Abstract

The National Institute of Standards and Technology (NIST) has developed a nominally quarter-wave linear retarder for wavelengths near 1.3 µm that is stable within ±0.1° retardance over a range of wavelength, input angle, temperature, and environmental variations. The device consists of two concatenated Fresnel rhombs made from a low stress-optic-coefficient glass that minimizes the residual birefringence from machining and packaging. Device machining, assembly, and antireflection coating tolerances are discussed, and the theoretical performance is compared with measurements. Humidity can modify retardance of the total-internal-reflection surfaces; we discuss packaging that mitigates this effect and provides an estimated 10-year lifetime for the device. Several measurement methods were intercompared to ensure that the device retardance can be measured with an uncertainty less than 0.1°. Similar retarders will be certified by NIST and made available as Standard Reference Materials.

© 1997 Optical Society of America

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References

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  1. P. D. Hale, G. W. Day, “Stability of birefringent linear retarders (waveplates),” Appl. Opt. 27, 5146–5153 (1988).
    [Crossref] [PubMed]
  2. K. B. Rochford, P. A. Williams, A. H. Rose, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
    [Crossref]
  3. J. M. Bennett, “A critical evaluation of rhomb-type quarterwave retarders,” Appl. Opt. 9, 2123–2129 (1970).
    [Crossref] [PubMed]
  4. N. N. Nagib, S. A. Khodier, “Optimization of a rhomb-type quarter-wave phase retarder,” Appl. Opt. 34, 2927–2930 (1995).
  5. R. M. A. Azzam, M. M. K. Howlader, “Silicon-based polarization optics for the 1.3 and 1.55-µm communications wavelengths,” J. Lightwave Technol. 14, 873–878 (1996).
    [Crossref]
  6. N. N. Nagib, M. S. El-Bahrawy, S. E. Demian, S. Khodier, “Oblique incidence in total internal reflection phase retarders. II - Application,” Opt. Pura Apl. 27, 111–116 (1994).
  7. R. J. King, “Quarter-wave retardation systems based on the Fresnel rhomb principle,” J. Sci. Instrum. 43, 617–622 (1966).
    [Crossref]
  8. A. E. Oxley, “On apparatus for the production of circularly polarized light,” Philos. Mag. 21, 517–532 (1911).
    [Crossref]
  9. V. A. Kizel, Y. I. Krasilov, V. N. Shamraev, “Achromatic λ/4 device,” Opt. Spectrosc. 17, 248, 249 (1964).
  10. SF-57 the Schott Glaswerke (Mainz Germany) trade name for its 847238 glass, was used in our rhomb. NIST does not endorse this glass, and similar glasses produced by other manufacturers might work as well or better. The glass code (in the form xxxyyy, where xxx = nd - 1, and yyy = νD × 10) gives the refractive index and Abbé number at 587.6 nm.
  11. “The stress-optical coefficients of optical glasses,” Technical Note 15, 1984 (Schott Glaswerke, Mainz, Germany).
  12. G. W. Day, A. H. Rose, “Faraday effect sensors: the state of the art,” in Fiber Optic and Laser Sensors, R. P. DePaula, E. Udd, eds., Proc. SPIE985, 138–150 (1998).
  13. The following Sellmeier equation for use in the near infrared was supplied by Schott Glass Technologies, Inc., Duryea, Pa.: n2 = 1 + B1λ2/(λ2 - C1) + B2λ2/(λ2 - C2) + B3λ2/(λ2 - C3), where λ is the wavelength (in micrometers) B1 = 1.816 513 71, B2= 0.428 893 6, B3 = 1.071 862 78, C1 = 1.437 041 98 × 10-2, C2 = 5.928 011 72 × 10-2, and C3 = 121.419 941.
  14. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980) p. 50.
  15. E. Hecht, A. Zajak, Optics (Addison-Wesley, Reading, Mass., 1974), pp. 301–305.
  16. A. M. Mood, F. A. Graybill, D. C. Boes, Introduction to the Theory of Statistics (McGraw-Hill, New York, 1974), p. 175.
  17. B. N. Taylor, C. E. Kuyatt, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurements, NIST Technical Note 1297, 1993 (U.S. Government Printing Office, Washington, D.C.).
  18. P. A. Williams, A. H. Rose, C. M. Wang, Rotating-polarizer polarimeter for accurate retardance measurement,” Appl. Opt. 36, 6466–6472 (1997).
    [Crossref]
  19. H. G. Tompkins, A User’s Guide to Ellipsometry (Academic, New York, 1993), pp. 26–30.
  20. K. B. Rochford, C. M. Wang, “Uncertainty in null polarimeter measurements,” IR 5055, 1996 (National Institute of Standards and Technology, Gaithersburg, Md.).
  21. K. B. Rochford, C. M. Wang, “Accurate interferometric retardance measurements,” Appl. Opt. 36, 6473–6479 (1997).
    [Crossref]
  22. Measurements were made at Hewlett-Packard, Santa Rosa Lightwave Operation (Santa Rosa, Calif.) with a commercial polarimeter and at Meadowlark Optics (Longmont, Colo.) with a filtered-source ellipsometer.
  23. Lord Rayleigh, “The surface layer of polished silica and glass with further studies on optical contact,” Proc. R. Soc. London Ser. A, 160, 507–525 (1937).
    [Crossref]
  24. R. W. Ditchburn, G. A. Orchard, “The polarization of totally reflected light,” Proc. R. Soc. London Ser. B 67, 608–614 (1954).
    [Crossref]
  25. Standard Reference Materials Program, National Institute of Standards and Technology, Gaithersburg, Md. 20899.
  26. J. Comyn, “Introduction to polymer permeability and the mathematics of diffusion,” in Polymer Permeability, J. Comyn, ed. (Chapman and Hall, London, 1994), pp. 1–10.
  27. M. Tencer, “Moisture ingress into nonhermetic enclosures and packages. A quasi-steady-state model for diffusion and attenuation of ambient humidity variations,” in Proceedings of the 44th Electronic Components and Technology Conference (IEEE, New York, 1994), pp. 196–209.
  28. S. Pauly, “Permeability and diffusion data,” in Polymer Handbook, 3rd ed., J. Brandrup, E. H. Immergut, eds. (Wiley Interscience, New York, 1989), pp. VI:435–449.
  29. UOP, Inc., “Molecular sieve water and air data sheets,” Technical sheet F-43C-6, 1991 (Houston, Tex.).

