Abstract

A complete matrix description of ray-optic propagation in an astigmatic multipass cell is presented, with expressions for the coupling of coordinates. A pair of transforms puts the 4×4 paraxial system matrix into block-diagonal form, allowing a solution using Sylvester's theorem. A variation on the Jones matrix calculus is employed wherein the ray coordinates on both resonator mirrors are simultaneously represented as a single state of the system. The formulations are applicable to resonators with any degree of astigmatism and axial twist. Examples are presented of beam paths and the boundary shapes of beam spots. The shape of the pattern boundaries, as a function of the coordinate coupling coefficient, influences the practical availability of patterns.

© 2007 Optical Society of America

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References

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  1. J. B. McManus, P. L Kebabian, and M. S. Zahniser, "Astigmatic mirror multiple pass absorption cells for long pathlength spectroscopy," Appl. Opt. 34, 3336-3348 (1995).
    [CrossRef] [PubMed]
  2. D. R. Herriott and H. J. Schulte, "Folded optical delay lines," Appl. Opt. 4, 883-889 (1965).
    [CrossRef]
  3. J. A. Silver, "Simple dense pattern optical multipass cells," Appl. Opt. 44, 6545-6556 (2005).
    [CrossRef] [PubMed]
  4. L.-Y. Hao, S. Qiang, G.-R. Wu, L. Qi, D. Feng, and Q.-S. Zhu, "Cylindrical mirror multipass Lissajous system for laser photoacoustic spectroscopy," Rev. Sci. Instrum. 73, 2079-3085 (2002).
    [CrossRef]
  5. J. B. McManus, D. D. Nelson, J. H. Shorter, R. Jimenez, S. Herndon, S. Saleska, and M. Zahniser, "A high precision pulsed quantum cascade laser spectrometer for measurements of stable isotopes of carbon dioxide," J. Mod. Opt. 52, 2309-2321 (2005).
    [CrossRef]
  6. J. B. McManus, M. S. Zahniser, D. D. Nelson, L. R. Williams, and C. E. Kolb, "Infrared laser spectrometer with balanced absorption for measurement of isotopic ratios of carbon gases," Spectrochim. Acta 58, 2465-2479 (2002).
    [CrossRef]
  7. D. D. Nelson, J. H. Shorter, J. B. McManus, and M. S. Zahniser, "Sub-part-per-billion detection of nitric oxide in air using a thermoelectrically cooled mid-infrared quantum cascade laser spectrometer," Appl. Phys. B 75, 343-350 (2002).
    [CrossRef]
  8. D. Weidmann, G. Wysocki, C. Oppenheimer, and F. K. Tittel, "Development of a compact quantum cascade laser spectrometer for field measurements of CO2 isotopes," Appl. Phys. B 80, 255-260 (2005).
    [CrossRef]
  9. R. Korman, R. Königstedt, U. Parchatka, J. Lelieveld, and H. Fisher, "QUALITAS: a mid-infrared spectrometer for sensitive trace gas measurements based on quantum cascade lasers in CW operation," Rev. Sci. Instrum. 76, 075102 (2005).
    [CrossRef]
  10. D. Richter, A. Fried, B. P. Wert, J. G. Walega, and F. K. Tittel, "Development of a tunable mid-IR difference frequency laser source for highly sensitive airborne trace gas detection," Appl. Phys B 75, 281-288 (2002).
    [CrossRef]
  11. J. B. McManus, D. Nelson, M. Zahniser, L. Mechold, M. Osiac, J. Röpcke, and A. Rousseau, "TOBI: a two-laser beam infrared system for time-resolved plasma diagnostics of infrared active compounds," Rev. Sci. Instrum. 74, 2709-2713 (2003).
    [CrossRef]
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    [CrossRef]
  13. A. E. Siegman, Lasers (University Science Books, 1986).
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    [CrossRef]
  15. H. Kogelnik and T. Li, "Laser beams and resonators," Appl. Opt. 5, 1550-1567 (1966).
    [CrossRef] [PubMed]

2005 (4)

J. B. McManus, D. D. Nelson, J. H. Shorter, R. Jimenez, S. Herndon, S. Saleska, and M. Zahniser, "A high precision pulsed quantum cascade laser spectrometer for measurements of stable isotopes of carbon dioxide," J. Mod. Opt. 52, 2309-2321 (2005).
[CrossRef]

D. Weidmann, G. Wysocki, C. Oppenheimer, and F. K. Tittel, "Development of a compact quantum cascade laser spectrometer for field measurements of CO2 isotopes," Appl. Phys. B 80, 255-260 (2005).
[CrossRef]

