Abstract

We demonstrate an extremely simple frequency-resolved-optical-gating (GRENOUILLE) device for measuring the intensity and phase of relatively long—ps—pulses. In order to achieve the required high spectral resolution and large temporal range, it uses a few-cm-thick second-harmonic-generation crystal in the shape of a pentagon. This has the additional advantage of reducing the device’s total number of components to as few as three simple easily aligned optics, making it the simplest device ever developed for complete pulse measurement. We report complete intensity-and-phase measurements of pulses up to 15ps long with a time-bandwidth product of 21.

© 2010 OSA

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References

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  1. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
    [CrossRef]
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    [CrossRef]
  3. P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett. 26(12), 932–934 (2001).
    [CrossRef]
  4. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic Publishers, 2002).
  5. C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23(10), 792–794 (1998).
    [CrossRef]
  6. C. Dorrer, B. de Beauvoir, C. Le Blanc, S. Ranc, J. P. Rousseau, P. Rousseau, J. P. Chambaret, and F. Salin, “Single-shot real-time characterization of chirped-pulse amplification systems by spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 24(22), 1644–1646 (1999).
    [CrossRef]
  7. C. Dorrer, N. Belabas, J.-P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17(10), 1795–1802 (2000).
    [CrossRef]
  8. J. R. Birge, R. Ell, and F. X. Kärtner, “Two-dimensional spectral shearing interferometry for few-cycle pulse characterization,” Opt. Lett. 31(13), 2063–2065 (2006).
    [CrossRef] [PubMed]
  9. M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
    [CrossRef] [PubMed]
  10. D. H. Broaddus, M. A. Foster, O. Kuzucu, A. C. Turner-Foster, K. W. Koch, M. Lipson, and A. L. Gaeta, “Temporal-imaging system with simple external-clock triggering,” Opt. Express 18(13), 14262–14269 (2010).
    [CrossRef] [PubMed]
  11. C. Froehly, A. Lacourt, and J. C. Vienot, “Time impulse response and time frequency response of optical pupils: experimental confirmations and applications,” Nouv. Rev. Opt. 4(4), 183–196 (1973).
    [CrossRef]
  12. L. Lepetit, G. Cheriaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12(12), 2467–2474 (1995).
    [CrossRef]
  13. P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, “Crossed-beam spectral interferometry: a simple, high-spectral-resolution method for completely characterizing complex ultrashort pulses in real time,” Opt. Express 14(24), 11892–11900 (2006).
    [CrossRef] [PubMed]
  14. J. Cohen, P. Bowlan, V. Chauhan, and R. Trebino, “Measuring temporally complex ultrashort pulses using multiple-delay crossed-beam spectral interferometry,” Opt. Express 18(7), 6583–6597 (2010).
    [CrossRef] [PubMed]
  15. C. Dorrer and I. Kang, “Linear self-referencing techniques for short-optical-pulse characterization [Invited],” J. Opt. Soc. Am. B 25(6), A1–A12 (2008).
    [CrossRef]
  16. X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27(13), 1174–1176 (2002).
    [CrossRef]
  17. H. Fuchs, D. Woll, T. Ulm, and J. A. L'Huillier, “High resolution FROG system for the characterization of ps laser pulses,” Appl. Phys. B 88(3), 393–396 (2007).
    [CrossRef]
  18. C. Radzewicz, P. Wasylczyk, and J. S. Krasinski, “A poor man's FROG,” Opt. Commun. 186(4-6), 329–333 (2000).
    [CrossRef]
  19. S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Extremely simple device for measuring 20-fs pulses,” Opt. Lett. 29(9), 1025–1027 (2004).
    [CrossRef] [PubMed]
  20. S. Akturk, M. Kimmel, and R. Trebino, “Extremely simple device for measuring 1.5-microm ultrashort laser pulses,” Opt. Express 12(19), 4483–4489 (2004).
    [CrossRef] [PubMed]
  21. S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE,” Opt. Express 11(5), 491–501 (2003).
    [CrossRef] [PubMed]
  22. D. Lee, and R. Trebino, “Extremely simple device for measuring ultrashort pulses in the visible,” in Lasers and Electro-Optics, 2009 and 2009 Conference on Quantum electronics and Laser Science Conference. CLEO/QELS 2009. Conference on, 2009), 1–2.
  23. D. Lee, Z. Wang, X. Gu, and R. Trebino, “Effect—and removal—of an ultrashort pulse’s spatial profile on the single-shot measurement of its temporal profile,” J. Opt. Soc. Am. B 25(6), A93 (2008).
    [CrossRef]
  24. P. O’Shea, S. Akturk, M. Kimmel, and R. Trebino, “Practical issues in ultra-short-pulse measurements with ‘GRENOUILLE’,” Appl. Phys. B 79(6), 683–691 (2004).
    [CrossRef]
  25. D. Lee, Z. Wang, X. Gu, and R. Trebino, “Effect—and removal—of an ultrashort pulse’s spatial profile on the single-shot measurement of its temporal profile,” J. Opt. Soc. Am. B 25(6), A93–A100 (2008).
    [CrossRef]
  26. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of nonlinear optical crystals, 3rd ed. (Springer-Verlag, Berlin, Germany, 1991).
  27. X. Liu, R. Trebino, and A. V. Smith, “Numerical Simulations of the Ultrasimple Ultrashort-Laser-Pulse Measurement Technique, GRENOUILLE,” in OSA Technical Digest Series (CD) (Optical Society of America, 2007), JThD8.
  28. P. A. Jansson and M. Richardson, “Deconvolution of Images and Spectra, 2nd Edition,” Opt. Eng. 36(11), 3224–3225 (1997).
    [CrossRef]

