Abstract

A simple, self-referenced technique for the determination of meridional surface slope errors of toroidal mirrors (TMs) used in synchrotron radiation is presented. An expanded, collimated, linearly polarized laser beam is allowed to fall on a TM, and the reflected toric wavefront from the TM is allowed to enter a Sagnac interferometer based lateral shearing interferometer setup, which duplicates the beam into identical wavefront components having orthogonal linear polarizations, i.e., p and s polarizations, and introduces appropriate lateral shear between the identical pair of wavefronts along the meridional direction. The laterally sheared identical wavefronts interfere when brought to the same state of polarization. The optical path difference (OPD) variation along the direction of the lateral shear in the interference field is a map of the slope variation. Polarization phase shifting interferometery has been applied to find the OPD variation and thus the slope distribution and errors. Results obtained for a TM of moderately long average radius of curvature along the central meridional section and a relatively shorter radius of curvature along the sagittal section are presented and compared with that obtained with a long trace profiler.

© 2012 Optical Society of America

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References

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  1. A. Freund, “Mirrors for synchrotron beamlines,” in Handbook of Optics III, M. Bass, J. M. Enoch, E. W. V. Stryland, and W. L. Wolfe, eds. (McGraw-Hill, 2001), pp. 26.3–26.8.
  2. R. A. Paquin and M. R. Howells, “Mirror materials for synchrotron radiation optics,” Proc. SPIE 3152, 2–6 (1997).
    [CrossRef]
  3. P. Z. Takacs, S. Qian, and J. Colbert, “Design of a long-trace profiler,” Proc. SPIE 749, 59–64 (1987).
  4. G. Ludbrook, S. Alcock, and K. J. S. Sawhney, “A Fizeau interferometer system with double-pass and stitching for characterizing the figure error of large (>1  m) synchrotron optics,” Proc. SPIE 7389, 738939 (2009).
    [CrossRef]
  5. S. Morozumi, “Toric-wavefront testing holographic interferometry for concave grating,” Appl. Opt. 23, 3082–3090 (1984).
    [CrossRef]
  6. T. M. Jeong, M. Menon, and G. Yoon, “Measurement of wave-front aberration in soft contact lens by use of a Shack-Hartmann wave-front sensor,” Appl. Opt. 44, 4523–4527 (2005).
    [CrossRef]
  7. M. Strojnik, G. Paez, and M. Mantravadi, “Lateral shear interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 122–184.
  8. M. V. R. K. Murty, “The use of a single plane parallel plate as a lateral shearing interferometer with a visible gas laser source,” Appl. Opt. 3, 531–534 (1964).
    [CrossRef]
  9. Y. H. Lo, A. R. D. Somervell, and T. H. Barnes, “Wavefront measurement with a phase-shifting, lateral-shearing Sagnac interferometer operating in broadband light,” Opt. Laser Eng. 43, 33–42 (2005).
    [CrossRef]
  10. M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24, 4439–444 (1985).
    [CrossRef]
  11. S. Chatterjee, Y. P. Kumar, and B. Bhaduri, “Measurement of surface figure of plane optical surfaces with polarization phase-shifting Fizeau interferometer,” Opt. Laser Technol. 39, 268–274 (2007).
    [CrossRef]
  12. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometery: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef]
  13. K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1988), Vol. 26, pp. 349–393.
  14. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998), pp. 248–255.

2009 (1)

G. Ludbrook, S. Alcock, and K. J. S. Sawhney, “A Fizeau interferometer system with double-pass and stitching for characterizing the figure error of large (>1  m) synchrotron optics,” Proc. SPIE 7389, 738939 (2009).
[CrossRef]

2007 (1)

S. Chatterjee, Y. P. Kumar, and B. Bhaduri, “Measurement of surface figure of plane optical surfaces with polarization phase-shifting Fizeau interferometer,” Opt. Laser Technol. 39, 268–274 (2007).
[CrossRef]

2005 (2)

T. M. Jeong, M. Menon, and G. Yoon, “Measurement of wave-front aberration in soft contact lens by use of a Shack-Hartmann wave-front sensor,” Appl. Opt. 44, 4523–4527 (2005).
[CrossRef]

Y. H. Lo, A. R. D. Somervell, and T. H. Barnes, “Wavefront measurement with a phase-shifting, lateral-shearing Sagnac interferometer operating in broadband light,” Opt. Laser Eng. 43, 33–42 (2005).
[CrossRef]

1997 (1)

R. A. Paquin and M. R. Howells, “Mirror materials for synchrotron radiation optics,” Proc. SPIE 3152, 2–6 (1997).
[CrossRef]

1987 (2)

1985 (1)

1984 (1)

1964 (1)

Alcock, S.

