Abstract

We have investigated two approximation methods for estimating the normalized point source sensitivity (PSSN), which is a recently developed optical performance metric for telescopes. One is an approximation based on the power spectral density (PSD) of the wavefront error. The other is the root-square-sum of the wavefront slope. We call these approximations β approximation and SlopeRMS approximation, respectively. Our analysis shows that for the Thirty Meter Telescope (TMT), the uncertainty of the β approximation is less than 1×103 if the PSSN is better than 0.95, assuming the input PSD estimation is accurate. In addition, we find that the SlopeRMS approximation is a simple method for estimating the worst-case PSSN value in the specific situation when the PSSN is dominated by low-frequency aberrations. Therefore, the SlopeRMS approximation is expected to be useful for specifying a mirror surface for mirror vendors. Accordingly, TMT has a plan to adopt the SlopeRMS approximation for its M2 and M3 polishing specification.

© 2013 Optical Society of America

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References

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  1. G. Z. Angeli, S. Roberts, and K. Vogiatzis, “Systems engineering for the preliminary design of the thirty meter telescope,” Proc. SPIE 7017, 701704 (2008).
    [CrossRef]
  2. C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, N. Sigrist, and L. Wang, “High-resolution optical modeling of the thirty meter telescope for systematic performance trades,” Proc. SPIE 7017, 70170U (2008).
    [CrossRef]
  3. I. R. King, “Accuracy of measurement of star images on a pixel array,” Publ. Astron. Soc. Pac. 95, 163–168 (1983).
    [CrossRef]
  4. B.-J. Seo, C. Nissly, G. Angeli, B. Ellerbroek, J. Nelson, N. Sigrist, and M. Troy, “Analysis of normalized point source sensitivity as performance metric for large telescopes,” Appl. Opt. 48, 5997–6007 (2009).
    [CrossRef]
  5. TMT, “Requirement for secondary mirror system (M2S),” TMT Project Communication TMT.OPT.DRD.07.004.CCR28 ( http://www.tmt.org/sites/default/files/documents/application/pdf/design%20requirements%20document%20for%20the%20m2s%2020121017.pdf , 2007).
  6. TMT, “Requirement for tertiary mirror system (M3S),” TMT Project Communication TMT.OPT.DRD.07.006.CCR28 http://www.tmt.org/sites/default/files/documents/application/pdf/design%20requirements%20document%202012-07.pdf , (2007).
  7. D. Redding, “MACOS manual (modeling and analysis for controlled optical systems),” NASA JPL D-9816, internal document5 (1999).
  8. B.-J. Seo, C. Nissly, M. Troy, G. Angeli, R. Bernier, L. Stepp, and E. Williams, “Estimation of normalized point source sensitivity for segment surface specifications for extremely large telescope (to be published, June 2013.),” Appl. Opt.52.
  9. J. A. Dooley and P. R. Lawson, Technology Plan for the Terrestrial Planet Finder Coronagraph (JPL, 2005), pp. 33–34.
  10. TMT, “Specification for finished 1.44-meter primary mirror segments,” TMT Project Communication TMT.OPT.SPE.07.002.CCR03 http://www.tmt.org/sites/default/files/TMT-OPT-SPE-07-002-CCR03-Specification-for-Finished-Primary-Mirror-Segments-Final.pdf , (2008).
  11. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  12. B. Seo, C. Nissly, G. Angeli, D. MacMynowski, N. Sigrist, M. Troy, and E. Williams, “Investigation of primary mirror segment’s residual errors for the thirty meter telescope,” Proc. SPIE 7427, 74270F (2009).
    [CrossRef]
  13. B. Ellerbroek, “OTF and structure function methods for (Seeing-Limited) TMT error budgets,” TMT Project Communication TMT.SEN.PRE.05.002.REL01 (2005).

