Abstract

A multipass absorption cell, based on an astigmatic variant of the off-axis resonator (Herriott) configuration, has been designed to obtain long path lengths in small volumes. Rotation of the mirror axes is used to obtain an effective adjustability in the two mirror radii. This allows one to compensate for errors in mirror radii that are encountered in manufacture, thereby generating the desired reentrant patterns with less-precise mirrors. A combination of mirror rotation and separation changes can be used to reach a variety of reentrant patterns and path lengths with a fixed set of astigmatic mirrors. The accessible patterns can be determined from trajectories, as a function of rotation and separation, through a general map of reentrant solutions. Desirable patterns for long-path spectroscopy can be chosen on the basis of path length, distance of the closest beam spot from the coupling hole, and tilt insensitivity. We describe the mathematics and analysis methods for the astigmatic cell with mirror rotation and then describe the design and test of prototype cells with this concept. Two cell designs are presented, a cell with 100-m path length in a volume of 3 L and a cell with 36-m path length in a volume of 0.3 L. Tests of low-volume absorption cells that use mirror rotation, designed for fast-flow atmospheric sampling, show the validity and the usefulness of the techniques that we have developed.

© 1995 Optical Society of America

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  1. D. R. Herriott, H. Kogelnik, R. Kompfner, “Off-axis paths in spherical mirror resonators,” Appl. Opt. 3, 523–526 (1964).
    [CrossRef]
  2. D. R. Herriott, H. J. Schulte, “Folded optical delay lines,” Appl. Opt. 4, 883–889 (1965).
    [CrossRef]
  3. W. R. Trutna, R. L. Byer, “Multiple-pass Raman gain cell,” Appl. Opt. 19, 301–312 (1980).
    [CrossRef] [PubMed]
  4. J. Altmann, R. Baumgart, D. C. Weitkamp, “Two-mirror multipass absorption cell,” Appl. Opt. 20, 995–999 (1981).
    [CrossRef] [PubMed]
  5. D. Kaur, A. M. de Souza, J. Wanna, S. A. Hammad, L. Mercorelli, D. S. Perry, “Multipass cell for molecular beam absorption spectroscopy,” Appl. Opt. 29, 119–124 (1990).
    [CrossRef] [PubMed]
  6. J. B. McManus, P. L. Kebabian, “Narrow optical interference fringes for certain set-up conditions in multipass absorption cells of the Herriott type,” Appl. Opt. 29, 898–900 (1990).
    [CrossRef] [PubMed]
  7. J. A. Silver, A. C. Stanton, “Optical interference fringe reduction in laser absorption experiments,” Appl. Opt. 27, 1914–1916 (1988).
    [CrossRef] [PubMed]
  8. J. B. McManus, P. L. Kebabian, C. E. Kolb, “Atmospheric methane measurement instrument using a Zeeman-split He–Ne laser,” Appl. Opt. 28, 5016–5023 (1989).
    [CrossRef] [PubMed]
  9. J. B. McManus, P. L. Kebabian, C. E. Kolb, “The Aerodyne Research Mobile Methane Monitor,” in Measurement of Atmospheric Gases, H. I. Schiff, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1433, 330–339 (1991).
  10. S. M. Anderson, M. S. Zahniser, “Open-path tunable diode laser absorption for eddy correlation flux measurements of atmospheric trace gases,” in Measurement of Atmospheric Gases, in H. I. Schiff, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1433, 167–178 (1991).
  11. C. R. Webster, R. D. May, C. A. Trimble, R. G. Chave, J. Kendal, “Aircraft (ER-2) laser infrared absorption spectrometer (ALIAS) for in-situ stratospheric measurements of HCl, N2O, CH4, NO2, and HNO3,” Appl. Opt. 33, 454–472 (1994).
    [CrossRef] [PubMed]
  12. P. Werle, F. Slemr, “Signal-to-noise ratio analysis in laser absorption spectrometers using optical multipass cells,” Appl. Opt. 30, 430–434 (1991).
    [CrossRef] [PubMed]
  13. P. L. Kebabian, “Off-axis cavity absorption cell,” U.S. patent5,291,265 (1March1994).
  14. J. U. White, “Long optical paths of large aperture,” J. Opt. Soc. Am. 32, 285–288 (1942).
    [CrossRef]
  15. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  16. M. R. Schroeder, Number Theory in Science and Communication, Vol. 7 of Springer Series in Information Sciences (Springer-Verlag, Berlin, 1986).
  17. matlab (Math Works, Inc., Natick, Mass., 1990).
  18. R. E. Mickens, Difference Equations (Van Nostrand Reinhold, New York, 1987).
  19. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 103.
  20. D. D. Nelson, M. S. Zahniser, “Air-broadened linewidth measurements in the ν2 vibrational band of the hydroperoxyl radical,” J. Mol. Spectrosc. 166, 273–279 (1994).
    [CrossRef]
  21. J. Wormhoudt, M. S. Zahniser, D. D. Nelson, J. B. McManus, R. C. Miake-Lye, C. E. Kolb, “Infrared tunable diode laser diagnostics for aircraft exhaust characterization,” in Laser Applications in Combustion and Combustion Diagnostics II, R. J. Locke, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2122, 49–60 (1994).
  22. C. E. Kolb, J. C. Wormhoudt, M. S. Zahnizer, “Recent advances in spectroscopic instrumentation for measuring stable gases in the natural environment,” in Methods in Ecology: Trace Gases, P. A. Matson, R. C. Harriss, eds. (Blackwell, Boston), to be published.
  23. D. L. Gonzalez, O. Piro, “Symmetric kicked self oscillators: iterated maps, strange attractors and symmetry of the phase-locking Farey sequence,” Phys. Rev. Lett. 55, 17–19 (1985).
    [CrossRef] [PubMed]
  24. T. Allen, “On the arithmetic of phase locking: coupled neurons as a lattice on R2,” Physica 6, 305–320 (1983).