G. Ludbrook, S. Alcock, and K. J. S. Sawhney, “A Fizeau interferometer system with double-pass and stitching for characterizing the figure error of large (>1  m) synchrotron optics,” Proc. SPIE 7389, 738939 (2009).
[CrossRef]

Barnes, T. H.

Y. H. Lo, A. R. D. Somervell, and T. H. Barnes, “Wavefront measurement with a phase-shifting, lateral-shearing Sagnac interferometer operating in broadband light,” Opt. Laser Eng. 43, 33–42 (2005).
[CrossRef]

Bhaduri, B.

S. Chatterjee, Y. P. Kumar, and B. Bhaduri, “Measurement of surface figure of plane optical surfaces with polarization phase-shifting Fizeau interferometer,” Opt. Laser Technol. 39, 268–274 (2007).
[CrossRef]

Chatterjee, S.

S. Chatterjee, Y. P. Kumar, and B. Bhaduri, “Measurement of surface figure of plane optical surfaces with polarization phase-shifting Fizeau interferometer,” Opt. Laser Technol. 39, 268–274 (2007).
[CrossRef]

Colbert, J.

P. Z. Takacs, S. Qian, and J. Colbert, “Design of a long-trace profiler,” Proc. SPIE 749, 59–64 (1987).

Creath, K.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1988), Vol. 26, pp. 349–393.

Delisle, C.

Eiju, T.

Freund, A.

A. Freund, “Mirrors for synchrotron beamlines,” in Handbook of Optics III, M. Bass, J. M. Enoch, E. W. V. Stryland, and W. L. Wolfe, eds. (McGraw-Hill, 2001), pp. 26.3–26.8.

Hariharan, P.

Howells, M. R.

R. A. Paquin and M. R. Howells, “Mirror materials for synchrotron radiation optics,” Proc. SPIE 3152, 2–6 (1997).
[CrossRef]

Jeong, T. M.

Kothiyal, M. P.

Kumar, Y. P.

S. Chatterjee, Y. P. Kumar, and B. Bhaduri, “Measurement of surface figure of plane optical surfaces with polarization phase-shifting Fizeau interferometer,” Opt. Laser Technol. 39, 268–274 (2007).
[CrossRef]

Lo, Y. H.

Y. H. Lo, A. R. D. Somervell, and T. H. Barnes, “Wavefront measurement with a phase-shifting, lateral-shearing Sagnac interferometer operating in broadband light,” Opt. Laser Eng. 43, 33–42 (2005).
[CrossRef]

Ludbrook, G.

G. Ludbrook, S. Alcock, and K. J. S. Sawhney, “A Fizeau interferometer system with double-pass and stitching for characterizing the figure error of large (>1  m) synchrotron optics,” Proc. SPIE 7389, 738939 (2009).
[CrossRef]

Malacara, D.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998), pp. 248–255.

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998), pp. 248–255.

Mantravadi, M.

M. Strojnik, G. Paez, and M. Mantravadi, “Lateral shear interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 122–184.

Menon, M.

Morozumi, S.

Murty, M. V. R. K.

Oreb, B. F.

Paez, G.

M. Strojnik, G. Paez, and M. Mantravadi, “Lateral shear interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 122–184.

Paquin, R. A.

R. A. Paquin and M. R. Howells, “Mirror materials for synchrotron radiation optics,” Proc. SPIE 3152, 2–6 (1997).
[CrossRef]

Qian, S.

P. Z. Takacs, S. Qian, and J. Colbert, “Design of a long-trace profiler,” Proc. SPIE 749, 59–64 (1987).

Sawhney, K. J. S.

G. Ludbrook, S. Alcock, and K. J. S. Sawhney, “A Fizeau interferometer system with double-pass and stitching for characterizing the figure error of large (>1  m) synchrotron optics,” Proc. SPIE 7389, 738939 (2009).
[CrossRef]

Servin, M.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998), pp. 248–255.