2010

2008

2007

H. Fuchs, D. Woll, T. Ulm, and J. A. L'Huillier, “High resolution FROG system for the characterization of ps laser pulses,” Appl. Phys. B 88(3), 393–396 (2007).
[CrossRef]

2006

2004

2003

2002

2001

2000

C. Dorrer, N. Belabas, J.-P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17(10), 1795–1802 (2000).
[CrossRef]

C. Radzewicz, P. Wasylczyk, and J. S. Krasinski, “A poor man's FROG,” Opt. Commun. 186(4-6), 329–333 (2000).
[CrossRef]

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
[CrossRef]

1999

1998

1997

P. A. Jansson and M. Richardson, “Deconvolution of Images and Spectra, 2nd Edition,” Opt. Eng. 36(11), 3224–3225 (1997).
[CrossRef]

1995

1973

C. Froehly, A. Lacourt, and J. C. Vienot, “Time impulse response and time frequency response of optical pupils: experimental confirmations and applications,” Nouv. Rev. Opt. 4(4), 183–196 (1973).
[CrossRef]

Akturk, S.

Belabas, N.

Birge, J. R.

Bowlan, P.

Broaddus, D. H.

Chambaret, J. P.

Chauhan, V.

Cheriaux, G.

Cohen, J.

de Beauvoir, B.

Dorrer, C.

Ell, R.

Foster, M. A.

D. H. Broaddus, M. A. Foster, O. Kuzucu, A. C. Turner-Foster, K. W. Koch, M. Lipson, and A. L. Gaeta, “Temporal-imaging system with simple external-clock triggering,” Opt. Express 18(13), 14262–14269 (2010).
[CrossRef] [PubMed]

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
[CrossRef] [PubMed]

Froehly, C.

C. Froehly, A. Lacourt, and J. C. Vienot, “Time impulse response and time frequency response of optical pupils: experimental confirmations and applications,” Nouv. Rev. Opt. 4(4), 183–196 (1973).
[CrossRef]

Fuchs, H.

H. Fuchs, D. Woll, T. Ulm, and J. A. L'Huillier, “High resolution FROG system for the characterization of ps laser pulses,” Appl. Phys. B 88(3), 393–396 (2007).
[CrossRef]

Gabolde, P.

Gaeta, A. L.

D. H. Broaddus, M. A. Foster, O. Kuzucu, A. C. Turner-Foster, K. W. Koch, M. Lipson, and A. L. Gaeta, “Temporal-imaging system with simple external-clock triggering,” Opt. Express 18(13), 14262–14269 (2010).
[CrossRef] [PubMed]

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
[CrossRef] [PubMed]

Geraghty, D. F.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
[CrossRef] [PubMed]

Gu, X.