1994 (2)

1991 (1)

1990 (2)

1989 (1)

1988 (1)

1985 (1)

D. L. Gonzalez, O. Piro, “Symmetric kicked self oscillators: iterated maps, strange attractors and symmetry of the phase-locking Farey sequence,” Phys. Rev. Lett. 55, 17–19 (1985).
[CrossRef] [PubMed]

1983 (1)

T. Allen, “On the arithmetic of phase locking: coupled neurons as a lattice on R2,” Physica 6, 305–320 (1983).

1981 (1)

1980 (1)

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1965 (1)

1964 (1)

1942 (1)

Allen, T.

T. Allen, “On the arithmetic of phase locking: coupled neurons as a lattice on R2,” Physica 6, 305–320 (1983).

Altmann, J.

Anderson, S. M.

S. M. Anderson, M. S. Zahniser, “Open-path tunable diode laser absorption for eddy correlation flux measurements of atmospheric trace gases,” in Measurement of Atmospheric Gases, in H. I. Schiff, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1433, 167–178 (1991).

Baumgart, R.

Byer, R. L.

Chave, R. G.

de Souza, A. M.

Gonzalez, D. L.

D. L. Gonzalez, O. Piro, “Symmetric kicked self oscillators: iterated maps, strange attractors and symmetry of the phase-locking Farey sequence,” Phys. Rev. Lett. 55, 17–19 (1985).
[CrossRef] [PubMed]

Hammad, S. A.

Herriott, D. R.

Kaur, D.

Kebabian, P. L.

J. B. McManus, P. L. Kebabian, “Narrow optical interference fringes for certain set-up conditions in multipass absorption cells of the Herriott type,” Appl. Opt. 29, 898–900 (1990).
[CrossRef] [PubMed]

J. B. McManus, P. L. Kebabian, C. E. Kolb, “Atmospheric methane measurement instrument using a Zeeman-split He–Ne laser,” Appl. Opt. 28, 5016–5023 (1989).
[CrossRef] [PubMed]

J. B. McManus, P. L. Kebabian, C. E. Kolb, “The Aerodyne Research Mobile Methane Monitor,” in Measurement of Atmospheric Gases, H. I. Schiff, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1433, 330–339 (1991).

P. L. Kebabian, “Off-axis cavity absorption cell,” U.S. patent5,291,265 (1March1994).

Kendal, J.

Kogelnik, H.

Kolb, C. E.

