Abstract

Optical space domain reflectometry (OSDR) implemented with a novel interferometric characterization method is demonstrated to be an accurate technique for characterizing the complex coupling coefficient of strong fiber Bragg gratings. A theoretical model is also presented, incorporating the effect of the heat perturbation shape, which accurately predicts the measurement behavior. It is shown that the measurement accuracy and spatial resolution are dramatically improved by removing the effect of the heat perturbation shape on the reconstructed profile using a deconvolution technique. The improvement in accuracy is illustrated by the excellent agreement between a weak grating reconstructed with OSDR and the same grating reconstructed with optical frequency domain reflectometry and a layer-peeling method. Reconstruction of a strong grating with an integrated coupling coefficient, q¯L=8.14, demonstrates the utility of this technique.

© 2009 Optical Society of America

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  1. R. Feced and M. N. Zervas, “Effects of random phase and amplitude errors in optical fiber Bragg gratings,” J. Lightwave Technol. 18, 90-101 (2000).
    [CrossRef]
  2. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277-1294 (1997).
    [CrossRef]
  3. P. Giaccari, H. G. Limberger, and R. P. Salathe, “Local coupling-coefficient characterization in fiber Bragg gratings,” Opt. Lett. 28, 598-600 (2003).
    [CrossRef] [PubMed]
  4. X. Chapeleau, D. Leduc, C. Lupi, F. Lopez-Gejo, M. Douay, R. Le Ny, and C. Boisrobert, “Local characterization of fiber-Bragg gratings through combined use of low-coherence interferometry and a layer-peeling algorithm,” Appl. Opt. 45, 728-735 (2006).
    [CrossRef] [PubMed]
  5. O. H. Waagaard, “Spatial characterization of strong fiber Bragg gratings using thermal chirp and opticalfrequency- domain reflectometry,” J. Lightwave Technol. 23, 909-914(2005).
    [CrossRef]
  6. P. A. Krug, R. Stolte, and R. Ulrich, “Measurement of index modulation along an optical fiber Bragg grating,” Opt. Lett. 20, 1767-1769 (1995).
    [CrossRef] [PubMed]
  7. F. El-Diasty, A. Heaney, and T. Erdogan, “Analysis of fiber Bragg gratings by a side-diffraction interference technique,” Appl. Opt. 40, 890-896 (2001).
    [CrossRef]
  8. I. Petermann, S. Helmfrid, and A. T. Friberg, “Limitations of the interferometric side diffraction technique for fibre Bragg grating characterization,” Opt. Commun. 201, 301-308 (2002).
    [CrossRef]
  9. I. Petermann, S. Helmfrid, and P. Y. Fonjallaz, “Fibre Bragg grating characterization with ultraviolet-based interferometric side diffraction,” J. Opt. A Pure Appl. Opt. 5, 437-441(2003).
    [CrossRef]
  10. J. Canning, M. Janos, D. Y. Stepanov, and M. G. Sceats, “Direct measurement of grating chirp using resonant side scatter spectra,” Electron. Lett. 32, 1608-1610 (1996).
    [CrossRef]
  11. J. Canning, D. C. Psaila, Z. Brodzeli, A. Higley, and M. Janos, “Characterization of apodized fiber Bragg gratings for rejection filter applications,” Appl. Opt. 36, 9378-9382(1997).
    [CrossRef]
  12. C. J. S. de Matos, P. Torres, L. C. G. Valente, W. Margulis, and R. Stubbe, “Fiber Bragg grating (FBG) characterization and shaping by local pressure,” J. Lightwave Technol. 19, 1206-1211 (2001).
    [CrossRef]
  13. N. Roussel, S. Magne, C. Martinez, and P. Ferdinand, “Measurement of index modulation along fiber Bragg gratings by side scattering and local heating techniques,” Opt. Fiber Technol. 5, 119-132 (1999).
    [CrossRef]
  14. E. Brinkmeyer, G. Stolze, and D. Johlen, “Optical space domain reflectometry (OSDR) for determination of strength and chirp distribution along optical fiber gratings,” in Proceedings of the Optical Society of America Conference on Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides (BGPP) (OSA, 1997), paper BsuC2-1.
  15. I. G. Korolev, S. A. Vasil'ev, O. I. Medvedkov, and E. M. Dianov, “Study of local properties of fibre Bragg gratings by the method of optical space-domain reflectometry,” Quantum Electron. 33, 704-710 (2003).
    [CrossRef]
  16. This is derived from the definition of the argument of the coupling coefficient, arg(q(z))=−(4ηπ/λ)∫0zΔndc(z′)dz′, where Δndc is the DC component of the index change and Δndc(z)=∂n/∂T·ΔT(z)
  17. J. Skaar, “Synthesis and characterization of fiber Bragg gratings,” Ph.D. dissertation (The Norwegian University of Science and Technology, 2000) Chap. 2.
  18. H. Hsu, Applied Fourier Analysis (Harcourt Brace Jovanovich, 1984), Chaps. 5 and 7.
  19. W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39, 693-695(1981).
    [CrossRef]
  20. M. Froggatt, “Distributed measurement of the complex modulation of a photoinduced Bragg grating in an optical fiber,” Appl. Opt. 35, 5162-5164 (1996).
    [CrossRef] [PubMed]
  21. J. Skaar, L. G. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165-173 (2001).
    [CrossRef]
  22. R. J. Espejo and S. D. Dyer, “Practical spatial resolution limits of high-resolution fibre Bragg grating sensors using layer peeling,” Meas. Sci. Technol. 18, 1661-1666 (2007), Fig. 9.
    [CrossRef]
  23. This can also be used as an alternative means of characterizing strong gratings ; however, in this case the temperature perturbation must be large (>100 K) to separate the spectrum of the locally perturbed region from the main grating spectrum and phase information is generally lost.
  24. A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18, 1647-1653(1982).
    [CrossRef]
  25. G. A. Cranch, “Reconstruction of a strong fiber Bragg grating's complex coupling coefficient in erbium doped fiber with optical space domain reflectometry,” presented at the Conference on Lasers and Electro-Optics (CLEO), Baltimore, Maryland, 31 May-5 June2009, paper CThE5.