R. Korman, R. Königstedt, U. Parchatka, J. Lelieveld, and H. Fisher, "QUALITAS: a mid-infrared spectrometer for sensitive trace gas measurements based on quantum cascade lasers in CW operation," Rev. Sci. Instrum. 76, 075102 (2005).
[CrossRef]

J. A. Silver, "Simple dense pattern optical multipass cells," Appl. Opt. 44, 6545-6556 (2005).
[CrossRef] [PubMed]

2003 (1)

J. B. McManus, D. Nelson, M. Zahniser, L. Mechold, M. Osiac, J. Röpcke, and A. Rousseau, "TOBI: a two-laser beam infrared system for time-resolved plasma diagnostics of infrared active compounds," Rev. Sci. Instrum. 74, 2709-2713 (2003).
[CrossRef]

2002 (4)

L.-Y. Hao, S. Qiang, G.-R. Wu, L. Qi, D. Feng, and Q.-S. Zhu, "Cylindrical mirror multipass Lissajous system for laser photoacoustic spectroscopy," Rev. Sci. Instrum. 73, 2079-3085 (2002).
[CrossRef]

D. Richter, A. Fried, B. P. Wert, J. G. Walega, and F. K. Tittel, "Development of a tunable mid-IR difference frequency laser source for highly sensitive airborne trace gas detection," Appl. Phys B 75, 281-288 (2002).
[CrossRef]

J. B. McManus, M. S. Zahniser, D. D. Nelson, L. R. Williams, and C. E. Kolb, "Infrared laser spectrometer with balanced absorption for measurement of isotopic ratios of carbon gases," Spectrochim. Acta 58, 2465-2479 (2002).
[CrossRef]

D. D. Nelson, J. H. Shorter, J. B. McManus, and M. S. Zahniser, "Sub-part-per-billion detection of nitric oxide in air using a thermoelectrically cooled mid-infrared quantum cascade laser spectrometer," Appl. Phys. B 75, 343-350 (2002).
[CrossRef]

1995 (2)

1966 (1)

1965 (1)

1941 (1)

Casperson, L. W.

Feng, D.

L.-Y. Hao, S. Qiang, G.-R. Wu, L. Qi, D. Feng, and Q.-S. Zhu, "Cylindrical mirror multipass Lissajous system for laser photoacoustic spectroscopy," Rev. Sci. Instrum. 73, 2079-3085 (2002).
[CrossRef]

Fisher, H.

R. Korman, R. Königstedt, U. Parchatka, J. Lelieveld, and H. Fisher, "QUALITAS: a mid-infrared spectrometer for sensitive trace gas measurements based on quantum cascade lasers in CW operation," Rev. Sci. Instrum. 76, 075102 (2005).
[CrossRef]

Fried, A.

D. Richter, A. Fried, B. P. Wert, J. G. Walega, and F. K. Tittel, "Development of a tunable mid-IR difference frequency laser source for highly sensitive airborne trace gas detection," Appl. Phys B 75, 281-288 (2002).
[CrossRef]

Hao, L.-Y.

L.-Y. Hao, S. Qiang, G.-R. Wu, L. Qi, D. Feng, and Q.-S. Zhu, "Cylindrical mirror multipass Lissajous system for laser photoacoustic spectroscopy," Rev. Sci. Instrum. 73, 2079-3085 (2002).
[CrossRef]

Herndon, S.

J. B. McManus, D. D. Nelson, J. H. Shorter, R. Jimenez, S. Herndon, S. Saleska, and M. Zahniser, "A high precision pulsed quantum cascade laser spectrometer for measurements of stable isotopes of carbon dioxide," J. Mod. Opt. 52, 2309-2321 (2005).
[CrossRef]

Herriott, D. R.

Jimenez, R.

J. B. McManus, D. D. Nelson, J. H. Shorter, R. Jimenez, S. Herndon, S. Saleska, and M. Zahniser, "A high precision pulsed quantum cascade laser spectrometer for measurements of stable isotopes of carbon dioxide," J. Mod. Opt. 52, 2309-2321 (2005).
[CrossRef]

Jones, R. C.

Kebabian, P. L

Kogelnik, H.

Kolb, C. E.

J. B. McManus, M. S. Zahniser, D. D. Nelson, L. R. Williams, and C. E. Kolb, "Infrared laser spectrometer with balanced absorption for measurement of isotopic ratios of carbon gases," Spectrochim. Acta 58, 2465-2479 (2002).
[CrossRef]

Königstedt, R.