J. B. McManus, P. L. Kebabian, C. E. Kolb, “Atmospheric methane measurement instrument using a Zeeman-split He–Ne laser,” Appl. Opt. 28, 5016–5023 (1989).
[CrossRef] [PubMed]

J. Wormhoudt, M. S. Zahniser, D. D. Nelson, J. B. McManus, R. C. Miake-Lye, C. E. Kolb, “Infrared tunable diode laser diagnostics for aircraft exhaust characterization,” in Laser Applications in Combustion and Combustion Diagnostics II, R. J. Locke, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2122, 49–60 (1994).

J. B. McManus, P. L. Kebabian, C. E. Kolb, “The Aerodyne Research Mobile Methane Monitor,” in Measurement of Atmospheric Gases, H. I. Schiff, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1433, 330–339 (1991).

C. E. Kolb, J. C. Wormhoudt, M. S. Zahnizer, “Recent advances in spectroscopic instrumentation for measuring stable gases in the natural environment,” in Methods in Ecology: Trace Gases, P. A. Matson, R. C. Harriss, eds. (Blackwell, Boston), to be published.

Kompfner, R.

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

May, R. D.

McManus, J. B.

J. B. McManus, P. L. Kebabian, “Narrow optical interference fringes for certain set-up conditions in multipass absorption cells of the Herriott type,” Appl. Opt. 29, 898–900 (1990).
[CrossRef] [PubMed]

J. B. McManus, P. L. Kebabian, C. E. Kolb, “Atmospheric methane measurement instrument using a Zeeman-split He–Ne laser,” Appl. Opt. 28, 5016–5023 (1989).
[CrossRef] [PubMed]

J. Wormhoudt, M. S. Zahniser, D. D. Nelson, J. B. McManus, R. C. Miake-Lye, C. E. Kolb, “Infrared tunable diode laser diagnostics for aircraft exhaust characterization,” in Laser Applications in Combustion and Combustion Diagnostics II, R. J. Locke, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2122, 49–60 (1994).

J. B. McManus, P. L. Kebabian, C. E. Kolb, “The Aerodyne Research Mobile Methane Monitor,” in Measurement of Atmospheric Gases, H. I. Schiff, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1433, 330–339 (1991).

Mercorelli, L.

Miake-Lye, R. C.

J. Wormhoudt, M. S. Zahniser, D. D. Nelson, J. B. McManus, R. C. Miake-Lye, C. E. Kolb, “Infrared tunable diode laser diagnostics for aircraft exhaust characterization,” in Laser Applications in Combustion and Combustion Diagnostics II, R. J. Locke, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2122, 49–60 (1994).

Mickens, R. E.

R. E. Mickens, Difference Equations (Van Nostrand Reinhold, New York, 1987).

Nelson, D. D.

D. D. Nelson, M. S. Zahniser, “Air-broadened linewidth measurements in the ν2 vibrational band of the hydroperoxyl radical,” J. Mol. Spectrosc. 166, 273–279 (1994).
[CrossRef]

J. Wormhoudt, M. S. Zahniser, D. D. Nelson, J. B. McManus, R. C. Miake-Lye, C. E. Kolb, “Infrared tunable diode laser diagnostics for aircraft exhaust characterization,” in Laser Applications in Combustion and Combustion Diagnostics II, R. J. Locke, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2122, 49–60 (1994).

Perry, D. S.

Piro, O.

D. L. Gonzalez, O. Piro, “Symmetric kicked self oscillators: iterated maps, strange attractors and symmetry of the phase-locking Farey sequence,” Phys. Rev. Lett. 55, 17–19 (1985).
[CrossRef] [PubMed]

Schroeder, M. R.

M. R. Schroeder, Number Theory in Science and Communication, Vol. 7 of Springer Series in Information Sciences (Springer-Verlag, Berlin, 1986).

Schulte, H. J.

Silver, J. A.

Slemr, F.

Stanton, A. C.

Trimble, C. A.

Trutna, W. R.

Wanna, J.

Webster, C. R.

Weitkamp, D. C.

Werle, P.

White, J. U.

Wormhoudt, J.

J. Wormhoudt, M. S. Zahniser, D. D. Nelson, J. B. McManus, R. C. Miake-Lye, C. E. Kolb, “Infrared tunable diode laser diagnostics for aircraft exhaust characterization,” in Laser Applications in Combustion and Combustion Diagnostics II, R. J. Locke, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2122, 49–60 (1994).