1997 (2)

1996 (1)

R. M. A. Azzam, M. M. K. Howlader, “Silicon-based polarization optics for the 1.3 and 1.55-µm communications wavelengths,” J. Lightwave Technol. 14, 873–878 (1996).
[Crossref]

1995 (1)

1994 (1)

N. N. Nagib, M. S. El-Bahrawy, S. E. Demian, S. Khodier, “Oblique incidence in total internal reflection phase retarders. II - Application,” Opt. Pura Apl. 27, 111–116 (1994).

1988 (1)

1970 (1)

1966 (1)

R. J. King, “Quarter-wave retardation systems based on the Fresnel rhomb principle,” J. Sci. Instrum. 43, 617–622 (1966).
[Crossref]

1964 (1)

V. A. Kizel, Y. I. Krasilov, V. N. Shamraev, “Achromatic λ/4 device,” Opt. Spectrosc. 17, 248, 249 (1964).

1954 (1)

R. W. Ditchburn, G. A. Orchard, “The polarization of totally reflected light,” Proc. R. Soc. London Ser. B 67, 608–614 (1954).
[Crossref]

1937 (1)

Lord Rayleigh, “The surface layer of polished silica and glass with further studies on optical contact,” Proc. R. Soc. London Ser. A, 160, 507–525 (1937).
[Crossref]

1911 (1)

A. E. Oxley, “On apparatus for the production of circularly polarized light,” Philos. Mag. 21, 517–532 (1911).
[Crossref]

Azzam, R. M. A.

R. M. A. Azzam, M. M. K. Howlader, “Silicon-based polarization optics for the 1.3 and 1.55-µm communications wavelengths,” J. Lightwave Technol. 14, 873–878 (1996).
[Crossref]

Bennett, J. M.