2009 (2)

B. Seo, C. Nissly, G. Angeli, D. MacMynowski, N. Sigrist, M. Troy, and E. Williams, “Investigation of primary mirror segment’s residual errors for the thirty meter telescope,” Proc. SPIE 7427, 74270F (2009).
[CrossRef]

B.-J. Seo, C. Nissly, G. Angeli, B. Ellerbroek, J. Nelson, N. Sigrist, and M. Troy, “Analysis of normalized point source sensitivity as performance metric for large telescopes,” Appl. Opt. 48, 5997–6007 (2009).
[CrossRef]

2008 (2)

G. Z. Angeli, S. Roberts, and K. Vogiatzis, “Systems engineering for the preliminary design of the thirty meter telescope,” Proc. SPIE 7017, 701704 (2008).
[CrossRef]

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, N. Sigrist, and L. Wang, “High-resolution optical modeling of the thirty meter telescope for systematic performance trades,” Proc. SPIE 7017, 70170U (2008).
[CrossRef]

1983 (1)

I. R. King, “Accuracy of measurement of star images on a pixel array,” Publ. Astron. Soc. Pac. 95, 163–168 (1983).
[CrossRef]

1976 (1)

Angeli, G.

B.-J. Seo, C. Nissly, G. Angeli, B. Ellerbroek, J. Nelson, N. Sigrist, and M. Troy, “Analysis of normalized point source sensitivity as performance metric for large telescopes,” Appl. Opt. 48, 5997–6007 (2009).
[CrossRef]

B. Seo, C. Nissly, G. Angeli, D. MacMynowski, N. Sigrist, M. Troy, and E. Williams, “Investigation of primary mirror segment’s residual errors for the thirty meter telescope,” Proc. SPIE 7427, 74270F (2009).
[CrossRef]

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, N. Sigrist, and L. Wang, “High-resolution optical modeling of the thirty meter telescope for systematic performance trades,” Proc. SPIE 7017, 70170U (2008).
[CrossRef]

B.-J. Seo, C. Nissly, M. Troy, G. Angeli, R. Bernier, L. Stepp, and E. Williams, “Estimation of normalized point source sensitivity for segment surface specifications for extremely large telescope (to be published, June 2013.),” Appl. Opt.52.

Angeli, G. Z.

G. Z. Angeli, S. Roberts, and K. Vogiatzis, “Systems engineering for the preliminary design of the thirty meter telescope,” Proc. SPIE 7017, 701704 (2008).
[CrossRef]

Angione, J.

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, N. Sigrist, and L. Wang, “High-resolution optical modeling of the thirty meter telescope for systematic performance trades,” Proc. SPIE 7017, 70170U (2008).
[CrossRef]

Bernier, R.

B.-J. Seo, C. Nissly, M. Troy, G. Angeli, R. Bernier, L. Stepp, and E. Williams, “Estimation of normalized point source sensitivity for segment surface specifications for extremely large telescope (to be published, June 2013.),” Appl. Opt.52.

Crossfield, I.

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, N. Sigrist, and L. Wang, “High-resolution optical modeling of the thirty meter telescope for systematic performance trades,” Proc. SPIE 7017, 70170U (2008).
[CrossRef]

Dooley, J. A.

J. A. Dooley and P. R. Lawson, Technology Plan for the Terrestrial Planet Finder Coronagraph (JPL, 2005), pp. 33–34.

Ellerbroek, B.

B.-J. Seo, C. Nissly, G. Angeli, B. Ellerbroek, J. Nelson, N. Sigrist, and M. Troy, “Analysis of normalized point source sensitivity as performance metric for large telescopes,” Appl. Opt. 48, 5997–6007 (2009).
[CrossRef]

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, N. Sigrist, and L. Wang, “High-resolution optical modeling of the thirty meter telescope for systematic performance trades,” Proc. SPIE 7017, 70170U (2008).
[CrossRef]

B. Ellerbroek, “OTF and structure function methods for (Seeing-Limited) TMT error budgets,” TMT Project Communication TMT.SEN.PRE.05.002.REL01 (2005).

