Abstract

The signal expression measured by an OCT system is derived, which reveals the possibility of tissue dispersion compensation by introducing the required amount of dispersion in the reference arm and may be implemented by incorporating the grating- based rapid scanning optical delay (RSOD) lines in the reference arm of OCT. The more accurate expressions for the group-delay dispersion (GDD) and the second-order GDD are derived for the grating-based RSOD lines. A comparison of our results with previous ones is done, which shows that when only the GDD is of concern, the previouslyreported results are accurate. However, when it becomes necessary to take the effect of the second-order GDD into account, the more accurate formula must be used. The obtained results may be of great importance for maximizing the imaging properties of OCT in tissue imaging.

© 2007 Optical Society of America

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  1. R. C. Youngquist, S. Carr, and D. E. N. Davies, "Optical coherence-domain reflectommetry: a new optical evaluation technique," Opt. Lett. 12, 158-160 (1987).
    [CrossRef] [PubMed]
  2. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
    [CrossRef] [PubMed]
  3. E. A. Swanson, D. Huang, M. R, Hce, J. G. Fujimoto, C. P. Lin, and C. A. Puliafito, "High-speed optical coherence domain reflectometry," Opt. Lett. 17, 151-153 (1992).
    [CrossRef] [PubMed]
  4. C. K. Hitzenberger, A. Baumgartner, and A. F. Fercher, "Dispersion induced multiple peak splitting in partial coherence interferometry," Opt. Commun. 154, 179-195 (1998).
    [CrossRef]
  5. B. L. Danielson and C. Y. Boisrobert, "Absolute optical ranging using low-coherence interferometry," Appl. Opt. 30, 2975-2979 (1991).
    [CrossRef] [PubMed]
  6. C. K. Hitzenberger, W. Drexler, A. Baumgartner, and A. F. Fercher, "Dispersion effects in partial coherence interferometry," in Proc. SPIE 2981, 29-36 (1998).
    [CrossRef]
  7. G. J. Tearney, M. E. Brezinski, J. F. Southern, B. E. Bouma, M. R. Hee, and J. G. Fujimoto, "Determination of the refractive index of highly scattering human tissue by optical coherence tomography," Opt. Lett. 20, 2258-2260 (1995).
    [CrossRef] [PubMed]
  8. A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, "Numerical dispersion compensation for partial coherence interferometry and optical coherence tomography," Opt. Express 9, 610-615 (2001).
    [CrossRef] [PubMed]
  9. B. Bouma, G. J. Tearney, S. A. Boppart, M. R. Hee, M. E. Brezinski, and J. G. Fujimoto, "High-resolution optical coherence tomographic imaging using a mode-locked Ti:Al2O3 laser source," Opt. Lett. 20, 1486-1488 (1995).
    [CrossRef] [PubMed]
  10. G. J. Tearney, B. E. Bouma, and J. G. Fujimoto, "High-speed phase- and group-delay scanning with a grating-based phase control delay line," Opt. Lett. 22, 1811-1813 (1997).
    [CrossRef]
  11. W. K. Niblack, J. Ott. Schenk, B. Liu, and M. E. Brezinski, "Dispersion in a grating-based optical delay line for optical coherence tomography," Appl. Opt. 42, 4115-4118 (2003).
    [CrossRef] [PubMed]
  12. E. D. J. Smith, A. V. Zvyagin, and D. D. Sampson, "Real-time dispersion compensation in scanning interferometry," Opt. Lett. 27, 1998-2000 (2002).
    [CrossRef]
  13. K. F. Kwong, D. Yankelevich, K. C. Chu, J. P. Heritage, and A. Dinees, "400-Hz mechanical scanning optical delay line," Opt. Lett. 18, 558-561 (1993).
    [CrossRef] [PubMed]
  14. A. M. Rollins, M. D. Kulkarni, S. Yazdanfar, R. Ung-arunyawee, and J. A. Izatt, "In vivo video rate optical coherence tomography," Opt. Express 3, 219-229 (1998).
    [CrossRef] [PubMed]
  15. A. V. Zvyagin, E. D. J. Smith, and D. D. Sampson, "Delay and dispersion characteristics of a frequency-domain optical delay line for scanning interferometry," J. Opt. Soc. Am. A 20, 333-341 (2003).
    [CrossRef]
  16. W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, and A. F. Fercher, "Investigation of dispersion effects in ocular media by multiple wavelength partical coherence interferometry," Exp. Eye Res. 66, 25-33 (1998).
    [CrossRef] [PubMed]
  17. A. G. Van Engen, S. A. Diddams, and T. S. Clement, "Dispersion measurements of water with white-light interferometry," Appl. Opt. 37, 5679-5686 (1998).
    [CrossRef]
  18. R. L. Fork, C. H. Brito Cruz, P. C. Becker, and C. V. Shank, "Compression of optical pulses to six femtoseconds by using cubic phase compensation," Opt. Lett. 12, 483-485 (1987).
    [CrossRef] [PubMed]