Boes, D. C.

A. M. Mood, F. A. Graybill, D. C. Boes, Introduction to the Theory of Statistics (McGraw-Hill, New York, 1974), p. 175.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980) p. 50.

Clarke, I. G.

K. B. Rochford, P. A. Williams, A. H. Rose, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
[Crossref]

Comyn, J.

J. Comyn, “Introduction to polymer permeability and the mathematics of diffusion,” in Polymer Permeability, J. Comyn, ed. (Chapman and Hall, London, 1994), pp. 1–10.

Day, G. W.

P. D. Hale, G. W. Day, “Stability of birefringent linear retarders (waveplates),” Appl. Opt. 27, 5146–5153 (1988).
[Crossref] [PubMed]

K. B. Rochford, P. A. Williams, A. H. Rose, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
[Crossref]

G. W. Day, A. H. Rose, “Faraday effect sensors: the state of the art,” in Fiber Optic and Laser Sensors, R. P. DePaula, E. Udd, eds., Proc. SPIE985, 138–150 (1998).

Demian, S. E.

N. N. Nagib, M. S. El-Bahrawy, S. E. Demian, S. Khodier, “Oblique incidence in total internal reflection phase retarders. II - Application,” Opt. Pura Apl. 27, 111–116 (1994).

Ditchburn, R. W.

R. W. Ditchburn, G. A. Orchard, “The polarization of totally reflected light,” Proc. R. Soc. London Ser. B 67, 608–614 (1954).
[Crossref]

El-Bahrawy, M. S.

N. N. Nagib, M. S. El-Bahrawy, S. E. Demian, S. Khodier, “Oblique incidence in total internal reflection phase retarders. II - Application,” Opt. Pura Apl. 27, 111–116 (1994).

Graybill, F. A.

A. M. Mood, F. A. Graybill, D. C. Boes, Introduction to the Theory of Statistics (McGraw-Hill, New York, 1974), p. 175.

Hale, P. D.

P. D. Hale, G. W. Day, “Stability of birefringent linear retarders (waveplates),” Appl. Opt. 27, 5146–5153 (1988).
[Crossref] [PubMed]

K. B. Rochford, P. A. Williams, A. H. Rose, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
[Crossref]

Hecht, E.

E. Hecht, A. Zajak, Optics (Addison-Wesley, Reading, Mass., 1974), pp. 301–305.

Howlader, M. M. K.

R. M. A. Azzam, M. M. K. Howlader, “Silicon-based polarization optics for the 1.3 and 1.55-µm communications wavelengths,” J. Lightwave Technol. 14, 873–878 (1996).
[Crossref]

Khodier, S.

N. N. Nagib, M. S. El-Bahrawy, S. E. Demian, S. Khodier, “Oblique incidence in total internal reflection phase retarders. II - Application,” Opt. Pura Apl. 27, 111–116 (1994).

Khodier, S. A.

King, R. J.

R. J. King, “Quarter-wave retardation systems based on the Fresnel rhomb principle,” J. Sci. Instrum. 43, 617–622 (1966).
[Crossref]

Kizel, V. A.

V. A. Kizel, Y. I. Krasilov, V. N. Shamraev, “Achromatic λ/4 device,” Opt. Spectrosc. 17, 248, 249 (1964).

Krasilov, Y. I.

V. A. Kizel, Y. I. Krasilov, V. N. Shamraev, “Achromatic λ/4 device,” Opt. Spectrosc. 17, 248, 249 (1964).

Kuyatt, C. E.

B. N. Taylor, C. E. Kuyatt, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurements, NIST Technical Note 1297, 1993 (U.S. Government Printing Office, Washington, D.C.).

Mood, A. M.

A. M. Mood, F. A. Graybill, D. C. Boes, Introduction to the Theory of Statistics (McGraw-Hill, New York, 1974), p. 175.

Nagib, N. N.