Iaconis, C.

Jansson, P. A.

P. A. Jansson and M. Richardson, “Deconvolution of Images and Spectra, 2nd Edition,” Opt. Eng. 36(11), 3224–3225 (1997).
[CrossRef]

Joffre, M.

Kang, I.

Kärtner, F. X.

Kimmel, M.

Koch, K. W.

Krasinski, J. S.

C. Radzewicz, P. Wasylczyk, and J. S. Krasinski, “A poor man's FROG,” Opt. Commun. 186(4-6), 329–333 (2000).
[CrossRef]

Kuzucu, O.

Lacourt, A.

C. Froehly, A. Lacourt, and J. C. Vienot, “Time impulse response and time frequency response of optical pupils: experimental confirmations and applications,” Nouv. Rev. Opt. 4(4), 183–196 (1973).
[CrossRef]

Le Blanc, C.

Lee, D.

Lepetit, L.

L'Huillier, J. A.

H. Fuchs, D. Woll, T. Ulm, and J. A. L'Huillier, “High resolution FROG system for the characterization of ps laser pulses,” Appl. Phys. B 88(3), 393–396 (2007).
[CrossRef]

Likforman, J.-P.

Lipson, M.

D. H. Broaddus, M. A. Foster, O. Kuzucu, A. C. Turner-Foster, K. W. Koch, M. Lipson, and A. L. Gaeta, “Temporal-imaging system with simple external-clock triggering,” Opt. Express 18(13), 14262–14269 (2010).
[CrossRef] [PubMed]

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
[CrossRef] [PubMed]

McGresham, K.

O’Shea, P.

Radzewicz, C.

C. Radzewicz, P. Wasylczyk, and J. S. Krasinski, “A poor man's FROG,” Opt. Commun. 186(4-6), 329–333 (2000).
[CrossRef]

Ranc, S.

Richardson, M.

P. A. Jansson and M. Richardson, “Deconvolution of Images and Spectra, 2nd Edition,” Opt. Eng. 36(11), 3224–3225 (1997).
[CrossRef]

Rousseau, J. P.

Rousseau, P.

Salem, R.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
[CrossRef] [PubMed]

Salin, F.

Shreenath, A.

Shreenath, A. P.

Trebino, R.

J. Cohen, P. Bowlan, V. Chauhan, and R. Trebino, “Measuring temporally complex ultrashort pulses using multiple-delay crossed-beam spectral interferometry,” Opt. Express 18(7), 6583–6597 (2010).
[CrossRef] [PubMed]

D. Lee, Z. Wang, X. Gu, and R. Trebino, “Effect—and removal—of an ultrashort pulse’s spatial profile on the single-shot measurement of its temporal profile,” J. Opt. Soc. Am. B 25(6), A93 (2008).
[CrossRef]

D. Lee, Z. Wang, X. Gu, and R. Trebino, “Effect—and removal—of an ultrashort pulse’s spatial profile on the single-shot measurement of its temporal profile,” J. Opt. Soc. Am. B 25(6), A93–A100 (2008).
[CrossRef]

P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, “Crossed-beam spectral interferometry: a simple, high-spectral-resolution method for completely characterizing complex ultrashort pulses in real time,” Opt. Express 14(24), 11892–11900 (2006).
[CrossRef] [PubMed]

S. Akturk, M. Kimmel, and R. Trebino, “Extremely simple device for measuring 1.5-microm ultrashort laser pulses,” Opt. Express 12(19), 4483–4489 (2004).
[CrossRef] [PubMed]

P. O’Shea, S. Akturk, M. Kimmel, and R. Trebino, “Practical issues in ultra-short-pulse measurements with ‘GRENOUILLE’,” Appl. Phys. B 79(6), 683–691 (2004).
[CrossRef]

S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Extremely simple device for measuring 20-fs pulses,” Opt. Lett. 29(9), 1025–1027 (2004).
[CrossRef] [PubMed]

S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE,” Opt. Express 11(5), 491–501 (2003).
[CrossRef] [PubMed]

X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27(13), 1174–1176 (2002).
[CrossRef]

P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett. 26(12), 932–934 (2001).
[CrossRef]

Turner-Foster, A. C.