2003 (2)

2002 (1)

2001 (1)

1998 (5)

C. K. Hitzenberger, W. Drexler, A. Baumgartner, and A. F. Fercher, "Dispersion effects in partial coherence interferometry," in Proc. SPIE 2981, 29-36 (1998).
[CrossRef]

C. K. Hitzenberger, A. Baumgartner, and A. F. Fercher, "Dispersion induced multiple peak splitting in partial coherence interferometry," Opt. Commun. 154, 179-195 (1998).
[CrossRef]

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, and A. F. Fercher, "Investigation of dispersion effects in ocular media by multiple wavelength partical coherence interferometry," Exp. Eye Res. 66, 25-33 (1998).
[CrossRef] [PubMed]

A. G. Van Engen, S. A. Diddams, and T. S. Clement, "Dispersion measurements of water with white-light interferometry," Appl. Opt. 37, 5679-5686 (1998).
[CrossRef]

A. M. Rollins, M. D. Kulkarni, S. Yazdanfar, R. Ung-arunyawee, and J. A. Izatt, "In vivo video rate optical coherence tomography," Opt. Express 3, 219-229 (1998).
[CrossRef] [PubMed]

1997 (1)

1995 (2)

1993 (1)

1992 (1)

1991 (2)

B. L. Danielson and C. Y. Boisrobert, "Absolute optical ranging using low-coherence interferometry," Appl. Opt. 30, 2975-2979 (1991).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

1987 (2)

Baumgartner, A.

C. K. Hitzenberger, A. Baumgartner, and A. F. Fercher, "Dispersion induced multiple peak splitting in partial coherence interferometry," Opt. Commun. 154, 179-195 (1998).
[CrossRef]

C. K. Hitzenberger, W. Drexler, A. Baumgartner, and A. F. Fercher, "Dispersion effects in partial coherence interferometry," in Proc. SPIE 2981, 29-36 (1998).
[CrossRef]

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, and A. F. Fercher, "Investigation of dispersion effects in ocular media by multiple wavelength partical coherence interferometry," Exp. Eye Res. 66, 25-33 (1998).
[CrossRef] [PubMed]

Becker, P. C.

Boisrobert, C. Y.

Boppart, S. A.

Bouma, B.

Bouma, B. E.

Brezinski, M. E.

Brito Cruz, C. H.

Carr, S.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Chu, K. C.

Clement, T. S.

Danielson, B. L.

Davies, D. E. N.

Diddams, S. A.

Dinees, A.

Drexler, W.

C. K. Hitzenberger, W. Drexler, A. Baumgartner, and A. F. Fercher, "Dispersion effects in partial coherence interferometry," in Proc. SPIE 2981, 29-36 (1998).
[CrossRef]

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, and A. F. Fercher, "Investigation of dispersion effects in ocular media by multiple wavelength partical coherence interferometry," Exp. Eye Res. 66, 25-33 (1998).
[CrossRef] [PubMed]

Fercher, A. F.

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, "Numerical dispersion compensation for partial coherence interferometry and optical coherence tomography," Opt. Express 9, 610-615 (2001).
[CrossRef] [PubMed]

C. K. Hitzenberger, W. Drexler, A. Baumgartner, and A. F. Fercher, "Dispersion effects in partial coherence interferometry," in Proc. SPIE 2981, 29-36 (1998).
[CrossRef]

C. K. Hitzenberger, A. Baumgartner, and A. F. Fercher, "Dispersion induced multiple peak splitting in partial coherence interferometry," Opt. Commun. 154, 179-195 (1998).
[CrossRef]

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, and A. F. Fercher, "Investigation of dispersion effects in ocular media by multiple wavelength partical coherence interferometry," Exp. Eye Res. 66, 25-33 (1998).
[CrossRef] [PubMed]

Findl, O.