R. Korman, R. Königstedt, U. Parchatka, J. Lelieveld, and H. Fisher, "QUALITAS: a mid-infrared spectrometer for sensitive trace gas measurements based on quantum cascade lasers in CW operation," Rev. Sci. Instrum. 76, 075102 (2005).
[CrossRef]

Korman, R.

R. Korman, R. Königstedt, U. Parchatka, J. Lelieveld, and H. Fisher, "QUALITAS: a mid-infrared spectrometer for sensitive trace gas measurements based on quantum cascade lasers in CW operation," Rev. Sci. Instrum. 76, 075102 (2005).
[CrossRef]

Lelieveld, J.

R. Korman, R. Königstedt, U. Parchatka, J. Lelieveld, and H. Fisher, "QUALITAS: a mid-infrared spectrometer for sensitive trace gas measurements based on quantum cascade lasers in CW operation," Rev. Sci. Instrum. 76, 075102 (2005).
[CrossRef]

Li, T.

McManus, J. B.

J. B. McManus, D. D. Nelson, J. H. Shorter, R. Jimenez, S. Herndon, S. Saleska, and M. Zahniser, "A high precision pulsed quantum cascade laser spectrometer for measurements of stable isotopes of carbon dioxide," J. Mod. Opt. 52, 2309-2321 (2005).
[CrossRef]

J. B. McManus, D. Nelson, M. Zahniser, L. Mechold, M. Osiac, J. Röpcke, and A. Rousseau, "TOBI: a two-laser beam infrared system for time-resolved plasma diagnostics of infrared active compounds," Rev. Sci. Instrum. 74, 2709-2713 (2003).
[CrossRef]

D. D. Nelson, J. H. Shorter, J. B. McManus, and M. S. Zahniser, "Sub-part-per-billion detection of nitric oxide in air using a thermoelectrically cooled mid-infrared quantum cascade laser spectrometer," Appl. Phys. B 75, 343-350 (2002).
[CrossRef]

J. B. McManus, M. S. Zahniser, D. D. Nelson, L. R. Williams, and C. E. Kolb, "Infrared laser spectrometer with balanced absorption for measurement of isotopic ratios of carbon gases," Spectrochim. Acta 58, 2465-2479 (2002).
[CrossRef]

J. B. McManus, P. L Kebabian, and M. S. Zahniser, "Astigmatic mirror multiple pass absorption cells for long pathlength spectroscopy," Appl. Opt. 34, 3336-3348 (1995).
[CrossRef] [PubMed]

Mechold, L.

J. B. McManus, D. Nelson, M. Zahniser, L. Mechold, M. Osiac, J. Röpcke, and A. Rousseau, "TOBI: a two-laser beam infrared system for time-resolved plasma diagnostics of infrared active compounds," Rev. Sci. Instrum. 74, 2709-2713 (2003).
[CrossRef]

Nelson, D.

J. B. McManus, D. Nelson, M. Zahniser, L. Mechold, M. Osiac, J. Röpcke, and A. Rousseau, "TOBI: a two-laser beam infrared system for time-resolved plasma diagnostics of infrared active compounds," Rev. Sci. Instrum. 74, 2709-2713 (2003).
[CrossRef]

Nelson, D. D.

J. B. McManus, D. D. Nelson, J. H. Shorter, R. Jimenez, S. Herndon, S. Saleska, and M. Zahniser, "A high precision pulsed quantum cascade laser spectrometer for measurements of stable isotopes of carbon dioxide," J. Mod. Opt. 52, 2309-2321 (2005).
[CrossRef]

D. D. Nelson, J. H. Shorter, J. B. McManus, and M. S. Zahniser, "Sub-part-per-billion detection of nitric oxide in air using a thermoelectrically cooled mid-infrared quantum cascade laser spectrometer," Appl. Phys. B 75, 343-350 (2002).
[CrossRef]

J. B. McManus, M. S. Zahniser, D. D. Nelson, L. R. Williams, and C. E. Kolb, "Infrared laser spectrometer with balanced absorption for measurement of isotopic ratios of carbon gases," Spectrochim. Acta 58, 2465-2479 (2002).
[CrossRef]

Oppenheimer, C.

D. Weidmann, G. Wysocki, C. Oppenheimer, and F. K. Tittel, "Development of a compact quantum cascade laser spectrometer for field measurements of CO2 isotopes," Appl. Phys. B 80, 255-260 (2005).
[CrossRef]

Osiac, M.

J. B. McManus, D. Nelson, M. Zahniser, L. Mechold, M. Osiac, J. Röpcke, and A. Rousseau, "TOBI: a two-laser beam infrared system for time-resolved plasma diagnostics of infrared active compounds," Rev. Sci. Instrum. 74, 2709-2713 (2003).
[CrossRef]

Parchatka, U.

