Abstract

The homogeneity test of glass plates in a Fizeau interferometer requires the measurement of the glass sample in reflected as well as in transmitted light. For the measurement in transmitted light, the sample has to be inserted into the ray path of a Fizeau or Twyman–Green interferometer, which leads to a nested cavity setup. To separate the interference signals from the different cavities, we illuminate a Fizeau interferometer with an adaptive frequency comb. In this way, rigid glass plates can be measured, and linear variations in the homogeneity can also be detected. The adaptive frequency comb is provided by a variable Fabry–Perot filter under broadband illumination from a superluminescence diode. Compared to approaches using a two-beam interferometer as a filter for the broadband light source, the visibility of the fringe system is considerably higher.

© 2013 Optical Society of America

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References

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  1. J. Schwider, “Interferometrische homogenitaetspruefung mit kompensation,” Opt. Commun. 6, 106–110 (1972).
    [CrossRef]
  2. G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed., Vol. XIII, (Elsevier, 1976), pp. 93–167.
  3. F. Twyman and J. W. Perry, “The determination of Poisson’s ratio and of the absolute stress-variation of refractive index,” Proc. Phys. Soc. London 34, 151–154 (1921).
    [CrossRef]
  4. F. W. Rosberry, “The measurement of homogeneity of optical materials in the visible and near infrared,” Appl. Opt. 5, 961–966 (1966).
    [CrossRef]
  5. F. E. Roberts and P. Langenbeck, “Homogeneity evaluation of very large disks,” Appl. Opt. 8, 2311–2314 (1969).
    [CrossRef]
  6. W. Kowalik, “Interference measurement of continuous heterogeneities in optical materials,” Appl. Opt. 17, 2956–2966 (1978).
    [CrossRef]
  7. J. Schwider, R. Burow, K.-E. Elßner, R. Spolaczyk, and J. Grzanna, “Homogeneity testing by phase sampling interferometry,” Appl. Opt. 24, 3059–3061 (1985).
    [CrossRef]
  8. M. K. Okada, H. Sakuta, T. Ose, and J. Tsujiuchi, “Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable source phase shifting interferometry,” Appl. Opt. 29, 3280–3285 (1990).
    [CrossRef]
  9. M. Suematsu and M. Takeda, “Wavelength-shift interferometry for distance measurements using the Fourier transform technique of fringe analysis,” Appl. Opt. 30, 4046–4055 (1991).
    [CrossRef]
  10. P. de Groot, “Measurement of transparent plates with wavelength-tuned phase shifting interferometry,” Appl. Opt. 39, 2658–2663 (2000).
    [CrossRef]
  11. L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. 42, 2354–2365 (2003).
    [CrossRef]
  12. D. Battistoni, “FT interferometry measures homogeneity,” Photonic Spectra (2004), http://www.photonics.com/Article.aspx?AID=18379 .
  13. K. Hibino, J. Burke, R. Hanayama, and B. F. Oreb, “Multiple-surface testing by a wavelength-scanning interferometer for refractive index inhomogeneity measurement,” Opt. Spectrosc. 101, 18–22 (2006).
    [CrossRef]
  14. J. Schwider, “Coarse frequency comb interferometry,” Proc. SPIE 7063, 706304 (2008).
    [CrossRef]
  15. I. Harder, G. Leuchs, K. Mantel, and J. Schwider, “Adaptive frequency comb illumination for interferometry in the case of nested two-beam cavities,” Appl. Opt. 50, 4942–4956 (2011).
    [CrossRef]
  16. Z. Bor, K. Osvay, B. Racz, and G. Szabo, “Group refractive index measurement Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
    [CrossRef]
  17. J. Schwider and K. Mantel, “Homogeneity test of glass plates using adaptive frequency comb illumination in Fizeau interferometry,” in EOS Topical Meeting on Optical Microsystems (OμS11) (European Optical Society, 2011) paper 4556.
  18. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
    [CrossRef]
  19. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef]
  20. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef]
  21. P. Hariharan, “Digital phase stepping interferometry: effects of multiply reflected beams,” Appl. Opt. 26, 2506–2507(1987).
    [CrossRef]
  22. J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 3889–3892 (1989).
    [CrossRef]
  23. J. Schwider, T. Dresel, and B. Manzke, “Some considerations of reduction of reference phase error in phase-stepping interferometry,” Appl. Opt. 38, 655–659 (1999).
    [CrossRef]
  24. http://www.optik.uni-erlangen.de/odem/index.php?lang=e&type=6&topic=raytrace .