D. H. Broaddus, M. A. Foster, O. Kuzucu, A. C. Turner-Foster, K. W. Koch, M. Lipson, and A. L. Gaeta, “Temporal-imaging system with simple external-clock triggering,” Opt. Express 18(13), 14262–14269 (2010).
[CrossRef] [PubMed]

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
[CrossRef] [PubMed]

Ulm, T.

H. Fuchs, D. Woll, T. Ulm, and J. A. L'Huillier, “High resolution FROG system for the characterization of ps laser pulses,” Appl. Phys. B 88(3), 393–396 (2007).
[CrossRef]

Vienot, J. C.

C. Froehly, A. Lacourt, and J. C. Vienot, “Time impulse response and time frequency response of optical pupils: experimental confirmations and applications,” Nouv. Rev. Opt. 4(4), 183–196 (1973).
[CrossRef]

Walmsley, I. A.

Wang, Z.

Wasylczyk, P.

C. Radzewicz, P. Wasylczyk, and J. S. Krasinski, “A poor man's FROG,” Opt. Commun. 186(4-6), 329–333 (2000).
[CrossRef]

Weiner, A. M.

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
[CrossRef]

Windeler, R. S.

Woll, D.

H. Fuchs, D. Woll, T. Ulm, and J. A. L'Huillier, “High resolution FROG system for the characterization of ps laser pulses,” Appl. Phys. B 88(3), 393–396 (2007).
[CrossRef]

Xu, L.

Zeek, E.

Appl. Phys. B

H. Fuchs, D. Woll, T. Ulm, and J. A. L'Huillier, “High resolution FROG system for the characterization of ps laser pulses,” Appl. Phys. B 88(3), 393–396 (2007).
[CrossRef]

P. O’Shea, S. Akturk, M. Kimmel, and R. Trebino, “Practical issues in ultra-short-pulse measurements with ‘GRENOUILLE’,” Appl. Phys. B 79(6), 683–691 (2004).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

C. Dorrer, “High-speed measurements for optical telecommunication systems,” IEEE J. Sel. Top. Quantum Electron. 12(4), 843–858 (2006).
[CrossRef]

J. Opt. Soc. Am. B

Nature

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
[CrossRef] [PubMed]

Nouv. Rev. Opt.

C. Froehly, A. Lacourt, and J. C. Vienot, “Time impulse response and time frequency response of optical pupils: experimental confirmations and applications,” Nouv. Rev. Opt. 4(4), 183–196 (1973).
[CrossRef]

Opt. Commun.

C. Radzewicz, P. Wasylczyk, and J. S. Krasinski, “A poor man's FROG,” Opt. Commun. 186(4-6), 329–333 (2000).
[CrossRef]

Opt. Eng.

P. A. Jansson and M. Richardson, “Deconvolution of Images and Spectra, 2nd Edition,” Opt. Eng. 36(11), 3224–3225 (1997).
[CrossRef]

Opt. Express

Opt. Lett.

Rev. Sci. Instrum.

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
[CrossRef]

Other

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic Publishers, 2002).

D. Lee, and R. Trebino, “Extremely simple device for measuring ultrashort pulses in the visible,” in Lasers and Electro-Optics, 2009 and 2009 Conference on Quantum electronics and Laser Science Conference. CLEO/QELS 2009. Conference on, 2009), 1–2.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of nonlinear optical crystals, 3rd ed. (Springer-Verlag, Berlin, Germany, 1991).

X. Liu, R. Trebino, and A. V. Smith, “Numerical Simulations of the Ultrasimple Ultrashort-Laser-Pulse Measurement Technique, GRENOUILLE,” in OSA Technical Digest Series (CD) (Optical Society of America, 2007), JThD8.