2007

R. J. Espejo and S. D. Dyer, “Practical spatial resolution limits of high-resolution fibre Bragg grating sensors using layer peeling,” Meas. Sci. Technol. 18, 1661-1666 (2007), Fig. 9.
[CrossRef]

2006

2005

2003

P. Giaccari, H. G. Limberger, and R. P. Salathe, “Local coupling-coefficient characterization in fiber Bragg gratings,” Opt. Lett. 28, 598-600 (2003).
[CrossRef] [PubMed]

I. Petermann, S. Helmfrid, and P. Y. Fonjallaz, “Fibre Bragg grating characterization with ultraviolet-based interferometric side diffraction,” J. Opt. A Pure Appl. Opt. 5, 437-441(2003).
[CrossRef]

I. G. Korolev, S. A. Vasil'ev, O. I. Medvedkov, and E. M. Dianov, “Study of local properties of fibre Bragg gratings by the method of optical space-domain reflectometry,” Quantum Electron. 33, 704-710 (2003).
[CrossRef]

2002

I. Petermann, S. Helmfrid, and A. T. Friberg, “Limitations of the interferometric side diffraction technique for fibre Bragg grating characterization,” Opt. Commun. 201, 301-308 (2002).
[CrossRef]

2001

2000

1999

N. Roussel, S. Magne, C. Martinez, and P. Ferdinand, “Measurement of index modulation along fiber Bragg gratings by side scattering and local heating techniques,” Opt. Fiber Technol. 5, 119-132 (1999).
[CrossRef]

1997

1996

M. Froggatt, “Distributed measurement of the complex modulation of a photoinduced Bragg grating in an optical fiber,” Appl. Opt. 35, 5162-5164 (1996).
[CrossRef] [PubMed]

J. Canning, M. Janos, D. Y. Stepanov, and M. G. Sceats, “Direct measurement of grating chirp using resonant side scatter spectra,” Electron. Lett. 32, 1608-1610 (1996).
[CrossRef]

1995

1982

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18, 1647-1653(1982).
[CrossRef]

1981

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39, 693-695(1981).
[CrossRef]

Boisrobert, C.