2011 (1)

2008 (1)

J. Schwider, “Coarse frequency comb interferometry,” Proc. SPIE 7063, 706304 (2008).
[CrossRef]

2006 (1)

K. Hibino, J. Burke, R. Hanayama, and B. F. Oreb, “Multiple-surface testing by a wavelength-scanning interferometer for refractive index inhomogeneity measurement,” Opt. Spectrosc. 101, 18–22 (2006).
[CrossRef]

2003 (1)

2000 (1)

1999 (1)

1995 (1)

1991 (1)

1990 (2)

1989 (1)

1987 (1)

1985 (1)

1983 (1)

1978 (1)

1974 (1)

1972 (1)

J. Schwider, “Interferometrische homogenitaetspruefung mit kompensation,” Opt. Commun. 6, 106–110 (1972).
[CrossRef]

1969 (1)

1966 (1)

1921 (1)

F. Twyman and J. W. Perry, “The determination of Poisson’s ratio and of the absolute stress-variation of refractive index,” Proc. Phys. Soc. London 34, 151–154 (1921).
[CrossRef]

Battistoni, D.

D. Battistoni, “FT interferometry measures homogeneity,” Photonic Spectra (2004), http://www.photonics.com/Article.aspx?AID=18379 .

Bor, Z.

Z. Bor, K. Osvay, B. Racz, and G. Szabo, “Group refractive index measurement Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
[CrossRef]

Brangaccio, D. J.

Bruning, J. H.

Burke, J.

K. Hibino, J. Burke, R. Hanayama, and B. F. Oreb, “Multiple-surface testing by a wavelength-scanning interferometer for refractive index inhomogeneity measurement,” Opt. Spectrosc. 101, 18–22 (2006).
[CrossRef]

Burow, R.

Creath, K.

de Groot, P.

Deck, L.

Dresel, T.

Elßner, K.-E.

Elssner, K.-E.

Gallagher, J. E.

Grzanna, J.

Hanayama, R.

K. Hibino, J. Burke, R. Hanayama, and B. F. Oreb, “Multiple-surface testing by a wavelength-scanning interferometer for refractive index inhomogeneity measurement,” Opt. Spectrosc. 101, 18–22 (2006).
[CrossRef]

Harder, I.

Hariharan, P.

Herriott, D. R.

Hibino, K.

K. Hibino, J. Burke, R. Hanayama, and B. F. Oreb, “Multiple-surface testing by a wavelength-scanning interferometer for refractive index inhomogeneity measurement,” Opt. Spectrosc. 101, 18–22 (2006).
[CrossRef]

Kowalik, W.

Langenbeck, P.

Leuchs, G.

Mantel, K.

I. Harder, G. Leuchs, K. Mantel, and J. Schwider, “Adaptive frequency comb illumination for interferometry in the case of nested two-beam cavities,” Appl. Opt. 50, 4942–4956 (2011).
[CrossRef]

J. Schwider and K. Mantel, “Homogeneity test of glass plates using adaptive frequency comb illumination in Fizeau interferometry,” in EOS Topical Meeting on Optical Microsystems (OμS11) (European Optical Society, 2011) paper 4556.

Manzke, B.

Merkel, K.

Okada, M. K.

Oreb, B. F.