Gilles, L.

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, N. Sigrist, and L. Wang, “High-resolution optical modeling of the thirty meter telescope for systematic performance trades,” Proc. SPIE 7017, 70170U (2008).
[CrossRef]

King, I. R.

I. R. King, “Accuracy of measurement of star images on a pixel array,” Publ. Astron. Soc. Pac. 95, 163–168 (1983).
[CrossRef]

Lawson, P. R.

J. A. Dooley and P. R. Lawson, Technology Plan for the Terrestrial Planet Finder Coronagraph (JPL, 2005), pp. 33–34.

MacMynowski, D.

B. Seo, C. Nissly, G. Angeli, D. MacMynowski, N. Sigrist, M. Troy, and E. Williams, “Investigation of primary mirror segment’s residual errors for the thirty meter telescope,” Proc. SPIE 7427, 74270F (2009).
[CrossRef]

Nelson, J.

Nissly, C.

B.-J. Seo, C. Nissly, G. Angeli, B. Ellerbroek, J. Nelson, N. Sigrist, and M. Troy, “Analysis of normalized point source sensitivity as performance metric for large telescopes,” Appl. Opt. 48, 5997–6007 (2009).
[CrossRef]

B. Seo, C. Nissly, G. Angeli, D. MacMynowski, N. Sigrist, M. Troy, and E. Williams, “Investigation of primary mirror segment’s residual errors for the thirty meter telescope,” Proc. SPIE 7427, 74270F (2009).
[CrossRef]

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, N. Sigrist, and L. Wang, “High-resolution optical modeling of the thirty meter telescope for systematic performance trades,” Proc. SPIE 7017, 70170U (2008).
[CrossRef]

B.-J. Seo, C. Nissly, M. Troy, G. Angeli, R. Bernier, L. Stepp, and E. Williams, “Estimation of normalized point source sensitivity for segment surface specifications for extremely large telescope (to be published, June 2013.),” Appl. Opt.52.

Noll, R. J.

Redding, D.

D. Redding, “MACOS manual (modeling and analysis for controlled optical systems),” NASA JPL D-9816, internal document5 (1999).

Roberts, S.

G. Z. Angeli, S. Roberts, and K. Vogiatzis, “Systems engineering for the preliminary design of the thirty meter telescope,” Proc. SPIE 7017, 701704 (2008).
[CrossRef]

Seo, B.

B. Seo, C. Nissly, G. Angeli, D. MacMynowski, N. Sigrist, M. Troy, and E. Williams, “Investigation of primary mirror segment’s residual errors for the thirty meter telescope,” Proc. SPIE 7427, 74270F (2009).
[CrossRef]

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, N. Sigrist, and L. Wang, “High-resolution optical modeling of the thirty meter telescope for systematic performance trades,” Proc. SPIE 7017, 70170U (2008).
[CrossRef]

Seo, B.-J.

B.-J. Seo, C. Nissly, G. Angeli, B. Ellerbroek, J. Nelson, N. Sigrist, and M. Troy, “Analysis of normalized point source sensitivity as performance metric for large telescopes,” Appl. Opt. 48, 5997–6007 (2009).
[CrossRef]

B.-J. Seo, C. Nissly, M. Troy, G. Angeli, R. Bernier, L. Stepp, and E. Williams, “Estimation of normalized point source sensitivity for segment surface specifications for extremely large telescope (to be published, June 2013.),” Appl. Opt.52.

Sigrist, N.

B. Seo, C. Nissly, G. Angeli, D. MacMynowski, N. Sigrist, M. Troy, and E. Williams, “Investigation of primary mirror segment’s residual errors for the thirty meter telescope,” Proc. SPIE 7427, 74270F (2009).
[CrossRef]

B.-J. Seo, C. Nissly, G. Angeli, B. Ellerbroek, J. Nelson, N. Sigrist, and M. Troy, “Analysis of normalized point source sensitivity as performance metric for large telescopes,” Appl. Opt. 48, 5997–6007 (2009).
[CrossRef]

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, N. Sigrist, and L. Wang, “High-resolution optical modeling of the thirty meter telescope for systematic performance trades,” Proc. SPIE 7017, 70170U (2008).
[CrossRef]

Stepp, L.