N. N. Nagib, S. A. Khodier, “Optimization of a rhomb-type quarter-wave phase retarder,” Appl. Opt. 34, 2927–2930 (1995).

N. N. Nagib, M. S. El-Bahrawy, S. E. Demian, S. Khodier, “Oblique incidence in total internal reflection phase retarders. II - Application,” Opt. Pura Apl. 27, 111–116 (1994).

Orchard, G. A.

R. W. Ditchburn, G. A. Orchard, “The polarization of totally reflected light,” Proc. R. Soc. London Ser. B 67, 608–614 (1954).
[Crossref]

Oxley, A. E.

A. E. Oxley, “On apparatus for the production of circularly polarized light,” Philos. Mag. 21, 517–532 (1911).
[Crossref]

Pauly, S.

S. Pauly, “Permeability and diffusion data,” in Polymer Handbook, 3rd ed., J. Brandrup, E. H. Immergut, eds. (Wiley Interscience, New York, 1989), pp. VI:435–449.

Rayleigh, Lord

Lord Rayleigh, “The surface layer of polished silica and glass with further studies on optical contact,” Proc. R. Soc. London Ser. A, 160, 507–525 (1937).
[Crossref]

Rochford, K. B.

K. B. Rochford, C. M. Wang, “Accurate interferometric retardance measurements,” Appl. Opt. 36, 6473–6479 (1997).
[Crossref]

K. B. Rochford, C. M. Wang, “Uncertainty in null polarimeter measurements,” IR 5055, 1996 (National Institute of Standards and Technology, Gaithersburg, Md.).

K. B. Rochford, P. A. Williams, A. H. Rose, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
[Crossref]

Rose, A. H.

P. A. Williams, A. H. Rose, C. M. Wang, Rotating-polarizer polarimeter for accurate retardance measurement,” Appl. Opt. 36, 6466–6472 (1997).
[Crossref]

G. W. Day, A. H. Rose, “Faraday effect sensors: the state of the art,” in Fiber Optic and Laser Sensors, R. P. DePaula, E. Udd, eds., Proc. SPIE985, 138–150 (1998).

K. B. Rochford, P. A. Williams, A. H. Rose, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
[Crossref]

Shamraev, V. N.

V. A. Kizel, Y. I. Krasilov, V. N. Shamraev, “Achromatic λ/4 device,” Opt. Spectrosc. 17, 248, 249 (1964).

Taylor, B. N.

B. N. Taylor, C. E. Kuyatt, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurements, NIST Technical Note 1297, 1993 (U.S. Government Printing Office, Washington, D.C.).

Tencer, M.

M. Tencer, “Moisture ingress into nonhermetic enclosures and packages. A quasi-steady-state model for diffusion and attenuation of ambient humidity variations,” in Proceedings of the 44th Electronic Components and Technology Conference (IEEE, New York, 1994), pp. 196–209.

Tompkins, H. G.

H. G. Tompkins, A User’s Guide to Ellipsometry (Academic, New York, 1993), pp. 26–30.

Wang, C. M.

Williams, P. A.

P. A. Williams, A. H. Rose, C. M. Wang, Rotating-polarizer polarimeter for accurate retardance measurement,” Appl. Opt. 36, 6466–6472 (1997).
[Crossref]

K. B. Rochford, P. A. Williams, A. H. Rose, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980) p. 50.

Zajak, A.

E. Hecht, A. Zajak, Optics (Addison-Wesley, Reading, Mass., 1974), pp. 301–305.

Appl. Opt. (5)

J. Lightwave Technol. (1)

R. M. A. Azzam, M. M. K. Howlader, “Silicon-based polarization optics for the 1.3 and 1.55-µm communications wavelengths,” J. Lightwave Technol. 14, 873–878 (1996).
[Crossref]

J. Sci. Instrum. (1)

R. J. King, “Quarter-wave retardation systems based on the Fresnel rhomb principle,” J. Sci. Instrum. 43, 617–622 (1966).
[Crossref]

Opt. Pura Apl. (1)

N. N. Nagib, M. S. El-Bahrawy, S. E. Demian, S. Khodier, “Oblique incidence in total internal reflection phase retarders. II - Application,” Opt. Pura Apl. 27, 111–116 (1994).