R. Korman, R. Königstedt, U. Parchatka, J. Lelieveld, and H. Fisher, "QUALITAS: a mid-infrared spectrometer for sensitive trace gas measurements based on quantum cascade lasers in CW operation," Rev. Sci. Instrum. 76, 075102 (2005).
[CrossRef]

Qi, L.

L.-Y. Hao, S. Qiang, G.-R. Wu, L. Qi, D. Feng, and Q.-S. Zhu, "Cylindrical mirror multipass Lissajous system for laser photoacoustic spectroscopy," Rev. Sci. Instrum. 73, 2079-3085 (2002).
[CrossRef]

Qiang, S.

L.-Y. Hao, S. Qiang, G.-R. Wu, L. Qi, D. Feng, and Q.-S. Zhu, "Cylindrical mirror multipass Lissajous system for laser photoacoustic spectroscopy," Rev. Sci. Instrum. 73, 2079-3085 (2002).
[CrossRef]

Richter, D.

D. Richter, A. Fried, B. P. Wert, J. G. Walega, and F. K. Tittel, "Development of a tunable mid-IR difference frequency laser source for highly sensitive airborne trace gas detection," Appl. Phys B 75, 281-288 (2002).
[CrossRef]

Röpcke, J.

J. B. McManus, D. Nelson, M. Zahniser, L. Mechold, M. Osiac, J. Röpcke, and A. Rousseau, "TOBI: a two-laser beam infrared system for time-resolved plasma diagnostics of infrared active compounds," Rev. Sci. Instrum. 74, 2709-2713 (2003).
[CrossRef]

Rousseau, A.

J. B. McManus, D. Nelson, M. Zahniser, L. Mechold, M. Osiac, J. Röpcke, and A. Rousseau, "TOBI: a two-laser beam infrared system for time-resolved plasma diagnostics of infrared active compounds," Rev. Sci. Instrum. 74, 2709-2713 (2003).
[CrossRef]

Saleska, S.

J. B. McManus, D. D. Nelson, J. H. Shorter, R. Jimenez, S. Herndon, S. Saleska, and M. Zahniser, "A high precision pulsed quantum cascade laser spectrometer for measurements of stable isotopes of carbon dioxide," J. Mod. Opt. 52, 2309-2321 (2005).
[CrossRef]

Schulte, H. J.

Shorter, J. H.

J. B. McManus, D. D. Nelson, J. H. Shorter, R. Jimenez, S. Herndon, S. Saleska, and M. Zahniser, "A high precision pulsed quantum cascade laser spectrometer for measurements of stable isotopes of carbon dioxide," J. Mod. Opt. 52, 2309-2321 (2005).
[CrossRef]

D. D. Nelson, J. H. Shorter, J. B. McManus, and M. S. Zahniser, "Sub-part-per-billion detection of nitric oxide in air using a thermoelectrically cooled mid-infrared quantum cascade laser spectrometer," Appl. Phys. B 75, 343-350 (2002).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Silver, J. A.

Tittel, F. K.

D. Weidmann, G. Wysocki, C. Oppenheimer, and F. K. Tittel, "Development of a compact quantum cascade laser spectrometer for field measurements of CO2 isotopes," Appl. Phys. B 80, 255-260 (2005).
[CrossRef]

D. Richter, A. Fried, B. P. Wert, J. G. Walega, and F. K. Tittel, "Development of a tunable mid-IR difference frequency laser source for highly sensitive airborne trace gas detection," Appl. Phys B 75, 281-288 (2002).
[CrossRef]

Tovar, A. A.

Walega, J. G.

D. Richter, A. Fried, B. P. Wert, J. G. Walega, and F. K. Tittel, "Development of a tunable mid-IR difference frequency laser source for highly sensitive airborne trace gas detection," Appl. Phys B 75, 281-288 (2002).
[CrossRef]

Weidmann, D.

D. Weidmann, G. Wysocki, C. Oppenheimer, and F. K. Tittel, "Development of a compact quantum cascade laser spectrometer for field measurements of CO2 isotopes," Appl. Phys. B 80, 255-260 (2005).
[CrossRef]

Wert, B. P.

D. Richter, A. Fried, B. P. Wert, J. G. Walega, and F. K. Tittel, "Development of a tunable mid-IR difference frequency laser source for highly sensitive airborne trace gas detection," Appl. Phys B 75, 281-288 (2002).
[CrossRef]

Williams, L. R.