B.-J. Seo, C. Nissly, M. Troy, G. Angeli, R. Bernier, L. Stepp, and E. Williams, “Estimation of normalized point source sensitivity for segment surface specifications for extremely large telescope (to be published, June 2013.),” Appl. Opt.52.

Troy, M.

B. Seo, C. Nissly, G. Angeli, D. MacMynowski, N. Sigrist, M. Troy, and E. Williams, “Investigation of primary mirror segment’s residual errors for the thirty meter telescope,” Proc. SPIE 7427, 74270F (2009).
[CrossRef]

B.-J. Seo, C. Nissly, G. Angeli, B. Ellerbroek, J. Nelson, N. Sigrist, and M. Troy, “Analysis of normalized point source sensitivity as performance metric for large telescopes,” Appl. Opt. 48, 5997–6007 (2009).
[CrossRef]

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, N. Sigrist, and L. Wang, “High-resolution optical modeling of the thirty meter telescope for systematic performance trades,” Proc. SPIE 7017, 70170U (2008).
[CrossRef]

B.-J. Seo, C. Nissly, M. Troy, G. Angeli, R. Bernier, L. Stepp, and E. Williams, “Estimation of normalized point source sensitivity for segment surface specifications for extremely large telescope (to be published, June 2013.),” Appl. Opt.52.

Vogiatzis, K.

G. Z. Angeli, S. Roberts, and K. Vogiatzis, “Systems engineering for the preliminary design of the thirty meter telescope,” Proc. SPIE 7017, 701704 (2008).
[CrossRef]

Wang, L.

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, N. Sigrist, and L. Wang, “High-resolution optical modeling of the thirty meter telescope for systematic performance trades,” Proc. SPIE 7017, 70170U (2008).
[CrossRef]

Williams, E.

B. Seo, C. Nissly, G. Angeli, D. MacMynowski, N. Sigrist, M. Troy, and E. Williams, “Investigation of primary mirror segment’s residual errors for the thirty meter telescope,” Proc. SPIE 7427, 74270F (2009).
[CrossRef]

B.-J. Seo, C. Nissly, M. Troy, G. Angeli, R. Bernier, L. Stepp, and E. Williams, “Estimation of normalized point source sensitivity for segment surface specifications for extremely large telescope (to be published, June 2013.),” Appl. Opt.52.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Proc. SPIE (3)

B. Seo, C. Nissly, G. Angeli, D. MacMynowski, N. Sigrist, M. Troy, and E. Williams, “Investigation of primary mirror segment’s residual errors for the thirty meter telescope,” Proc. SPIE 7427, 74270F (2009).
[CrossRef]

G. Z. Angeli, S. Roberts, and K. Vogiatzis, “Systems engineering for the preliminary design of the thirty meter telescope,” Proc. SPIE 7017, 701704 (2008).
[CrossRef]

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, N. Sigrist, and L. Wang, “High-resolution optical modeling of the thirty meter telescope for systematic performance trades,” Proc. SPIE 7017, 70170U (2008).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

I. R. King, “Accuracy of measurement of star images on a pixel array,” Publ. Astron. Soc. Pac. 95, 163–168 (1983).
[CrossRef]

Other (7)

TMT, “Requirement for secondary mirror system (M2S),” TMT Project Communication TMT.OPT.DRD.07.004.CCR28 ( http://www.tmt.org/sites/default/files/documents/application/pdf/design%20requirements%20document%20for%20the%20m2s%2020121017.pdf , 2007).

