ ( k n ) f m ] J n ( m ) J k n ( m ) .
I ˜ out ( k f m ) = n = ( k 1 ) 2 + { [ H * ( n f m ) H ( ( n + k ) f m ) J n ( m ) J n + k ( m ) ] + [ H * ( n f m ) H ( ( n + k ) f m ) J n ( m ) J ( n + k ) ( m ) ] } .

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