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Figures (8)

Fig. 1
Fig. 1

Polar plots of SHG efficiency vs. output angle for various colors of a broadband beam impinging on a SHG crystal. Note that, for a thin crystal (upper left), the SHG efficiency varies slowly with angle for all colors, leading to a large phase-matching bandwidth for a given angle. As the crystal thickness increases, the polar plots become narrower, leading to very small phase-matching bandwidths. The thinnest crystal shown here would be required for all pulse-measurement techniques. GRENOUILLE, however, uses a thick crystal (lower right) to spectrally resolve the autocorrelation signal, yielding a FROG trace — without the need for a spectrometer.

Fig. 2
Fig. 2

Single-shot FROG measurements involve crossing large beams at a large angle, so that the relative delay between the two beams varies transversely across the crystal (right). The delay varies along x, the dimension in which the two beams are crossing. This can be accomplished more easily and without the need for alignment using a prism with a large apex angle (left).

Fig. 3
Fig. 3

Top and side views of GRENOUILLE. Use of the above anamorphic lens after the crystal yields a device with only four components.

Fig. 4
Fig. 4

The set-up for the ps GRENOUILLE. The schematic shows both the top view and the side view. The pentagon-shaped crystal both crosses the beams (top view) and angularly disperses the signal beam (side view). The collimating lens located a focal length, fc, from the camera maps the angularly dispersed beam to position on the camera. The imaging lens is chosen to provide the depth of field necessary to image the entire width of the crystal. It is placed a distance di from the camera satisfying the imaging condition.

Fig. 5
Fig. 5

a. The top and side views of the pentagonal SHG crystal. The top view also displays the axis about which the crystal can be rotated to tune the wavelength. b. The crossing of the signal beams inside of the crystal. This figure shows that by increasing the width of the beam, w, or the apex angle of the pentagon, α, the delay range is increased.

Fig. 6
Fig. 6

a. The ray tracing diagram used to calculate the crossing angle of the two beams inside the crystal. b. The diagram used to calculate the range of delay inside the crystal.

Fig. 7
Fig. 7

a. The measured ps GRENOUILLE trace for a 5.5ps double pulse. b. The retrieved GRENOUILLE trace. c. The GRENOUILLE retrieved spectrum compared with that of a high resolution spectrometer with 0.01nm spectral resolution. The high contrast fringes demonstrate the high resolution of GRENOUILLE compared to a conventional spectrometer. d. The retrieved temporal intensity and phase of the 5.5ps pulse. As expected, the retrieved temporal intensities of the pulses were equal.

Fig. 8
Fig. 8

a. The measured GRENOUILLE trace for a train of pulses. b. The retrieved GRENOUILLE trace. c. The GRENOUILLE retrieved spectrum compared with that of a spectrometer. The agreement between the two independent measurements confirms GRENOUILLE’s capabilities. d. The retrieved temporal intensity and phase of the train of pulses.

Tables (1)

Tables Icon

Table 1 Different possible crystal materials for varying wavelength ranges a

Equations (17)

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

G V M L > > τ p .
G V D L < < τ c ,
λ f w h m = .44 λ 0 / L | n ' ( λ 0 ) 1 2 n ' ( λ 0 / 2 ) | .
   θ 1 = 90 α 2 sin 1 ( 1 n cos ( α 2 ) ) .
D = s sin θ 1 ,
s = r sin ( α / 2 ) .
Δ τ = D n c ,
Δ τ = r c cot α 2 ( n 2 cos 2 ( α 2 ) sin ( α 2 ) ) .
n e ( 2 ω ) = n o ( ω ) cos ( θ 1 ) ,
ε = 2 d o r c / M d ,
E m i n = N h c λ = 5.7 pJ
P S H G = I S H G w x w y τ p r ,
I S H G = 8 π 2 η 0 d 2 I i n 2 L 2 λ 2 n 3 ,
I i n = E i n w x w y τ p ,
L = 2 π w y 2 λ .
P S H G = 8 π 2 η 0 d 2 E i n 2 L 3 / 2 2 π w x λ 5 / 2 n 3 τ p r .
E m i n = λ 5 / 2 n 3 w x τ p P m i n 8 π 2 η 0 d 2 L 3 / 2 r 2 π .

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