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, and A. F. Fercher, "Investigation of dispersion effects in ocular media by multiple wavelength partical coherence interferometry," Exp. Eye Res. 66, 25-33 (1998).
[CrossRef] [PubMed]

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Fork, R. L.

Fujimoto, J. G.

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Hce, M. R

Hee, M. R.

Heritage, J. P.

Hitzenberger, C. K.

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, "Numerical dispersion compensation for partial coherence interferometry and optical coherence tomography," Opt. Express 9, 610-615 (2001).
[CrossRef] [PubMed]

C. K. Hitzenberger, A. Baumgartner, and A. F. Fercher, "Dispersion induced multiple peak splitting in partial coherence interferometry," Opt. Commun. 154, 179-195 (1998).
[CrossRef]

C. K. Hitzenberger, W. Drexler, A. Baumgartner, and A. F. Fercher, "Dispersion effects in partial coherence interferometry," in Proc. SPIE 2981, 29-36 (1998).
[CrossRef]

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, and A. F. Fercher, "Investigation of dispersion effects in ocular media by multiple wavelength partical coherence interferometry," Exp. Eye Res. 66, 25-33 (1998).
[CrossRef] [PubMed]

Huang, D.

E. A. Swanson, D. Huang, M. R, Hce, J. G. Fujimoto, C. P. Lin, and C. A. Puliafito, "High-speed optical coherence domain reflectometry," Opt. Lett. 17, 151-153 (1992).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Izatt, J. A.

Karamata, B.

Kulkarni, M. D.

Kwong, K. F.

Lasser, T.

Lin, C. P.

E. A. Swanson, D. Huang, M. R, Hce, J. G. Fujimoto, C. P. Lin, and C. A. Puliafito, "High-speed optical coherence domain reflectometry," Opt. Lett. 17, 151-153 (1992).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Liu, B.

Niblack, W. K.

Ott. Schenk, J.

Puliafito, C. A.

E. A. Swanson, D. Huang, M. R, Hce, J. G. Fujimoto, C. P. Lin, and C. A. Puliafito, "High-speed optical coherence domain reflectometry," Opt. Lett. 17, 151-153 (1992).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Rollins, A. M.

Sampson, D. D.

Sattmann, H.

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, and A. F. Fercher, "Investigation of dispersion effects in ocular media by multiple wavelength partical coherence interferometry," Exp. Eye Res. 66, 25-33 (1998).
[CrossRef] [PubMed]

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Shank, C. V.

Smith, E. D. J.

Southern, J. F.

Sticker, M.

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

E. A. Swanson, D. Huang, M. R, Hce, J. G. Fujimoto, C. P. Lin, and C. A. Puliafito, "High-speed optical coherence domain reflectometry," Opt. Lett. 17, 151-153 (1992).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Tearney, G. J.

Ung-arunyawee, R.

Van Engen, A. G.

Yankelevich, D.

Yazdanfar, S.

Youngquist, R. C.

Zawadzki, R.

Zvyagin, A. V.

Appl. Opt. (3)

Exp. Eye Res. (1)

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, and A. F. Fercher, "Investigation of dispersion effects in ocular media by multiple wavelength partical coherence interferometry," Exp. Eye Res. 66, 25-33 (1998).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

C. K. Hitzenberger, A. Baumgartner, and A. F. Fercher, "Dispersion induced multiple peak splitting in partial coherence interferometry," Opt. Commun. 154, 179-195 (1998).
[CrossRef]

Opt. Express (2)

Opt. Lett. (8)

E. D. J. Smith, A. V. Zvyagin, and D. D. Sampson, "Real-time dispersion compensation in scanning interferometry," Opt. Lett. 27, 1998-2000 (2002).
[CrossRef]

R. C. Youngquist, S. Carr, and D. E. N. Davies, "Optical coherence-domain reflectommetry: a new optical evaluation technique," Opt. Lett. 12, 158-160 (1987).
[CrossRef] [PubMed]

R. L. Fork, C. H. Brito Cruz, P. C. Becker, and C. V. Shank, "Compression of optical pulses to six femtoseconds by using cubic phase compensation," Opt. Lett. 12, 483-485 (1987).
[CrossRef] [PubMed]