J. B. McManus, M. S. Zahniser, D. D. Nelson, L. R. Williams, and C. E. Kolb, "Infrared laser spectrometer with balanced absorption for measurement of isotopic ratios of carbon gases," Spectrochim. Acta 58, 2465-2479 (2002).
[CrossRef]

Wu, G.-R.

L.-Y. Hao, S. Qiang, G.-R. Wu, L. Qi, D. Feng, and Q.-S. Zhu, "Cylindrical mirror multipass Lissajous system for laser photoacoustic spectroscopy," Rev. Sci. Instrum. 73, 2079-3085 (2002).
[CrossRef]

Wysocki, G.

D. Weidmann, G. Wysocki, C. Oppenheimer, and F. K. Tittel, "Development of a compact quantum cascade laser spectrometer for field measurements of CO2 isotopes," Appl. Phys. B 80, 255-260 (2005).
[CrossRef]

Zahniser, M.

J. B. McManus, D. D. Nelson, J. H. Shorter, R. Jimenez, S. Herndon, S. Saleska, and M. Zahniser, "A high precision pulsed quantum cascade laser spectrometer for measurements of stable isotopes of carbon dioxide," J. Mod. Opt. 52, 2309-2321 (2005).
[CrossRef]

J. B. McManus, D. Nelson, M. Zahniser, L. Mechold, M. Osiac, J. Röpcke, and A. Rousseau, "TOBI: a two-laser beam infrared system for time-resolved plasma diagnostics of infrared active compounds," Rev. Sci. Instrum. 74, 2709-2713 (2003).
[CrossRef]

Zahniser, M. S.

D. D. Nelson, J. H. Shorter, J. B. McManus, and M. S. Zahniser, "Sub-part-per-billion detection of nitric oxide in air using a thermoelectrically cooled mid-infrared quantum cascade laser spectrometer," Appl. Phys. B 75, 343-350 (2002).
[CrossRef]

J. B. McManus, M. S. Zahniser, D. D. Nelson, L. R. Williams, and C. E. Kolb, "Infrared laser spectrometer with balanced absorption for measurement of isotopic ratios of carbon gases," Spectrochim. Acta 58, 2465-2479 (2002).
[CrossRef]

J. B. McManus, P. L Kebabian, and M. S. Zahniser, "Astigmatic mirror multiple pass absorption cells for long pathlength spectroscopy," Appl. Opt. 34, 3336-3348 (1995).
[CrossRef] [PubMed]

Zhu, Q.-S.

L.-Y. Hao, S. Qiang, G.-R. Wu, L. Qi, D. Feng, and Q.-S. Zhu, "Cylindrical mirror multipass Lissajous system for laser photoacoustic spectroscopy," Rev. Sci. Instrum. 73, 2079-3085 (2002).
[CrossRef]

Appl. Opt. (4)

Appl. Phys B (1)

D. Richter, A. Fried, B. P. Wert, J. G. Walega, and F. K. Tittel, "Development of a tunable mid-IR difference frequency laser source for highly sensitive airborne trace gas detection," Appl. Phys B 75, 281-288 (2002).
[CrossRef]

Appl. Phys. B (2)

D. D. Nelson, J. H. Shorter, J. B. McManus, and M. S. Zahniser, "Sub-part-per-billion detection of nitric oxide in air using a thermoelectrically cooled mid-infrared quantum cascade laser spectrometer," Appl. Phys. B 75, 343-350 (2002).
[CrossRef]

D. Weidmann, G. Wysocki, C. Oppenheimer, and F. K. Tittel, "Development of a compact quantum cascade laser spectrometer for field measurements of CO2 isotopes," Appl. Phys. B 80, 255-260 (2005).
[CrossRef]

J. Mod. Opt. (1)

J. B. McManus, D. D. Nelson, J. H. Shorter, R. Jimenez, S. Herndon, S. Saleska, and M. Zahniser, "A high precision pulsed quantum cascade laser spectrometer for measurements of stable isotopes of carbon dioxide," J. Mod. Opt. 52, 2309-2321 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Rev. Sci. Instrum. (3)

J. B. McManus, D. Nelson, M. Zahniser, L. Mechold, M. Osiac, J. Röpcke, and A. Rousseau, "TOBI: a two-laser beam infrared system for time-resolved plasma diagnostics of infrared active compounds," Rev. Sci. Instrum. 74, 2709-2713 (2003).
[CrossRef]

R. Korman, R. Königstedt, U. Parchatka, J. Lelieveld, and H. Fisher, "QUALITAS: a mid-infrared spectrometer for sensitive trace gas measurements based on quantum cascade lasers in CW operation," Rev. Sci. Instrum. 76, 075102 (2005).
[CrossRef]