K. Hibino, J. Burke, R. Hanayama, and B. F. Oreb, “Multiple-surface testing by a wavelength-scanning interferometer for refractive index inhomogeneity measurement,” Opt. Spectrosc. 101, 18–22 (2006).
[CrossRef]

Ose, T.

Osvay, K.

Z. Bor, K. Osvay, B. Racz, and G. Szabo, “Group refractive index measurement Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
[CrossRef]

Perry, J. W.

F. Twyman and J. W. Perry, “The determination of Poisson’s ratio and of the absolute stress-variation of refractive index,” Proc. Phys. Soc. London 34, 151–154 (1921).
[CrossRef]

Racz, B.

Z. Bor, K. Osvay, B. Racz, and G. Szabo, “Group refractive index measurement Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
[CrossRef]

Roberts, F. E.

Rosberry, F. W.

Rosenfeld, D. P.

Sakuta, H.

Schmit, J.

Schulz, G.

G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed., Vol. XIII, (Elsevier, 1976), pp. 93–167.

Schwider, J.

I. Harder, G. Leuchs, K. Mantel, and J. Schwider, “Adaptive frequency comb illumination for interferometry in the case of nested two-beam cavities,” Appl. Opt. 50, 4942–4956 (2011).
[CrossRef]

J. Schwider, “Coarse frequency comb interferometry,” Proc. SPIE 7063, 706304 (2008).
[CrossRef]

J. Schwider, T. Dresel, and B. Manzke, “Some considerations of reduction of reference phase error in phase-stepping interferometry,” Appl. Opt. 38, 655–659 (1999).
[CrossRef]

J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 3889–3892 (1989).
[CrossRef]

J. Schwider, R. Burow, K.-E. Elßner, R. Spolaczyk, and J. Grzanna, “Homogeneity testing by phase sampling interferometry,” Appl. Opt. 24, 3059–3061 (1985).
[CrossRef]

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef]

J. Schwider, “Interferometrische homogenitaetspruefung mit kompensation,” Opt. Commun. 6, 106–110 (1972).
[CrossRef]

G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed., Vol. XIII, (Elsevier, 1976), pp. 93–167.

J. Schwider and K. Mantel, “Homogeneity test of glass plates using adaptive frequency comb illumination in Fizeau interferometry,” in EOS Topical Meeting on Optical Microsystems (OμS11) (European Optical Society, 2011) paper 4556.

Spolaczyk, R.

Suematsu, M.

Szabo, G.

Z. Bor, K. Osvay, B. Racz, and G. Szabo, “Group refractive index measurement Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
[CrossRef]

Takeda, M.

Tsujiuchi, J.

Twyman, F.

F. Twyman and J. W. Perry, “The determination of Poisson’s ratio and of the absolute stress-variation of refractive index,” Proc. Phys. Soc. London 34, 151–154 (1921).
[CrossRef]

White, A. D.

Appl. Opt. (15)

F. W. Rosberry, “The measurement of homogeneity of optical materials in the visible and near infrared,” Appl. Opt. 5, 961–966 (1966).
[CrossRef]

F. E. Roberts and P. Langenbeck, “Homogeneity evaluation of very large disks,” Appl. Opt. 8, 2311–2314 (1969).
[CrossRef]

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef]

W. Kowalik, “Interference measurement of continuous heterogeneities in optical materials,” Appl. Opt. 17, 2956–2966 (1978).
[CrossRef]

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef]

J. Schwider, R. Burow, K.-E. Elßner, R. Spolaczyk, and J. Grzanna, “Homogeneity testing by phase sampling interferometry,” Appl. Opt. 24, 3059–3061 (1985).
[CrossRef]

J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 3889–3892 (1989).
[CrossRef]

M. K. Okada, H. Sakuta, T. Ose, and J. Tsujiuchi, “Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable source phase shifting interferometry,” Appl. Opt. 29, 3280–3285 (1990).
[CrossRef]

M. Suematsu and M. Takeda, “Wavelength-shift interferometry for distance measurements using the Fourier transform technique of fringe analysis,” Appl. Opt. 30, 4046–4055 (1991).
[CrossRef]