Wormhoudt, J. C.

C. E. Kolb, J. C. Wormhoudt, M. S. Zahnizer, “Recent advances in spectroscopic instrumentation for measuring stable gases in the natural environment,” in Methods in Ecology: Trace Gases, P. A. Matson, R. C. Harriss, eds. (Blackwell, Boston), to be published.

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 103.

Zahniser, M. S.

D. D. Nelson, M. S. Zahniser, “Air-broadened linewidth measurements in the ν2 vibrational band of the hydroperoxyl radical,” J. Mol. Spectrosc. 166, 273–279 (1994).
[CrossRef]

S. M. Anderson, M. S. Zahniser, “Open-path tunable diode laser absorption for eddy correlation flux measurements of atmospheric trace gases,” in Measurement of Atmospheric Gases, in H. I. Schiff, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1433, 167–178 (1991).

J. Wormhoudt, M. S. Zahniser, D. D. Nelson, J. B. McManus, R. C. Miake-Lye, C. E. Kolb, “Infrared tunable diode laser diagnostics for aircraft exhaust characterization,” in Laser Applications in Combustion and Combustion Diagnostics II, R. J. Locke, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2122, 49–60 (1994).

Zahnizer, M. S.

C. E. Kolb, J. C. Wormhoudt, M. S. Zahnizer, “Recent advances in spectroscopic instrumentation for measuring stable gases in the natural environment,” in Methods in Ecology: Trace Gases, P. A. Matson, R. C. Harriss, eds. (Blackwell, Boston), to be published.

Appl. Opt. (10)

D. R. Herriott, H. Kogelnik, R. Kompfner, “Off-axis paths in spherical mirror resonators,” Appl. Opt. 3, 523–526 (1964).
[CrossRef]

D. R. Herriott, H. J. Schulte, “Folded optical delay lines,” Appl. Opt. 4, 883–889 (1965).
[CrossRef]

W. R. Trutna, R. L. Byer, “Multiple-pass Raman gain cell,” Appl. Opt. 19, 301–312 (1980).
[CrossRef] [PubMed]

J. Altmann, R. Baumgart, D. C. Weitkamp, “Two-mirror multipass absorption cell,” Appl. Opt. 20, 995–999 (1981).
[CrossRef] [PubMed]

J. B. McManus, P. L. Kebabian, C. E. Kolb, “Atmospheric methane measurement instrument using a Zeeman-split He–Ne laser,” Appl. Opt. 28, 5016–5023 (1989).
[CrossRef] [PubMed]

D. Kaur, A. M. de Souza, J. Wanna, S. A. Hammad, L. Mercorelli, D. S. Perry, “Multipass cell for molecular beam absorption spectroscopy,” Appl. Opt. 29, 119–124 (1990).
[CrossRef] [PubMed]

J. B. McManus, P. L. Kebabian, “Narrow optical interference fringes for certain set-up conditions in multipass absorption cells of the Herriott type,” Appl. Opt. 29, 898–900 (1990).
[CrossRef] [PubMed]

P. Werle, F. Slemr, “Signal-to-noise ratio analysis in laser absorption spectrometers using optical multipass cells,” Appl. Opt. 30, 430–434 (1991).
[CrossRef] [PubMed]

C. R. Webster, R. D. May, C. A. Trimble, R. G. Chave, J. Kendal, “Aircraft (ER-2) laser infrared absorption spectrometer (ALIAS) for in-situ stratospheric measurements of HCl, N2O, CH4, NO2, and HNO3,” Appl. Opt. 33, 454–472 (1994).
[CrossRef] [PubMed]

J. A. Silver, A. C. Stanton, “Optical interference fringe reduction in laser absorption experiments,” Appl. Opt. 27, 1914–1916 (1988).
[CrossRef] [PubMed]

J. Mol. Spectrosc. (1)

D. D. Nelson, M. S. Zahniser, “Air-broadened linewidth measurements in the ν2 vibrational band of the hydroperoxyl radical,” J. Mol. Spectrosc. 166, 273–279 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

Phys. Rev. Lett. (1)

D. L. Gonzalez, O. Piro, “Symmetric kicked self oscillators: iterated maps, strange attractors and symmetry of the phase-locking Farey sequence,” Phys. Rev. Lett. 55, 17–19 (1985).
[CrossRef] [PubMed]

Physica (1)

T. Allen, “On the arithmetic of phase locking: coupled neurons as a lattice on R2,” Physica 6, 305–320 (1983).

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other (9)

M. R. Schroeder, Number Theory in Science and Communication, Vol. 7 of Springer Series in Information Sciences (Springer-Verlag, Berlin, 1986).

matlab (Math Works, Inc., Natick, Mass., 1990).