E. A. Swanson, D. Huang, M. R, Hce, J. G. Fujimoto, C. P. Lin, and C. A. Puliafito, "High-speed optical coherence domain reflectometry," Opt. Lett. 17, 151-153 (1992).
[CrossRef] [PubMed]

K. F. Kwong, D. Yankelevich, K. C. Chu, J. P. Heritage, and A. Dinees, "400-Hz mechanical scanning optical delay line," Opt. Lett. 18, 558-561 (1993).
[CrossRef] [PubMed]

B. Bouma, G. J. Tearney, S. A. Boppart, M. R. Hee, M. E. Brezinski, and J. G. Fujimoto, "High-resolution optical coherence tomographic imaging using a mode-locked Ti:Al2O3 laser source," Opt. Lett. 20, 1486-1488 (1995).
[CrossRef] [PubMed]

G. J. Tearney, M. E. Brezinski, J. F. Southern, B. E. Bouma, M. R. Hee, and J. G. Fujimoto, "Determination of the refractive index of highly scattering human tissue by optical coherence tomography," Opt. Lett. 20, 2258-2260 (1995).
[CrossRef] [PubMed]

G. J. Tearney, B. E. Bouma, and J. G. Fujimoto, "High-speed phase- and group-delay scanning with a grating-based phase control delay line," Opt. Lett. 22, 1811-1813 (1997).
[CrossRef]

Proc. SPIE (1)

C. K. Hitzenberger, W. Drexler, A. Baumgartner, and A. F. Fercher, "Dispersion effects in partial coherence interferometry," in Proc. SPIE 2981, 29-36 (1998).
[CrossRef]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Schematic of the fiber-optic-based OCT.

Fig. 2
Fig. 2

Schematic of the grating-based optical delay line. The collimated light from a broadband source is incident on the grating, which is displaced a distance (Lf) away from the front focal plane of the lens. The diffracted light with the wavelength λ is focused by the lens on the tilt Mirror 1.

Fig. 3
Fig. 3

Displacement of the grating from the front focal plane to a distance (Lf) away from it. For the same frequency component ω of the beam with the same incident angle θ i , the diffraction angle θλ is the same.

Fig. 4
Fig. 4

Changes of second-order GDD between our result and that of Smith et al. are compared for their dependence on grating tilt θ g (Ref. [11]). The selected parameters of the phase-control RSOD line are from Ref. [11]. The FWHM spectrum of the source was 75 nm, with a center wavelength of 1280 nm. The total scan angle γ of mirror 1 is approximately 4°. L = 100   mm . The grating spacing p = 150 lines / mm . The focal length of the Fourier lens is 100   mm . (a) Smith et al. results, (b) our results.

Fig. 5
Fig. 5

Changes of second-order GDD on distance (Lf) between our result and that of Zvyagin, et al. [see Ref. [15], Eq. (30)]. The selected parameters of the phase-control RSOD line are from Ref. [11]. The FWHM spectrum of the source was 75   nm , with a center wavelength of 1280   nm . The total scan angle γ of mirror 1 is approximately 4°. The θ λ 0 = θ g = 27 ° . The grating spacing p = 150 lines∕mm. The focal length of the Fourier lens is 100   nm .

Equations (23)