J. Schwider, T. Dresel, and B. Manzke, “Some considerations of reduction of reference phase error in phase-stepping interferometry,” Appl. Opt. 38, 655–659 (1999).
[CrossRef]

J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[CrossRef]

P. de Groot, “Measurement of transparent plates with wavelength-tuned phase shifting interferometry,” Appl. Opt. 39, 2658–2663 (2000).
[CrossRef]

L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. 42, 2354–2365 (2003).
[CrossRef]

P. Hariharan, “Digital phase stepping interferometry: effects of multiply reflected beams,” Appl. Opt. 26, 2506–2507(1987).
[CrossRef]

I. Harder, G. Leuchs, K. Mantel, and J. Schwider, “Adaptive frequency comb illumination for interferometry in the case of nested two-beam cavities,” Appl. Opt. 50, 4942–4956 (2011).
[CrossRef]

Opt. Commun. (2)

J. Schwider, “Interferometrische homogenitaetspruefung mit kompensation,” Opt. Commun. 6, 106–110 (1972).
[CrossRef]

Z. Bor, K. Osvay, B. Racz, and G. Szabo, “Group refractive index measurement Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
[CrossRef]

Opt. Spectrosc. (1)

K. Hibino, J. Burke, R. Hanayama, and B. F. Oreb, “Multiple-surface testing by a wavelength-scanning interferometer for refractive index inhomogeneity measurement,” Opt. Spectrosc. 101, 18–22 (2006).
[CrossRef]

Proc. Phys. Soc. London (1)

F. Twyman and J. W. Perry, “The determination of Poisson’s ratio and of the absolute stress-variation of refractive index,” Proc. Phys. Soc. London 34, 151–154 (1921).
[CrossRef]

Proc. SPIE (1)

J. Schwider, “Coarse frequency comb interferometry,” Proc. SPIE 7063, 706304 (2008).
[CrossRef]

Other (4)

J. Schwider and K. Mantel, “Homogeneity test of glass plates using adaptive frequency comb illumination in Fizeau interferometry,” in EOS Topical Meeting on Optical Microsystems (OμS11) (European Optical Society, 2011) paper 4556.

http://www.optik.uni-erlangen.de/odem/index.php?lang=e&type=6&topic=raytrace .

D. Battistoni, “FT interferometry measures homogeneity,” Photonic Spectra (2004), http://www.photonics.com/Article.aspx?AID=18379 .

G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed., Vol. XIII, (Elsevier, 1976), pp. 93–167.

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Figures (23)

Fig. 1.
Fig. 1.

Principle of the adaptive frequency comb interferometer in the time domain, where white light can be considered as a Fourier-limited pulse. A FP-filter generates a train of such pulses from a broadband light source. Upon reflection off a Fizeau cavity, two copies of the pulse train are generated. By varying the thickness of the FP-filter, the pulse trains can be matched, giving rise to high contrast interference fringes. If the resonator lengths are incommensurable, the signals from the other cavities give rise to an incoherent background only.

Fig. 2.
Fig. 2.

Refractive index variations Δn, i.e., the “inhomogeneity” of the specimen, has to be separated from the surface deviations x, y. t20 is the mean thickness of the specimen.

Fig. 3.
Fig. 3.

Combination of reflected and transmitted light tests can be used to determine the refractive index variations. A measurement of the Fizeau cavity alone eliminates the surface deviations of the Fizeau plates, leaving the index variations in an absolute sense.

Fig. 4.
Fig. 4.

Homogeneity test overcoming the multiple beam interference problem (schematic). The plate under test is provided with a wedge, and is adjusted that in one instance, the front surface, and in the other instance, the back surface of the specimen is oriented collinearly with the reference surface, during the course of the measurement. The respective parasitic reflections leave the cavity under a suitable angle and are screened by a stop in the imaging telescope.

Fig. 5.
Fig. 5.