R. E. Mickens, Difference Equations (Van Nostrand Reinhold, New York, 1987).

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 103.

J. Wormhoudt, M. S. Zahniser, D. D. Nelson, J. B. McManus, R. C. Miake-Lye, C. E. Kolb, “Infrared tunable diode laser diagnostics for aircraft exhaust characterization,” in Laser Applications in Combustion and Combustion Diagnostics II, R. J. Locke, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2122, 49–60 (1994).

C. E. Kolb, J. C. Wormhoudt, M. S. Zahnizer, “Recent advances in spectroscopic instrumentation for measuring stable gases in the natural environment,” in Methods in Ecology: Trace Gases, P. A. Matson, R. C. Harriss, eds. (Blackwell, Boston), to be published.

J. B. McManus, P. L. Kebabian, C. E. Kolb, “The Aerodyne Research Mobile Methane Monitor,” in Measurement of Atmospheric Gases, H. I. Schiff, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1433, 330–339 (1991).

S. M. Anderson, M. S. Zahniser, “Open-path tunable diode laser absorption for eddy correlation flux measurements of atmospheric trace gases,” in Measurement of Atmospheric Gases, in H. I. Schiff, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1433, 167–178 (1991).

P. L. Kebabian, “Off-axis cavity absorption cell,” U.S. patent5,291,265 (1March1994).

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Figures (13)

Fig. 1
Fig. 1

Symmetrical pattern {N = 182, M x = 80, M y = 76}, with even M x , y and with spots far from the coupling hole. Only spots on the input mirror are shown. For all figures, a circle indicates the coupling hole.

Fig. 2
Fig. 2

Pattern {190, 80, 76}, with even M x , y but with spots close to the hole. Only spots on the input mirror are shown.

Fig. 3
Fig. 3

Asymmetrical pattern {188, 86, 83}, with odd M y . Only spots on the input mirror are shown.

Fig. 4
Fig. 4

Geometry of the cell with mirror axis rotation.

Fig. 5
Fig. 5

Iterated solution for cell escape as a function of ϕ x , ϕ y , with normalized hole radius ρ = 0.1. The logarithm of the number of passes is encoded as tone. The location of pattern {N = 182, M x = 80, M y = 76} is indicated.

Fig. 6
Fig. 6

Plot of zones of nearly reentrant patterns constructed with the algorithm described in Appendix A. Patterns with the beam exiting in any direction up to a limiting number of 1000 round trips are shown. The location of our pattern with 182 passes is indicated.

Fig. 7
Fig. 7

Plot of zones of nearly reentrant patterns constructed with the algorithm described in Appendix A. Patterns with the beam exiting in the (−1, −1) quadrant only, up to a limiting number of 1000 passes, are shown. Trajectories showing the patterns accessible to the system as the mirror separation and rotation angle are changed are drawn. The three lines are for spacing changes (tics are placed at intervals of 0.2% of the base length) with fixed mirror axis angles of 0°, 22°, and 30°.

Fig. 8
Fig. 8

Cross-sectional view of the 100-m-path-length, 3.2-L fast-flow cell. The cell is shown to scale, with an overall length of 66.7 cm and a diameter of 15 cm.

Fig. 9
Fig. 9

Comparison of calculated and observed rear-mirror spot patterns for the pattern {182, 80, 76}. The calculated pattern is on the left and includes 17° of rotation between the mirror axes. The spot diameter decreases exponentially to simulate decreasing brightness. A photograph of the observed spot pattern, with a visible trace beam, is shown on the right.

Fig. 10
Fig. 10

Response of the 100-m-path-length, 3.2-L fast-flow cell to an impulse of N2O. With a pumping speed of 5 L/s, the flushing time of 0.33 s (a 3-s−1 rate).