Brinkmeyer, E.

E. Brinkmeyer, G. Stolze, and D. Johlen, “Optical space domain reflectometry (OSDR) for determination of strength and chirp distribution along optical fiber gratings,” in Proceedings of the Optical Society of America Conference on Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides (BGPP) (OSA, 1997), paper BsuC2-1.

Brodzeli, Z.

Canning, J.

J. Canning, D. C. Psaila, Z. Brodzeli, A. Higley, and M. Janos, “Characterization of apodized fiber Bragg gratings for rejection filter applications,” Appl. Opt. 36, 9378-9382(1997).
[CrossRef]

J. Canning, M. Janos, D. Y. Stepanov, and M. G. Sceats, “Direct measurement of grating chirp using resonant side scatter spectra,” Electron. Lett. 32, 1608-1610 (1996).
[CrossRef]

Chapeleau, X.

Cranch, G. A.

G. A. Cranch, “Reconstruction of a strong fiber Bragg grating's complex coupling coefficient in erbium doped fiber with optical space domain reflectometry,” presented at the Conference on Lasers and Electro-Optics (CLEO), Baltimore, Maryland, 31 May-5 June2009, paper CThE5.

Dandridge, A.

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18, 1647-1653(1982).
[CrossRef]

de Matos, C. J. S.

Dianov, E. M.

I. G. Korolev, S. A. Vasil'ev, O. I. Medvedkov, and E. M. Dianov, “Study of local properties of fibre Bragg gratings by the method of optical space-domain reflectometry,” Quantum Electron. 33, 704-710 (2003).
[CrossRef]

Douay, M.

Dyer, S. D.

R. J. Espejo and S. D. Dyer, “Practical spatial resolution limits of high-resolution fibre Bragg grating sensors using layer peeling,” Meas. Sci. Technol. 18, 1661-1666 (2007), Fig. 9.
[CrossRef]

Eickhoff, W.

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39, 693-695(1981).
[CrossRef]

El-Diasty, F.

Erdogan, T.

F. El-Diasty, A. Heaney, and T. Erdogan, “Analysis of fiber Bragg gratings by a side-diffraction interference technique,” Appl. Opt. 40, 890-896 (2001).
[CrossRef]

J. Skaar, L. G. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165-173 (2001).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

Espejo, R. J.

R. J. Espejo and S. D. Dyer, “Practical spatial resolution limits of high-resolution fibre Bragg grating sensors using layer peeling,” Meas. Sci. Technol. 18, 1661-1666 (2007), Fig. 9.
[CrossRef]

Feced, R.

Ferdinand, P.

N. Roussel, S. Magne, C. Martinez, and P. Ferdinand, “Measurement of index modulation along fiber Bragg gratings by side scattering and local heating techniques,” Opt. Fiber Technol. 5, 119-132 (1999).
[CrossRef]

Fonjallaz, P. Y.

I. Petermann, S. Helmfrid, and P. Y. Fonjallaz, “Fibre Bragg grating characterization with ultraviolet-based interferometric side diffraction,” J. Opt. A Pure Appl. Opt. 5, 437-441(2003).
[CrossRef]

Friberg, A. T.

I. Petermann, S. Helmfrid, and A. T. Friberg, “Limitations of the interferometric side diffraction technique for fibre Bragg grating characterization,” Opt. Commun. 201, 301-308 (2002).
[CrossRef]

Froggatt, M.

Giaccari, P.