Optical path lengths in a Fizeau cavity. The surface deviations are counted positive along the outer surface normal. The numbers on the left refer to the corresponding wave aberrations Wi. The quantities t10, t20, and t30 represent the distances between mathematical reference planes, located at A, D, E, and H.

Fig. 6.
Fig. 6.

Measuring the refractive index variations by determining the geometrical and optical thickness of the specimen separately.

Fig. 7.
Fig. 7.

Setup. Light source space (left part) and Fizeau space (right part) are connected by a multimode fiber.

Fig. 8.
Fig. 8.

Measurement result for the refractive index variation Δn (left) and thickness variations (right), measured via the r/t-scheme (all data in waves). The p/v-value for Δn corresponds to 6.86nm/cm. A Zernike fit of degree 12 has been applied.

Fig. 9.
Fig. 9.

Reproducibility of the homogeneity variations (left) and thickness variations (right) obtained via the r/t-scheme (all data in waves). The p/v-value for Δn corresponds to 2.26nm/cm. A Zernike fit of degree 12 has been applied.

Fig. 10.
Fig. 10.

Measurement result for the refractive index variation Δn (left) and thickness variations (right), measured via the g/o scheme (all data in waves). The p/v-value for Δn corresponds to 6.60nm/cm. A Zernike fit of degree 12 has been applied.

Fig. 11.
Fig. 11.

Reproducibility of the homogeneity variations (left) and thickness variations (right) obtained via the g/o scheme (all data in waves). The p/v-value for Δn corresponds to 2.27nm/cm. A Zernike fit of degree 12 has been applied.

Fig. 12.
Fig. 12.

Comparison between the homogeneity variations (left) and the thickness variations (right), obtained via the r/t-scheme and the g/o scheme (all data in waves). The p/v-value for Δn corresponds to 3.25nm/cm. A Zernike fit of degree 12 has been applied.

Fig. 13.
Fig. 13.

Consistency test. Difference in the homogeneity (left) and thickness variations (right), calculated via W2W1 and via W5, respectively (for the r/t-scheme, all data in waves). The p/v-value for Δn corresponds to 1.26nm/cm. A Zernike fit of degree 12 has been applied.

Fig. 14.
Fig. 14.

Experimental setup of the stacked Fizeau interferometer with kinematic alignment units.

Fig. 15.
Fig. 15.

Demonstration of the drift sensitivity by subtracting the unwrapped phase values of two consecutive measurements. Left: unwrapped phase difference. Right: linear part of the phase difference.

Fig. 16.
Fig. 16.

Left: in the central region, defined by the position of the specimen, the measurement data W1 to W4 are evaluated. The rim region outside the specimen is used for tracking the tilt, correcting W3 and W4. Right: interferogram showing central and rim region.

Fig. 17.
Fig. 17.

Measurement result for the refractive index variation Δn in the r/t-scheme, including the linear function (all data in waves). The p/v-value for Δn corresponds to 11.35nm/cm. A Zernike fit of degree 12 has been applied.

Fig. 18.
Fig. 18.

Reproducibility of the homogeneity variations obtained via the r/t-scheme including the linear function (all data in waves). The p/v-value for Δn corresponds to 3.62nm/cm. A Zernike fit of degree 12 has been applied.

Fig. 19.
Fig. 19.

Measurement result for the refractive index variation Δn in the g/o scheme, including the linear function (all data in waves). The p/v-value for Δn corresponds to 19.73nm/cm. A Zernike fit of degree 12 has been applied.

Fig. 20.
Fig. 20.

Difference between a homogeneity test in the r/t- and the g/o scheme including the linear function (all data in waves). The p/v-value for Δn corresponds to 21.74nm/cm. A Zernike fit of degree 12 has been applied.

Fig. 21.
Fig. 21.

Consistency test. Difference between W6 and W3W5W1.

Fig. 22.
Fig. 22.