L.-Y. Hao, S. Qiang, G.-R. Wu, L. Qi, D. Feng, and Q.-S. Zhu, "Cylindrical mirror multipass Lissajous system for laser photoacoustic spectroscopy," Rev. Sci. Instrum. 73, 2079-3085 (2002).
[CrossRef]

Spectrochim. Acta (1)

J. B. McManus, M. S. Zahniser, D. D. Nelson, L. R. Williams, and C. E. Kolb, "Infrared laser spectrometer with balanced absorption for measurement of isotopic ratios of carbon gases," Spectrochim. Acta 58, 2465-2479 (2002).
[CrossRef]

Other (1)

A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (12)

Fig. 1
Fig. 1

Geometry of the astigmatic cell.

Fig. 2
Fig. 2

Comparison of beam-spot locations on mirrors, computed using a ray-trace program (solid markers) and the matrix formulations (open markers), for astigmatic mirror radii of 246 and 269.4   mm . The pattern { N = 238 , M x = 142 , M y = 134 } appears at a mirror spacing of 321.21   mm and a twist angle of 12.82°. In this case, Ψ = 0.049 .

Fig. 3
Fig. 3

Comparison of beam-spot locations on mirrors, computed using a ray-trace program (solid markers) and the matrix formulations (open markers), for astigmatic mirror radii of 246 and 269.4   mm . The pattern { N = 262 , M x = 154 , M y = 148 } appears at a mirror spacing of 321.72   mm and a twist angle of 17.58°. In this case, Ψ = 0.066 .

Fig. 4
Fig. 4

Comparison of beam-spot locations on mirrors, computed using a ray-trace program (solid markers) and the matrix formulations (open markers), for a cylindrical mirror cell, a cylindrical radius of 200   mm . The pattern { N = 174 , M x = 44 , M y = 50 } appears at a mirror spacing of 113.19   mm and a twist angle of 49.12°. In this case, Ψ = 0.20 .

Fig. 5
Fig. 5

Variation in pattern borders as the aim-in point is changed, maintaining a constant radius on the back mirror, represented as the black dots on the half-circle, center. The pattern borders are shown surrounding, with the back border dashed, with a dot for the aim-in point, and the front border solid. The mirror parameters are: R x = 246 , R x = 269.4 , d = 320 , τ = 20 ° , Ψ = 0.075 .

Fig. 6
Fig. 6

Pattern map, N pass ( φ x , φ y ) , for mirror radii of 246 and 269.4   mm , the hole radius∕mirror radius = ρ = 0.1 . The map was iterated using a simplified function. Complete patterns ( M x and M y even) are in color, while incomplete patterns (either or both M x , M y odd) are in shades of gray.

Fig. 7
Fig. 7

Pattern map versus mirror separation and twist. The map was iterated using the matrix solution, for mirror radii of 246 and 269.4   mm , ρ = 0.1 , mirror separation of 197 205   mm and a twist of 0°–45°. The color coding is the same as in Fig. 6.

Fig. 8
Fig. 8

Map of pass number versus spacing and twist angle for a cylindrical mirror cell, (mirror radii of R x = , R y = 200   mm , and ρ = 0.1 ). The color coding is the same as in Fig. 6.

Fig. 9
Fig. 9

Coupling coefficient calculated for an astigmatic mirror cell with small difference in radii (246 and 269.4   mm ) and nominal twist near 0°.

Fig. 10
Fig. 10

Mirror filling efficiency, the ratio of the pattern area to that of a square, for an astigmatic cell (mirror radii of 246 and 269 .4   mm ) and nominal twist near 0°.

Fig. 11
Fig. 11

Coupling coefficient calculated for a cylindrical mirror cell (mirror radius of 200   mm ) and nominal twist near 45°.

Fig. 12
Fig. 12

Mirror filling efficiency, the ratio of the pattern area to that of a square, a cylindrical mirror cell (mirror radius of 200   mm ) and nominal twist near 45°.