Somervell, A. R. D.

Y. H. Lo, A. R. D. Somervell, and T. H. Barnes, “Wavefront measurement with a phase-shifting, lateral-shearing Sagnac interferometer operating in broadband light,” Opt. Laser Eng. 43, 33–42 (2005).
[CrossRef]

Strojnik, M.

M. Strojnik, G. Paez, and M. Mantravadi, “Lateral shear interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 122–184.

Takacs, P. Z.

P. Z. Takacs, S. Qian, and J. Colbert, “Design of a long-trace profiler,” Proc. SPIE 749, 59–64 (1987).

Yoon, G.

Appl. Opt. (5)

Opt. Laser Eng. (1)

Y. H. Lo, A. R. D. Somervell, and T. H. Barnes, “Wavefront measurement with a phase-shifting, lateral-shearing Sagnac interferometer operating in broadband light,” Opt. Laser Eng. 43, 33–42 (2005).
[CrossRef]

Opt. Laser Technol. (1)

S. Chatterjee, Y. P. Kumar, and B. Bhaduri, “Measurement of surface figure of plane optical surfaces with polarization phase-shifting Fizeau interferometer,” Opt. Laser Technol. 39, 268–274 (2007).
[CrossRef]

Proc. SPIE (3)

R. A. Paquin and M. R. Howells, “Mirror materials for synchrotron radiation optics,” Proc. SPIE 3152, 2–6 (1997).
[CrossRef]

P. Z. Takacs, S. Qian, and J. Colbert, “Design of a long-trace profiler,” Proc. SPIE 749, 59–64 (1987).

G. Ludbrook, S. Alcock, and K. J. S. Sawhney, “A Fizeau interferometer system with double-pass and stitching for characterizing the figure error of large (>1  m) synchrotron optics,” Proc. SPIE 7389, 738939 (2009).
[CrossRef]

Other (4)

M. Strojnik, G. Paez, and M. Mantravadi, “Lateral shear interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 122–184.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1988), Vol. 26, pp. 349–393.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998), pp. 248–255.

A. Freund, “Mirrors for synchrotron beamlines,” in Handbook of Optics III, M. Bass, J. M. Enoch, E. W. V. Stryland, and W. L. Wolfe, eds. (McGraw-Hill, 2001), pp. 26.3–26.8.

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Figures (8)

Fig. 1.
Fig. 1.

Schematic of the setup for the determination of the meridional slope errors of a TM.

Fig. 2.
Fig. 2.

Schematic of the setup for increasing the probe beam footprint on the TM.

Fig. 3.
Fig. 3.

Phase shifted lateral shearing interferograms grabbed for the five-step polarization phase shifting interferometry. (A)  φ = 0 , (B)  φ = π / 2 , (C)  φ = π , (D)  φ = 3 π / 2 , and (E)  φ = 2 π .

Fig. 4.
Fig. 4.

Unwrapped OPD values ( Δ W ) obtained along a central meridional section.

Fig. 5.
Fig. 5.

Variation of slope ( α ) along the meridional section of the TM along with the fitted curve.

Fig. 6.
Fig. 6.

Variation of slope error ( Δ α ) along the section of the TM.

Fig. 7.
Fig. 7.

LTP fringes due to superposition of laterally separated pencil beams from the TM.

Fig. 8.
Fig. 8.

Variation of slope error ( Δ α ) along the section of the TM as measured with the LTP.

Equations (6)

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

Δ W = W [ ( X S ) , Y ] W [ X , Y ] ,
I ( x , y ) = I 0 ( x , y ) { 1 + V ( x , y ) cos [ β ( x , y ) + φ j ] } ,
β = tan 1 [ 2 ( I 4 I 2 ) I 1 + I 5 2 I 3 ] .
Δ β = ( ε 2 / 4 ) sin 2 β ,
Δ W = ( λ 2 π ) β ,
α = a 1 + a 2 x + a 3 x 2 + a 4 x 3 + a 5 x 4 , with a 1 = 5.51377 × 10 4 , a 2 = 1.22512 × 10 4 , a 3 = 5.7798 × 10 7 , a 4 = 7.50616 × 10 8 , a 5 = 1.61284 × 10 9 .

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