Abstract

A shearing interferometer is presented that uses polarization control to shear the wavefront and to modulate the interference pattern. The shear is generated by spatial walk-off in a birefringent crystal. By adjusting the orientation of the birefringent crystal, the components of the wavefront gradient can be independently measured to allow determination of the full wavefront vector gradient as well as reconstruction of the wavefront. Further, the monolithic nature of the crystal used for shearing allows the interferometer to be set up without need for precise alignment of any components. An algorithm incorporating homodyne detection is presented, which analyzes the modulated interferograms to determine the components of the wavefront gradient, from which the wavefront is reconstructed. The thermal deformation of a mirror subject to heating from absorption of a Gaussian pump beam was accurately observed with a sensitivity better than λ/160. We show that this sensitivity is scale invariant, and present a method to account for the nonuniform spatial frequency response of the interferometer.

© 2012 Optical Society of America

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References

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    [CrossRef]
  2. L. Zhu Tuan and V. K. Kirillovski, “Computerized shearing interferometer,” J. Opt. Technol. 75, 156–160 (2008).
    [CrossRef]
  3. V. M. Murukeshan, O. Lin Seng, and A. Asundi, “Polarization phase shifting shearography for optical metrological applications,” Opt. Laser Technol. 30, 527–531 (1998).
    [CrossRef]
  4. V. Rosso, L. Zhang, F. Michel, Y. Renotte, Y. Lion, and A.-M. Habraken, “Out-of-plane displacement derivative measurement: comparison of results obtained by a shearographic interferometer using the separation of the polarization states and the finite element method,” Proc. SPIE 6343, 634327 (2006).
    [CrossRef]
  5. G. R. Fowels, Introduction to Modern Optics, 2nd ed. (Dover, 1975).
  6. D. N. Nikogosyan, “Beta barium borate (BBO) a review of its properties and applications,” Appl. Phys. A 52, 359–368 (1991).
    [CrossRef]
  7. A. Yariv and P. Yeh, Optical Waves in Crystals, Propagation and Control of Laser Radiation (Wiley, 2003).
  8. M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover, 1970).
  9. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998), MATLAB code posted by Bruce Spottiswoode.
  10. A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” in Proceedings of Tenth IEEE International Conference on Computer Vision (IEEE, 2005), 174–181.
  11. P. P. Lu, A. L. Bullington, P. Beyersdorf, S. Traeger, J. Mansell, R. Beausoleil, E. K. Gustafson, R. L. Byer, and M. M. Fejer, “Wavefront distortion of the reflected and diffracted beams produced by the thermoelastic deformation of a diffraction grating heated by a Gaussian laser beam,” J. Opt. Soc. Am. A 24, 659–668 (2007).
    [CrossRef]
  12. M. Servin, M. Cywiak, and A. Davila, “Extreme shearing interferometry: theoretical limits with practical consequences,” Opt. Express 15, 17805–17818 (2007).
    [CrossRef]
  13. P. Hello and J. Y. Vinet, “Analytical models of thermal aberrations in massive mirrors heated by high power laser beams,” French J. Phys. 51, 1267–1282 (1990).
    [CrossRef]
  14. P. Hello and J. Y. Vinet, “Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams,” French J. Phys. 51, 2243–2261 (1990).
    [CrossRef]
  15. W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1991).
    [CrossRef]
  16. A. F. Brooks, “Hartmann wavefront sensors for advanced gravitational wave interferometers,” Ph.D. thesis (University of Adelaide, 2007).
  17. S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, “Thermal compensation system description,” LIGO-T050064-00-R (2005).
  18. G. Harry, “Advanced LIGO test masses and core optics,” LIGO-G1000098 (2010).
  19. M. Flanigan and G. Billingsley, “Advanced LIGO input test mass (ITM),” LIGO-E080511-V3-D (2008).