Equations (37)

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X n = A x sin ( n θ x ) , Y n = A y sin ( n θ y ) ,
θ x = π M x / N , θ y = π M y / N ,
X = | x Δ x / Δ z y Δ y / Δ z | .
Space,   D = | 1 d 0 0 0 1 0 0 0 0 1 d 0 0 0 1 | ,
Reflection,   R = | 1 0 0 0 g x 1 0 0 0 0 1 0 0 0 g y 1 | , g x y = 2 / R x y = 1 / f x y ,
Rotation ,   τ = | cos ( τ ) 0 sin ( τ ) 0 0 cos ( τ ) 0 sin ( τ ) sin ( τ ) 0 cos ( τ ) 0 0 sin ( τ ) 0 cos ( τ ) | .
C r t = D τ f 1 R τ f D τ b 1 R τ b .
C r t = τ f 1 D R τ f τ b 1 D R τ b .
C 1 = | [ 1 d g U ] d s c d δ 0 g U 1 s c δ 0 s c d δ 0 [ 1 d g L ] d s c δ 0 g L 1 | .
C r t = | [ 1 d g U ] d s f c f d δ 0 g U 1 s f c f δ 0 s f c f d δ 0 [ 1 d g L ] d s f c f δ 0 g L 1 | | [ 1 d g U ] d s b c b d δ 0 g U 1 s b c b δ 0 s b c b d δ 0 [ 1 d g L ] d s b c b δ 0 g L 1 | .
Z = | x n x n 1 y n y n 1 | .
T Z = | 1 0 0 0 1 d 0 0 0 0 1 0 0 0 1 d | , T Z 1 = | 1 0 0 0 d 1 d 1 0 0 0 0 1 0 0 0 d 1 d 1 | .
D z = T Z D T Z 1 = | 2 1 0 0 1 0 0 0 0 0 2 1 0 0 1 0 | ,
R z = T Z R T Z 1 = | 1 0 0 0 g x d 0 0 0 0 0 1 0 0 0 g y d 0 | .
C 1 z = | β U 1 ε 0 1 0 0 0 ε 0 β L 1 0 0 1 0 | , with   β U = 2 d g U , β L = 2 d g L , ε = s c d δ .
C r t z = | ( β U 2 1 ε 2 ) β U ε ( β U β L ) ε β U 1 ε 0 + ε ( β U β L ) ε ( β L 2 1 ε 2 ) β L ε 0 β L 1 | .
T D = | 1 0 Ψ 0 0 1 0 Ψ Ψ 0 1 0 0 Ψ 0 1 | ( 1 Ψ 2 ) 1 / 2 ,
T D 1 = | 1 0 Ψ 0 0 1 0 Ψ Ψ 0 1 0 0 Ψ 0 1 | ( 1 Ψ 2 ) 1 / 2 .
Z n = T D Z n = ( 1 Ψ 2 ) 1 / 2 | x b Ψ y b x f + Ψ y f y b Ψ x b y f + Ψ x f | .
C r t z = ( 1 Ψ 2 ) 1 | [ Q U Ψ 2 Q L 2 Ψ ε Δ ] [ ( β U + Ψ 2 β L 2 Ψ ε ) ] [ Ψ ( Q U Q L ) ε Δ ( 1 + Ψ 2 ) ] [ Ψ ( β U + β L ) ε ( 1 + Ψ 2 ) ] [ β U + Ψ 2 β L 2 Ψ ε ] [ Ψ 2 1 ] [ Ψ ( β U + β L ) ε ( 1 + Ψ 2 ) ] [ 0 ] [ Ψ ( Q L Q U ) + ε Δ ( 1 + Ψ 2 ) ] [ Ψ ( β U + β L ) ε ( 1 + Ψ 2 ) ] [ Q L Ψ 2 Q U + 2 Ψ ε Δ ] [ ( β L + Ψ 2 β U 2 Ψ ε ) ] [ Ψ ( β U + β L ) ε ( 1 + Ψ 2 ) ] [ 0 ] [ β L + Ψ 2 β U 2 Ψ ε ] [ Ψ 2 1 ] | ,
Ψ ε ( 1 + Ψ 2 ) = 0 , Ψ = [ ± ( 2 4 ε 2 ) 1 / 2 ] / ( 2 ε ) ,
C r t z = ( 1 Ψ 2 ) 1 | [ Q U Ψ 2 Q L 2 Ψ ε Δ ] [ ( β U + Ψ 2 β L 2 Ψ ε ) ] [ 0 ] [ 0 ] [ β U + Ψ 2 β L 2 Ψ ε ] [ Ψ 2 1 ] [ 0 ] [ 0 ] [ 0 ] [ 0 ] [ Q L Ψ 2 Q U + 2 Ψ ε Δ ] [ ( β L + Ψ 2 β U 2 Ψ ε ) ] [ 0 ] [ 0 ] [ β L + Ψ 2 β U 2 Ψ ε ] [ Ψ 2 1 ] | .
| A B C D | N = ( sin θ ) 1 | [ A sin N θ sin ( N 1 ) θ ] [ C sin N θ ] [ B sin N θ ] [ D sin N θ sin ( N 1 ) θ ] | ,