Giallorenzi, T. G.

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18, 1647-1653(1982).
[CrossRef]

Heaney, A.

Helmfrid, S.

I. Petermann, S. Helmfrid, and P. Y. Fonjallaz, “Fibre Bragg grating characterization with ultraviolet-based interferometric side diffraction,” J. Opt. A Pure Appl. Opt. 5, 437-441(2003).
[CrossRef]

I. Petermann, S. Helmfrid, and A. T. Friberg, “Limitations of the interferometric side diffraction technique for fibre Bragg grating characterization,” Opt. Commun. 201, 301-308 (2002).
[CrossRef]

Higley, A.

Hsu, H.

H. Hsu, Applied Fourier Analysis (Harcourt Brace Jovanovich, 1984), Chaps. 5 and 7.

Janos, M.

J. Canning, D. C. Psaila, Z. Brodzeli, A. Higley, and M. Janos, “Characterization of apodized fiber Bragg gratings for rejection filter applications,” Appl. Opt. 36, 9378-9382(1997).
[CrossRef]

J. Canning, M. Janos, D. Y. Stepanov, and M. G. Sceats, “Direct measurement of grating chirp using resonant side scatter spectra,” Electron. Lett. 32, 1608-1610 (1996).
[CrossRef]

Johlen, D.

E. Brinkmeyer, G. Stolze, and D. Johlen, “Optical space domain reflectometry (OSDR) for determination of strength and chirp distribution along optical fiber gratings,” in Proceedings of the Optical Society of America Conference on Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides (BGPP) (OSA, 1997), paper BsuC2-1.

Korolev, I. G.

I. G. Korolev, S. A. Vasil'ev, O. I. Medvedkov, and E. M. Dianov, “Study of local properties of fibre Bragg gratings by the method of optical space-domain reflectometry,” Quantum Electron. 33, 704-710 (2003).
[CrossRef]

Krug, P. A.

Le Ny, R.

Leduc, D.

Limberger, H. G.

Lopez-Gejo, F.

Lupi, C.

Magne, S.

N. Roussel, S. Magne, C. Martinez, and P. Ferdinand, “Measurement of index modulation along fiber Bragg gratings by side scattering and local heating techniques,” Opt. Fiber Technol. 5, 119-132 (1999).
[CrossRef]

Margulis, W.

Martinez, C.

N. Roussel, S. Magne, C. Martinez, and P. Ferdinand, “Measurement of index modulation along fiber Bragg gratings by side scattering and local heating techniques,” Opt. Fiber Technol. 5, 119-132 (1999).
[CrossRef]

Medvedkov, O. I.

I. G. Korolev, S. A. Vasil'ev, O. I. Medvedkov, and E. M. Dianov, “Study of local properties of fibre Bragg gratings by the method of optical space-domain reflectometry,” Quantum Electron. 33, 704-710 (2003).
[CrossRef]

Petermann, I.

I. Petermann, S. Helmfrid, and P. Y. Fonjallaz, “Fibre Bragg grating characterization with ultraviolet-based interferometric side diffraction,” J. Opt. A Pure Appl. Opt. 5, 437-441(2003).
[CrossRef]

I. Petermann, S. Helmfrid, and A. T. Friberg, “Limitations of the interferometric side diffraction technique for fibre Bragg grating characterization,” Opt. Commun. 201, 301-308 (2002).
[CrossRef]

Psaila, D. C.

Roussel, N.

N. Roussel, S. Magne, C. Martinez, and P. Ferdinand, “Measurement of index modulation along fiber Bragg gratings by side scattering and local heating techniques,” Opt. Fiber Technol. 5, 119-132 (1999).
[CrossRef]

Salathe, R. P.

Sceats, M. G.

J. Canning, M. Janos, D. Y. Stepanov, and M. G. Sceats, “Direct measurement of grating chirp using resonant side scatter spectra,” Electron. Lett. 32, 1608-1610 (1996).
[CrossRef]

Skaar, J.