TMT, “Requirement for tertiary mirror system (M3S),” TMT Project Communication TMT.OPT.DRD.07.006.CCR28 http://www.tmt.org/sites/default/files/documents/application/pdf/design%20requirements%20document%202012-07.pdf , (2007).

D. Redding, “MACOS manual (modeling and analysis for controlled optical systems),” NASA JPL D-9816, internal document5 (1999).

B.-J. Seo, C. Nissly, M. Troy, G. Angeli, R. Bernier, L. Stepp, and E. Williams, “Estimation of normalized point source sensitivity for segment surface specifications for extremely large telescope (to be published, June 2013.),” Appl. Opt.52.

J. A. Dooley and P. R. Lawson, Technology Plan for the Terrestrial Planet Finder Coronagraph (JPL, 2005), pp. 33–34.

TMT, “Specification for finished 1.44-meter primary mirror segments,” TMT Project Communication TMT.OPT.SPE.07.002.CCR03 http://www.tmt.org/sites/default/files/TMT-OPT-SPE-07-002-CCR03-Specification-for-Finished-Primary-Mirror-Segments-Final.pdf , (2008).

B. Ellerbroek, “OTF and structure function methods for (Seeing-Limited) TMT error budgets,” TMT Project Communication TMT.SEN.PRE.05.002.REL01 (2005).

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Figures (13)

Fig. 1.
Fig. 1.

Conceptual procedure of the β approximation process: 1D space view (top row) and 2D space view (bottom row). We obtain first the PSSN density function, which is a multiplication of the PSD and the β function. Then, the overall PSSN is estimated by multiplying all PSSNs (or 1-PSSN density function) from all frequency regions. Due to the nature of wavefront aberration, the PSD at low spatial frequency is large and decreases as the spatial frequency increases. Therefore, overall shape of the PSSN density function becomes as shown. It has the dominant spatial frequency where the PSSN contribution is maximum, i.e., the PSSN density function becomes maximum and becomes negligible as the spatial frequency approaches either 0 or infinity.

Fig. 2.
Fig. 2.

(a) Entrance pupil (EnP) and example wavefront for the TMT. (b) and (c) 2D and 1D version of β function with baseline Fried parameter of ro=200mm. (d) log–log plot of (c). The β function is 2D in general, implying that the frequency response can be anisotropic unless the EnP is circularly symmetric. For this paper, we assume that the TMT aperture is circular enough. We first obtain the 1D β function for the response along the elevation coordinate reference system (ECRS) x direction. Then, we use the 1D β function for all direction to have 2D β function. Therefore, 2D β function here is isotropic.

Fig. 3.
Fig. 3.

Modeled β function in a log–log plot. The β function has two asymptotic regions: the constant high-frequency region with 2/rad2 and the quadratic low-frequency region with Eq. (5), which is derived in Appendix B. The two asymptotic lines intersect at the knee frequency fknee computed in Eq. (6), which is close to 1/ro. The breaking frequency is also defined with Eq. (8).

Fig. 4.
Fig. 4.

PSSN density functions using the von Karman PSD model. For each M1 segment, M2, and M3, a von Karman PSD with fo and γ values of 2.0cycle/m, and 3 are used. The amplitude parameters A are chosen 1.5nm2m2, 4.5nm2m2, and 8.0nm2m2, which generate the PSSN values of approximately 0.98, 0.99, and 0.99 for M1, M2, M3, respectively. For M3, two different PSSN density functions are obtained along the short or long axes since the input PSDs are considered isotropic. Note that the dominant spatial frequencies of M2 and M3 are located below the breaking frequency in the β function in Fig. 2(d). This means that the overall PSSN is dominated by the low spatial aberration below the breaking frequency in this example.

Fig. 5.
Fig. 5.

Setup for the numerical PSD study, which estimates the β approximation accuracy.

Fig. 6.
Fig. 6.

Result of the numerical PSD study. Accuracy of the β approximation is proportional to (1-PSSN) and it is within ±1×103 at PSSN of 0.95.

Fig. 7.
Fig. 7.