[ C r t z ] N = | Q U B U 0 0 C U D U 0 0 0 0 Q L B L 0 0 C L D L | N , = | [ Q U sin N θ U sin ( N 1 ) θ U ] / sin θ U [ B U sin N θ U / sin θ U ] 0 0 [ C U sin N θ U / sin θ U ] [ D U sin N θ U sin ( N 1 ) θ U ] / sin θ U 0 0 0 0 [ Q L sin N θ L sin ( N 1 ) θ L ] / sin θ L [ B L sin N θ L / sin θ L ] 0 0 [ C L sin N θ L / sin θ L ] [ D L sin N θ L sin ( N 1 ) θ L ] / sin θ L | .
cos ( θ U ) = { ( Q U Ψ 2 Q L 2 Ψ ε Δ ) / ( 1 Ψ 2 ) 1 } / 2 ,
cos ( θ L ) = { ( Q L Ψ 2 Q U + 2 Ψ ε Δ ) / ( 1 Ψ 2 ) 1 } / 2 .
cos ( θ U ) = ( β U 2 + Ψ 2 β L 2 ) / 2 ( 1 + Ψ 2 ) 1 ε 2 / 2 ,
cos ( θ L ) = ( β L 2 + Ψ 2 β U 2 ) / 2 ( 1 + Ψ 2 ) 1 ε 2 / 2 ,
cos ( θ U ) = { ( β U 2 + β L 2 ) + ( β U 2 β L 2 ) × [ 1 ( 2 ε / ) 2 ] 1 / 2 } 1 ε 2 / 2 ,
cos ( θ L ) = { ( β U 2 + β L 2 ) ( β U 2 β L 2 ) × [ 1 ( 2 ε / ) 2 ] 1 / 2 } 1 ε 2 / 2 .
C r t z = | [ 2 cos ( θ U ) + 1 ] 2 cos ( θ U / 2 ) 0 0 2 cos ( θ U / 2 ) 1 0 0 0 0 [ 2 cos ( θ L ) + 1 ] 2 cos ( θ L / 2 ) 0 0 2 cos ( θ L / 2 ) 1 | .
C r t z N = | [ ( 2 cos θ U + 1 ) sin N θ U sin ( N 1 ) θ U ] / sin θ U [ 2 cos ( θ U / 2 ) sin N θ U / sin θ U ] 0 0 [ 2 cos ( θ U / 2 ) sin N θ U / sin θ U ] [ sin N θ U sin ( N 1 ) θ U ] / sin θ U 0 0 0 0 [ ( 2 cos θ L + 1 ) sin N θ L sin ( N 1 ) θ L ] / sin θ L [ 2 cos ( θ L / 2 ) sin N θ L / sin θ L ] 0 0 [ 2 cos ( θ L / 2 ) sin N θ L / sin θ L ] [ sin N θ L sin ( N 1 ) θ L ] / sin θ L | .
C r t z N = | C u sin ( N + 1 / 2 ) θ U C u sin N θ U 0 0 C u sin N θ U C u sin ( N 1 / 2 ) θ U 0 0 0 0 C L sin ( N + 1 / 2 ) θ L C L sin N θ L 0 0 C L sin N θ L C L sin ( N 1 / 2 ) θ L | ,
Z 0 = | x b i 0 y b i 0 | , Z N = C r t z N Z 0 = | x b i C u sin ( N + 1 / 2 ) θ U x f i C u sin N θ U y b i C L sin ( N + 1 / 2 ) θ L y f i C L sin N θ L | ,
C r t z N Z 0 = | ( x b i Ψ y b i ) C u sin ( N + 1 / 2 ) θ U ( x b i Ψ y b i ) C u sin N θ U ( y b i Ψ x b i ) C L sin ( N + 1 / 2 ) θ L ( y b i Ψ x b i ) C L sin N θ L | ( 1 Ψ 2 ) 1 / 2 .
Z n = | ( x b i Ψ y b i ) C u sin ( N + 1 / 2 ) θ U ( x b i Ψ y b i ) C u sin N θ U ( y b i Ψ x b i ) C L sin ( N + 1 / 2 ) θ L ( y b i Ψ x b i ) C L sin N θ L + Ψ ( y b i Ψ x b i ) C L sin ( N + 1 / 2 ) θ L Ψ ( y b i Ψ x b i ) C L sin N θ L + Ψ ( x b i Ψ y b i ) C u sin ( N + 1 / 2 ) θ U Ψ ( x b i Ψ y b i ) C u sin N θ U | ( 1 Ψ 2 ) 1 .
R p min / R p max = ( 1 | Ψ | ) / ( 1 + | Ψ | ) .

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