J. Skaar, L. G. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165-173 (2001).
[CrossRef]

J. Skaar, “Synthesis and characterization of fiber Bragg gratings,” Ph.D. dissertation (The Norwegian University of Science and Technology, 2000) Chap. 2.

Stepanov, D. Y.

J. Canning, M. Janos, D. Y. Stepanov, and M. G. Sceats, “Direct measurement of grating chirp using resonant side scatter spectra,” Electron. Lett. 32, 1608-1610 (1996).
[CrossRef]

Stolte, R.

Stolze, G.

E. Brinkmeyer, G. Stolze, and D. Johlen, “Optical space domain reflectometry (OSDR) for determination of strength and chirp distribution along optical fiber gratings,” in Proceedings of the Optical Society of America Conference on Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides (BGPP) (OSA, 1997), paper BsuC2-1.

Stubbe, R.

Torres, P.

Tveten, A. B.

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18, 1647-1653(1982).
[CrossRef]

Ulrich, R.

P. A. Krug, R. Stolte, and R. Ulrich, “Measurement of index modulation along an optical fiber Bragg grating,” Opt. Lett. 20, 1767-1769 (1995).
[CrossRef] [PubMed]

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39, 693-695(1981).
[CrossRef]

Valente, L. C. G.

Vasil'ev, S. A.

I. G. Korolev, S. A. Vasil'ev, O. I. Medvedkov, and E. M. Dianov, “Study of local properties of fibre Bragg gratings by the method of optical space-domain reflectometry,” Quantum Electron. 33, 704-710 (2003).
[CrossRef]

Waagaard, O. H.

Wang, L. G.

J. Skaar, L. G. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165-173 (2001).
[CrossRef]

Zervas, M. N.

Appl. Opt.

Appl. Phys. Lett.

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39, 693-695(1981).
[CrossRef]

Electron. Lett.

J. Canning, M. Janos, D. Y. Stepanov, and M. G. Sceats, “Direct measurement of grating chirp using resonant side scatter spectra,” Electron. Lett. 32, 1608-1610 (1996).
[CrossRef]

IEEE J. Quantum Electron.

J. Skaar, L. G. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165-173 (2001).
[CrossRef]

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. 18, 1647-1653(1982).
[CrossRef]

J. Lightwave Technol.

J. Opt. A Pure Appl. Opt.

I. Petermann, S. Helmfrid, and P. Y. Fonjallaz, “Fibre Bragg grating characterization with ultraviolet-based interferometric side diffraction,” J. Opt. A Pure Appl. Opt. 5, 437-441(2003).
[CrossRef]

Meas. Sci. Technol.

R. J. Espejo and S. D. Dyer, “Practical spatial resolution limits of high-resolution fibre Bragg grating sensors using layer peeling,” Meas. Sci. Technol. 18, 1661-1666 (2007), Fig. 9.
[CrossRef]

Opt. Commun.

I. Petermann, S. Helmfrid, and A. T. Friberg, “Limitations of the interferometric side diffraction technique for fibre Bragg grating characterization,” Opt. Commun. 201, 301-308 (2002).
[CrossRef]

Opt. Fiber Technol.

N. Roussel, S. Magne, C. Martinez, and P. Ferdinand, “Measurement of index modulation along fiber Bragg gratings by side scattering and local heating techniques,” Opt. Fiber Technol. 5, 119-132 (1999).
[CrossRef]

Opt. Lett.

Quantum Electron.

I. G. Korolev, S. A. Vasil'ev, O. I. Medvedkov, and E. M. Dianov, “Study of local properties of fibre Bragg gratings by the method of optical space-domain reflectometry,” Quantum Electron. 33, 704-710 (2003).
[CrossRef]

Other

This is derived from the definition of the argument of the coupling coefficient, arg(q(z))=−(4ηπ/λ)∫0zΔndc(z′)dz′, where Δndc is the DC component of the index change and Δndc(z)=∂n/∂T·ΔT(z)

J. Skaar, “Synthesis and characterization of fiber Bragg gratings,” Ph.D. dissertation (The Norwegian University of Science and Technology, 2000) Chap. 2.