PSD computed from surface measurements of real mirrors from various vendors.

Fig. 8.
Fig. 8.

Setup for mirror sample PSD study. We consider real PSD inputs, LOA, piston, tip, and tilt (PTT), and warping harness (WH) control for TMT M1 segments. Finally, we obtain true PSSN [PSSN(T)], directly approximated PSSN [PSSN(D)], and approximated PSSN [PSSN(A)]. We compare them to estimate the accuracy of the β approximation.

Fig. 9.
Fig. 9.

β approximation error as a result for mirror sample PSD study. For each WH scheme and each PSD input, we have obtained the approximated PSSN value [PSSN(D)] from the retrieved PSD and PSSN(A) from the PSD estimation as described in Subsection 3.A.2. (a) Shows PSSN(D) and PSSN(A) with comparison to PSSN(T); their approximation errors, i.e., difference to the PSSN(T) are shown in (b).

Fig. 10.
Fig. 10.

Numerical test setup to study the relation between SlopeRMS and PSSN. See the text in Subsection 3.B.1 for detailed description of the steps.

Fig. 11.
Fig. 11.

Relation study result between the subaperture PSSN (PSSNi, y axis) and SlopeRMSi of the subaperture surface (bottom x axis) and SlopeRMSi of the subaperture WFE (top x axis) using the test setup described in Subsection 3.B.1. The blue dotted line represents the theoretical limit curve, which is obtained assuming the all aberration is below the breaking frequency. The theoretical limit curve is obtained using Eq. (18) for the wavefront and Eq. (19) for the surface.

Fig. 12.
Fig. 12.

β function computation methods.

Fig. 13.
Fig. 13.

Normalized difference ΔβN [Eq. (7)] using our numerically obtained η value of 2.75 at our default PSSN computation parameters, i.e., the Fried parameter ro of 200 mm and the wavelength of 0.5 μm.

Equations (28)

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

PSSNz(1αzCz2).
PSSNlimSi0,Ni(1f⃗=Siβ(f⃗)PSD(f⃗)df⃗),
PSSNlimΔf0,Ni(1f=fifi+1β(f)PSD(f)2πfdf).
PSSNdensity(f)=β(f)PSD(f).
β(f)=ηro2f2,
fknee=2/ηro1ro.
ΔβN(f)=ηro2f2β(f)β(f).
ΔβN(f)|f=fb<20%.
s⃗(r⃗)=∇⃗·w(r⃗),
Td(fi)=2πfi,
S⃗(f⃗)=x^2πfxW(f⃗)+y^2πfyW(f⃗),
|S⃗(f⃗)|2=|Td(f)|2|W(f)|2=4π2f2|W(f)|2,
PSDs(f⃗)=4π2f2PSDw(f⃗),
SlopeRMSw2=f=0PSDs(f)2πfdf=f=04π2f2PSDw(f)2πfdf,
1PSSNf=0β(f)PSDw,rad(f)2πfdf,
1PSSNf=0ηro2f2PSDw,rad(f)2πfdf.
1PSSNηro24π2SlopeRMSw,rad/m2ηro2λμm2SlopeRMSw,μrad2,
SlopeRMSwfe,μrad=rS·SlopeRMSsurf,μrad,
1PSSNηro2λμm2(rS)2SlopeRMSsurf,μrad2.
1PSSN=ηaro2λμm2SlopeRMSwfe,μrad2,
PSSN=|OTFe(f⃗)|2,
·(·)|OTFt+a(f⃗)|2df⃗|OTFt+a(f⃗)|2df⃗,
OTFe(f⃗)=exp(12k2D(λf⃗)),
OPD(r⃗)=Asin(2πfx+ϕ)
D(λf⃗)=2A2sin2(πλffx),
PSSN=12(πkλA)2fx2f2.
β(f)=(1PSSN)/(2πA2λ)2=4π2λ2fx2f2.
β(f)=ηro2fo2,

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