Opt. Spectrosc. (1)

V. A. Kizel, Y. I. Krasilov, V. N. Shamraev, “Achromatic λ/4 device,” Opt. Spectrosc. 17, 248, 249 (1964).

Philos. Mag. (1)

A. E. Oxley, “On apparatus for the production of circularly polarized light,” Philos. Mag. 21, 517–532 (1911).
[Crossref]

Proc. R. Soc. London Ser. A (1)

Lord Rayleigh, “The surface layer of polished silica and glass with further studies on optical contact,” Proc. R. Soc. London Ser. A, 160, 507–525 (1937).
[Crossref]

Proc. R. Soc. London Ser. B (1)

R. W. Ditchburn, G. A. Orchard, “The polarization of totally reflected light,” Proc. R. Soc. London Ser. B 67, 608–614 (1954).
[Crossref]

Other (17)

Standard Reference Materials Program, National Institute of Standards and Technology, Gaithersburg, Md. 20899.

J. Comyn, “Introduction to polymer permeability and the mathematics of diffusion,” in Polymer Permeability, J. Comyn, ed. (Chapman and Hall, London, 1994), pp. 1–10.

M. Tencer, “Moisture ingress into nonhermetic enclosures and packages. A quasi-steady-state model for diffusion and attenuation of ambient humidity variations,” in Proceedings of the 44th Electronic Components and Technology Conference (IEEE, New York, 1994), pp. 196–209.

S. Pauly, “Permeability and diffusion data,” in Polymer Handbook, 3rd ed., J. Brandrup, E. H. Immergut, eds. (Wiley Interscience, New York, 1989), pp. VI:435–449.

UOP, Inc., “Molecular sieve water and air data sheets,” Technical sheet F-43C-6, 1991 (Houston, Tex.).

Measurements were made at Hewlett-Packard, Santa Rosa Lightwave Operation (Santa Rosa, Calif.) with a commercial polarimeter and at Meadowlark Optics (Longmont, Colo.) with a filtered-source ellipsometer.

H. G. Tompkins, A User’s Guide to Ellipsometry (Academic, New York, 1993), pp. 26–30.

K. B. Rochford, C. M. Wang, “Uncertainty in null polarimeter measurements,” IR 5055, 1996 (National Institute of Standards and Technology, Gaithersburg, Md.).

K. B. Rochford, P. A. Williams, A. H. Rose, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
[Crossref]

SF-57 the Schott Glaswerke (Mainz Germany) trade name for its 847238 glass, was used in our rhomb. NIST does not endorse this glass, and similar glasses produced by other manufacturers might work as well or better. The glass code (in the form xxxyyy, where xxx = nd - 1, and yyy = νD × 10) gives the refractive index and Abbé number at 587.6 nm.

“The stress-optical coefficients of optical glasses,” Technical Note 15, 1984 (Schott Glaswerke, Mainz, Germany).

G. W. Day, A. H. Rose, “Faraday effect sensors: the state of the art,” in Fiber Optic and Laser Sensors, R. P. DePaula, E. Udd, eds., Proc. SPIE985, 138–150 (1998).

The following Sellmeier equation for use in the near infrared was supplied by Schott Glass Technologies, Inc., Duryea, Pa.: n2 = 1 + B1λ2/(λ2 - C1) + B2λ2/(λ2 - C2) + B3λ2/(λ2 - C3), where λ is the wavelength (in micrometers) B1 = 1.816 513 71, B2= 0.428 893 6, B3 = 1.071 862 78, C1 = 1.437 041 98 × 10-2, C2 = 5.928 011 72 × 10-2, and C3 = 121.419 941.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980) p. 50.

E. Hecht, A. Zajak, Optics (Addison-Wesley, Reading, Mass., 1974), pp. 301–305.

A. M. Mood, F. A. Graybill, D. C. Boes, Introduction to the Theory of Statistics (McGraw-Hill, New York, 1974), p. 175.

B. N. Taylor, C. E. Kuyatt, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurements, NIST Technical Note 1297, 1993 (U.S. Government Printing Office, Washington, D.C.).