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I = E R ( t τ ) E S * ( t ) + E R * ( t τ ) E S ( t ) ,
I ( ω ) = F { I ( τ ) } = E R ( ω ) E S * + c . c .
= { R ( z ) E i ( ω ) exp [ i Φ S ( ω ) ] } { E i * ( ω ) exp [ i Φ R ( ω ) ] } + c . c . , = R ( z ) | E i ( ω ) | 2 exp { i [ Φ S ( ω ) Φ R ( ω ) ] } + c . c . ,
Φ j ( Ω ) = Φ j ( ω 0 ) + Φ j ( ω 0 ) Ω + Φ j ( ω 0 ) Ω 2 2 + 1 6 Φ j ( ω 0 ) Ω 3 + , ( j = R , S ) ,
p ( sin θ i + sin θ λ ) = m λ ,
Δ l γ = 4 f tan ( Δ θ λ ) γ + 4 y 0 γ .
Δ l γ = 4 L tan ( Δ θ λ ) γ + 4 y 0 γ .
Δ l λ = 4 L cos Δ θ λ + 4 f cos Δ θ λ + 4 ( L f ) m λ sin Δ θ λ p cos θ λ + 4 ( δ n ) δ l lens 8 f ,
Φ R ( ω ) = 4 y 0 ω γ c + 4 L ω γ sin Δ θ λ c cos Δ θ λ + 4 L ω cos Δ θ λ c + 4 f ω c cos Δ θ λ + 8 π m ( L f ) sin Δ θ λ p cos θ λ + ω δ n δ l lens c 8 f ω c ,
p sin ( Δ θ λ θ g ) + sin θ g = m ( λ λ 0 ) .
d ( Δ θ λ ) d ω = 2 π m c p ω 2 cos ( Δ θ λ θ g ) ,
d 2 ( Δ θ λ ) d ω 2 = 4 π m c p ω 3 cos ( Δ θ λ θ g ) 4 π 2 m 2 c 2 p 2 ω 4 sin ( Δ θ λ θ g ) cos 3 ( Δ θ λ θ g ) ,
d 3 ( Δ θ λ ) d ω 3 = 12 π m c p ω cos ( Δ θ λ θ g ) + 8 π 2 m 2 c 2 p 2 ω 5 cos 3 ( Δ θ λ θ g ) × { sin ( Δ θ λ θ g ) + π m c p ω + 3 π m c sin 2 ( Δ θ λ θ g ) p ω cos 2 ( Δ θ λ θ g ) } .
Φ R ( ω 0 ) = 4 ω 0 y 0 γ c + 4 ω 0 ( L f ) c + 4 ω 0 δ n δ l lens c ,
Φ R ( ω 0 ) = 4 y 0 γ c 8 π m L γ p ω 0 cos θ g + 4 ( L f ) c 16 π 2 m 2 c ( L f ) p 2 ω 0 2 cos 2 θ g × δ n δ l lens c + ω 0 δ l lens c ( d ( δ n ) d ω ) | ω = ω 0 ,
Φ R ( ω 0 ) = 16 π 2 m 2 c sin θ g L γ p 2 ω 0 3 cos 3 θ g + 16 π 2 m 2 c p 2 ω 0 3 cos 2 θ g ( L f ) + 2 δ l lens c ( d δ n d ω ) | ω = ω 0 +   ω 0 δ l lens c ( d 2 δ n d ω 2 ) | ω = ω 0 ,
Φ R ( ω 0 ) = 16 π 2 m 2 c L γ p 2 ω 0 4 { sin θ g cos 3 θ g + 6 π m c sin θ g p ω cos 5 θ g } 32 π 3 m 3 c 2 L γ p 3 ω 0 5 cos 3 θ g + 16 π 2 m 2 c ( L f ) p 2 ω 4 cos 2 θ g × { 3 + 10 π m c sin θ g p ω cos 2 θ g } + 64 π 3 m 3 c 2 sin θ g ( L f ) p 3 ω 5 cos 3 θ g + 64 π 4 m 4 c 3 ( L f ) p 4 ω 6 cos θ g × { 1 4 sin 2 θ g + sin 2 θ g cos θ g + 3 sin 2 θ g cos 2 θ g + 3 sin θ g cos 2 θ g } + 3 δ l lens c d 2 ( δ n ) d ω 2 | ω = ω 0 + ω 0 δ l lens c ( d 3 ( δ n ) d ω 3 ) | ω = ω 0 .
Φ R ( ω 0 ) Smith = 16 π 2 L m 2 c γ sin θ g p 2 ω 0 3 cos 3 θ g .
Φ R ( ω 0 ) Niblack = 16 π 2 m 2 c p 2 ω 0 3 cos 2 θ g ( L f ) .
Φ R ( ω 0 ) Smith = 16 π 2 m 2 c L γ p 2 ω 0 4 { sin θ g cos 3 θ g + 6 π m c sin θ g p ω cos 5 θ g } ,
Φ R ( ω 0 ) Zvyagin = 16 π 2 m 2 c ( L f ) p 2 ω 4 cos 2 θ g { 3 + 10 π m c sin θ g p ω cos 2 θ g } .
Φ R ( ω 0 ) lens = 2 δ l lens c ( d δ n d ω ) | ω = ω 0 + ω 0 δ l lens c ( d 2 δ n d ω 2 ) | ω = ω 0 ,
Φ R ( ω 0 ) lens = 3 δ l lens c d 2 ( δ n ) d ω 2 | ω = ω 0 + ω 0 δ l lens c ( d 3 ( δ n ) d ω 3 ) | ω = ω 0 .

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