Fig. 11
Fig. 11

Response of the 100-m-path-length, 3.2-L fast-flow cell to a rectangular wave of N2O. The recorded data stream is shown by the dotted curve, and exponential fits to the rise and fall are shown as solid curves. With a pumping speed of 5 L/s, the flushing time is ~0.29 s (a 3.4–3.5-s−1 rate).

Fig. 12
Fig. 12

Response of the 100-m-path-length, 3.2-L fast-flow cell to a rectangular wave of H2O. The recorded stream data are shown by the dotted curve, and an exponential fit to the fall is shown as a solid curve. With a pumping speed of 5 L/s, the flushing time is 0.33 s (a 3-s−1 rate).

Fig. 13
Fig. 13

Response of the 36-m-path-length, 0.27-L fast-flow cell to a square wave of CH4. With a pumping speed of 5 L/s, the flushing time is 0.04 s (a 25-s−1 rate).

Equations (23)

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x i = X 0 sin ( i Θ x ) ,             y i = Y 0 sin ( i Θ y ) , Θ x = cos - 1 ( 1 - D R x ) ,             Θ y = cos - 1 ( 1 - D R y ) ,
N Θ x = M x π ,             N Θ y = M y π ,
Θ x = π M x / N ,             Θ y = π M y / N .
ϕ x = Θ x - π 2 ,             ϕ y = Θ y - π 2 .
ϕ x , y = sin - 1 ( D R x , y - 1 )             or , for D R , ϕ x , y ( D R x , y - 1 ) .
ϕ x , y = π ( M x , y N - 1 2 ) .
k x = N - 2 M x ,             k y = N - 2 M y .
ϕ x , y = - ( π / 2 ) k x , y / N .
x j = ( - 1 ) j sin ( j π k x / N ) ,             y j = ( - 1 ) y sin ( j π k y / N ) ,
C = R ( R a , R b ) D ( D ) R ( R a , R b ) D ( D ) .
D ( D ) = [ 1 D 0 0 0 1 0 0 0 0 1 D 0 0 0 1 ] , R ( R a , R b ) = [ 1 0 0 0 - 2 / R a 1 0 0 0 0 1 0 0 0 - 2 / R b 1 ] .
R ( R a , R b ) = T ( - θ t ) R ( R a , R b ) T ( θ t ) .
T ( θ t ) = [ cos θ t 0 sin θ t 0 0 cos θ t 0 sin θ t - sin θ t 0 cos θ t 0 0 - sin θ t 0 cos θ t ] .
x n = ( A 2 - 2 - e 2 ) x n - 1 - x n - 2 + ɛ y n - 1 , y n = ( B 2 - 2 - e 2 ) y n - 1 - y n - 2 - ɛ x n - 1 ,
A = 2 - [ ( α a D ) cos 2 τ + ( α b D ) sin 2 τ ] , B = 2 - [ ( α a D ) sin 2 τ + ( α b D ) cos 2 τ ] e = ( 1 / 2 ) ( α a - α b ) D sin ( 2 τ ) , ɛ = - ( 1 / 4 ) [ ( α a - α b ) D ] 2 sin ( 4 τ ) ,
( Δ 2 x n Δ 2 y n ) = [ ( 2 - f x ) ɛ - ɛ ( 2 - f y ) ] ( x n y n ) ,
cos θ x 2 = ( 1 / 2 ) ( A 2 - 2 - e 2 - ξ ) , cos θ y 2 = ( 1 / 2 ) ( B 2 - 2 - e 2 + ξ ) ,
v s = x ^ X 0 i cos ( i Θ x ) [ D ( 2 R x - D ) ] - 1 / 2 + y ^ Y 0 i cos ( i Θ y ) [ D ( 2 R y - D ) ] - 1 / 2 .
v s ( 1 / 2 ) N R 0 { [ D ( 2 R x - D ) ] - 1 + [ D ( 2 R y - D ) ] - 1 } 1 / 2 N R 0 / D .
sin 2 ( 2 j esc ϕ x ) + sin 2 ( 2 j esc ϕ y ) < ρ 2 .
ϕ y ( ϕ x ) = sin - 1 [ γ sin ϕ x + ( γ - 1 ) ] ,
N ϕ x = I x π ,             N ϕ y = I y π ,
ϕ y / ϕ x = I y / I x .

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