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Figures (15)

Fig. 1
Fig. 1

Schematic of double-rhomb retarder; for our deviced = 10 mm, α ≈ 76.4°, andL ≈ 80 mm.

Fig. 2
Fig. 2

TIR angle dependence of retardance for glass withn = 1.8055.

Fig. 3
Fig. 3

Theoretical plot of retardance for various input angles.

Fig. 4
Fig. 4

Wavelength dependence of retardance calculated from Sellmeier equation.

Fig. 5
Fig. 5

Transmission spectrum of a double-rhomb retarder.

Fig. 6
Fig. 6

Probability density function for retardance errors arising from coherent reflections in a 90° retarder. The probability that the retardance error is between δ1 andδ2 is found through integratingP(Δδ) from δ1 toδ2.

Fig. 7
Fig. 7

Outline drawing of rhomb and protective packaging.

Fig. 8
Fig. 8

Calculated change in retardance and internal humidity for two environmental conditions.

Fig. 9
Fig. 9

Schematic of the null polarimeter with quarter-wave plate (QWP). Orientations are for fast-retarder axes or polarizer-transmission axes.

Fig. 10
Fig. 10

Measured retardance for five prototype stable retarders.

Fig. 11
Fig. 11

Retardance across the input face of the device measured with a 2-mm-diameter beam. The thick square indicates the usable area of the device defined by the epoxy seal.

Fig. 12
Fig. 12

Measured retardance at various input angles.

Fig. 13
Fig. 13

Measured retardance at several wavelengths; compared with theory, the solid line is the slope expected from glass dispersion, offset to match the measured retardance at 1320 nm.

Fig. 14
Fig. 14

Response of packaged device to thermal variations.

Fig. 15
Fig. 15

Measured retardance of a packaged rhomb over a period of 18 months. After day 440, the system was modified to make numerous measurements during a single day

Equations (16)

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Γ θ = 2   tan - 1 cos   θ sin 2 θ - 1 / n 2 1 / 2 sin 2 θ ,
Δ δ β = 4 Γ α - 2 Γ α + sin - 1 sin   β / n - 2 Γ α - sin - 1 sin   β / n
Δ δ R ,   ϕ t = tan - 1 R   sin ϕ t - δ 0 1 - R   cos ϕ t - δ 0 - tan - 1 R   sin ϕ t + δ 0 1 - R   cos ϕ t + δ 0 ,
P Δ δ = 1 - R 2   sin δ 0 π sin δ 0 - Δ δ 4 R 2 sin 2 δ 0 - Δ δ - sin Δ δ - R 2   sin 2 δ 0 - Δ δ 2 1 / 2 ,
tan - 1 R 2   sin 2 δ 0 - 2 R   sin δ 0 1 + R 2   cos 2 δ 0 - 2 R   cos δ 0 < Δ δ < tan - 1 R 2   sin 2 δ 0 + 2 R   sin δ 0 1 + R 2   cos 2 δ 0 + 2 R   cos δ 0 .
P δ 0 = 90 ° ;   Δ δ = sec 2 Δ δ π κ 1 - tan Δ δ / κ 2 1 / 2 , - tan - 1 κ < Δ δ < tan - 1 κ ,
Δ δ c = Δ δ R e ,   ϕ 1 t + Δ δ R e ,   ϕ 2 t + Δ δ R 1 ,   ϕ 1 t + ϕ 2 t ,
Δ θ t = ϵ t 2   cos δ 0 / 2 - cos δ 0 - 1 cos δ 0 + 1 .
L = 2 d sin   α   tan   α .
d Δ δ t dt = Δ δ RH RH t ,
dm t dt = PA w T S × RH O - RH I t 100 ,
PA w T = P g A w gasket + 2   P w A W window .
m t PA w T S · RH O · t 100 .
RH I t = 8.8 × 10 - 4   exp 43 m t m d .
Δ δ t 6.48 × 10 - 12 m d Δ δ RH S · RH O PA w T × exp 0.43 m d   S · RH O PA w T t - 1 .
t 0.01 % = 5.652 m d S · RH O PA w T ,

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