Residual periodic errors in the data. Top left: raw phase with several fringes. Top right: unwrapped phase. Bottom left: phase difference of two such consecutive runs freed from local noise through convolution of the data with a 5×5 unit kernel (gray scale has been adjusted for better visibility). Bottom right: noise level: difference of the difference between the low pass filtered data and the actual difference of two consecutive runs.

Fig. 23.
Fig. 23.

Left column: phase difference between object and reference wave travelling on axis, relative tilt angle 0.0005° (top); residual phase difference without linear term (bottom). Right column: phase difference between object and reference wave travelling under an angle of 0.1° relative to the optical axis, relative tilt angle 0.0005° (top); residual phase difference without linear term (bottom).

Equations (22)

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Δν=c2nairtFiz,
nairtFP=tgl{nλ0nλ0}=NtgltFP=Nnairtgl,
t=t20+(x+y)n=n0+Δn,
h=0t20Δn(s)ds,
OPD1=2nt2.
OPD2=2(t1+t3)+2nt2.
OPD3=2(t1+t3+t2),
ΔOPD=OPD2OPD3=2(n1)t2.
W1=2nairBC¯=2nair{t10(z1+x)},W2=2nairBC¯+2nCF¯=2nair{t10(z1+x)}+2n{t20+(x+y)}=2nair{t10(z1+x)}+2n0{t20+(x+y)}+2Δn{t20+(x+y)}W3=2nairBC¯+2nCF¯+2nairFG¯=2nair{t10(z1+x)}+2n{t20+(x+y)}+2nair{t30(y+z2)}=2nair{t10(z1+x)}+2nair{t30(y+z2)}+2(n0+Δn){t20+(x+y)}W4=2nairBG¯=2nair{t10+t20+t30(z1+z2)}W5=2nCF¯=2n0{t20+(x+y)}+2Δn{t20+(x+y)}W6=2nairFG¯=2nair{t30(y+z2)}W7=2nCF¯+2nairFG¯=2(n0+Δn){t20+(x+y)}+2nair{t30(y+z2)}.
W1=2nair(z1+x),W2=2nair(z1+x)+2Δnt20+2n0(x+y),W3=2nair(z1+x)+2Δnt20+2n0(x+y)2nair(y+z2),W4=2nair(z1+z2),W5=2Δnt20+2n0(x+y),W6=2nair(y+z2),W7=2n0(x+y)+2Δnt20+2Δn(x+y)2nair(y+z2).
W2W1=2Δnt20+2n0(x+y)=W5W3W4=2Δnt20+2(n0nair)(x+y).
Δn=12t20{n0nair(W3W4)(n0nair1)(W2W1)}Δd=12nair{(W2W1)(W3W4)},
W4W1W6=2nair(x+y)W2W1=2Δnt20+2n0(x+y)=W5.
Δn=12t20{W5n0nair(W4W1W6)}Δd=12nair(W4W1W6).
ΔΦ=arctan{ε¯Ccos2ΦSsin2Φ1Csin2Φ+Scos2Φ},
ε¯=1Rr=1Rεr,C=1Rr=1Rεrcos2φr,S=1Rr=1Rεrsin2φr
Q:=(Csin2ΦScos2Φ)1,
ΔΦarctan{(ε¯Ccos2ΦSsin2Φ)(1+Q+Q2)},
ΔΦ(ε¯Ccos2ΦSsin2Φ)(1+Q+Q2).
ΔΦ(ε¯Ccos2ΦSsin2Φ)+Q(ε¯Ccos2ΦSsin2Φ)+Q2(ε¯Ccos2ΦSsin2Φ).
ΔΦε¯(1+12(C2+S2))+(ε¯CS14(C2S+S3))sin2Φ+(ε¯SC14(CS2+C3))cos2Φ+(ε¯CS+12(S2C2))sin4Φ+(CS+ε¯2(S2C2))cos4Φ+(14(3C2SS3))sin6Φ+(14(3CS2+C3))cos6Φ.
Δn=n02t20nairL,

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