H. Hsu, Applied Fourier Analysis (Harcourt Brace Jovanovich, 1984), Chaps. 5 and 7.

E. Brinkmeyer, G. Stolze, and D. Johlen, “Optical space domain reflectometry (OSDR) for determination of strength and chirp distribution along optical fiber gratings,” in Proceedings of the Optical Society of America Conference on Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides (BGPP) (OSA, 1997), paper BsuC2-1.

G. A. Cranch, “Reconstruction of a strong fiber Bragg grating's complex coupling coefficient in erbium doped fiber with optical space domain reflectometry,” presented at the Conference on Lasers and Electro-Optics (CLEO), Baltimore, Maryland, 31 May-5 June2009, paper CThE5.

This can also be used as an alternative means of characterizing strong gratings ; however, in this case the temperature perturbation must be large (>100 K) to separate the spectrum of the locally perturbed region from the main grating spectrum and phase information is generally lost.

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Figures (7)

Fig. 1
Fig. 1

Linear system representation of OSDR.

Fig. 2
Fig. 2

(a) Temperature profile of perturbation for a stationary beam and a beam moving at 1 mm / s and (b) spatial frequency spectra of heat perturbations.

Fig. 3
Fig. 3

Dependence of OSDR signal on (a) Δ T p k and (b) δ m .

Fig. 4
Fig. 4

Experimental setup to implement OSDR.

Fig. 5
Fig. 5

OSDR signal from a (a)  5 cm Bartlett apodized weak FBG and a (b)  5 cm phase-shifted, apodized strong FBG.

Fig. 6
Fig. 6

(a) Deconvolved OSDR measurement (solid curve) and the reconstructed coupling coefficient of a weak grating using OFDR and layer peeling (dotted curve) and (b) reflectivity calculated using the transfer matrix method from the coupling coefficient derived by OSDR (thick solid curve), corresponding reflectivity spectra measured with OFDR (thin solid curve) and reflectivity cal culated from OSDR measurement without deconvolution (dashed curve).

Fig. 7
Fig. 7

(a) Reconstructed coupling coefficient of a strong grating using OSDR measurement (dashed curve) and deconvolved OSDR (D-OSDR) measurement (solid curve) and (b) corresponding reflectivity calculated using the transfer matrix method from the coupling coefficient derived by OSDR (short-dashed curve) and deconvolved OSDR (solid curve) and reflectivity spectrum measured with OFDR (long-dashed curve).

Equations (11)

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

d u d z = i δ u + q ( z ) v ,
d v d z = i δ v + q * ( z ) u ,
q ¯ L = 0 L q ( z ) d z ,
r ( δ ) = 0 q * ( z ) exp ( 2 i δ z ) d z .
r ˜ ( δ m ; z ) = 0 q * ( z ) exp ( i ϕ 0 ( z z ) ) exp ( 2 i δ m z ) d z .
r ˜ ( δ m ; z ) = r ( δ m ) + i 0 q * ( z ) ϕ 0 ( z z ) exp ( 2 i δ m z ) d z .
d r ˜ ( δ m ; z ) d z = 2 i η 2 π λ n T 0 q * ( z ) Δ T ( z z ) exp ( 2 i δ m z ) d z ,
d ϕ 0 ( z z ) d z = 2 η 2 π λ n T Δ T ( z z ) ,
d r ˜ ( δ m ; z ) d z = 2 i η 2 π λ n T { q * ( z ) exp ( 2 i δ m z ) Δ T ( z ) } ,
d r ˜ ( δ m ; z ) d z = i ϕ 0 q * ( z ) exp ( 2 i δ m z ) ,
Δ z = π / 2 | δ 1 / 2 | ,

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