2008 (1)

2007 (2)

2006 (1)

V. Rosso, L. Zhang, F. Michel, Y. Renotte, Y. Lion, and A.-M. Habraken, “Out-of-plane displacement derivative measurement: comparison of results obtained by a shearographic interferometer using the separation of the polarization states and the finite element method,” Proc. SPIE 6343, 634327 (2006).
[CrossRef]

1998 (1)

V. M. Murukeshan, O. Lin Seng, and A. Asundi, “Polarization phase shifting shearography for optical metrological applications,” Opt. Laser Technol. 30, 527–531 (1998).
[CrossRef]

1991 (2)

D. N. Nikogosyan, “Beta barium borate (BBO) a review of its properties and applications,” Appl. Phys. A 52, 359–368 (1991).
[CrossRef]

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1991).
[CrossRef]

1990 (2)

P. Hello and J. Y. Vinet, “Analytical models of thermal aberrations in massive mirrors heated by high power laser beams,” French J. Phys. 51, 1267–1282 (1990).
[CrossRef]

P. Hello and J. Y. Vinet, “Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams,” French J. Phys. 51, 2243–2261 (1990).
[CrossRef]

1985 (1)

Agrawal, A.

A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” in Proceedings of Tenth IEEE International Conference on Computer Vision (IEEE, 2005), 174–181.

Asundi, A.

V. M. Murukeshan, O. Lin Seng, and A. Asundi, “Polarization phase shifting shearography for optical metrological applications,” Opt. Laser Technol. 30, 527–531 (1998).
[CrossRef]

Ballmer, S.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, “Thermal compensation system description,” LIGO-T050064-00-R (2005).

Beausoleil, R.

Beyersdorf, P.

Billingsley, G.

M. Flanigan and G. Billingsley, “Advanced LIGO input test mass (ITM),” LIGO-E080511-V3-D (2008).

Brooks, A. F.

A. F. Brooks, “Hartmann wavefront sensors for advanced gravitational wave interferometers,” Ph.D. thesis (University of Adelaide, 2007).

Bullington, A. L.

Byer, R. L.

Chellappa, R.

A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” in Proceedings of Tenth IEEE International Conference on Computer Vision (IEEE, 2005), 174–181.

Cywiak, M.

Danzmann, K.

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1991).
[CrossRef]

Davila, A.

Delisle, C.

Fejer, M. M.

Flanigan, M.

M. Flanigan and G. Billingsley, “Advanced LIGO input test mass (ITM),” LIGO-E080511-V3-D (2008).

Fowels, G. R.

G. R. Fowels, Introduction to Modern Optics, 2nd ed. (Dover, 1975).

Frolov, V.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, “Thermal compensation system description,” LIGO-T050064-00-R (2005).

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998), MATLAB code posted by Bruce Spottiswoode.

Gustafson, E. K.

Habraken, A.-M.

V. Rosso, L. Zhang, F. Michel, Y. Renotte, Y. Lion, and A.-M. Habraken, “Out-of-plane displacement derivative measurement: comparison of results obtained by a shearographic interferometer using the separation of the polarization states and the finite element method,” Proc. SPIE 6343, 634327 (2006).
[CrossRef]

Harry, G.

G. Harry, “Advanced LIGO test masses and core optics,” LIGO-G1000098 (2010).

Hello, P.

P. Hello and J. Y. Vinet, “Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams,” French J. Phys. 51, 2243–2261 (1990).
[CrossRef]

P. Hello and J. Y. Vinet, “Analytical models of thermal aberrations in massive mirrors heated by high power laser beams,” French J. Phys. 51, 1267–1282 (1990).
[CrossRef]

Kells, W.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, “Thermal compensation system description,” LIGO-T050064-00-R (2005).

Kirillovski, V. K.

Kothiyal, M. P.

Lawrence, R.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, “Thermal compensation system description,” LIGO-T050064-00-R (2005).

Lin Seng, O.

V. M. Murukeshan, O. Lin Seng, and A. Asundi, “Polarization phase shifting shearography for optical metrological applications,” Opt. Laser Technol. 30, 527–531 (1998).
[CrossRef]

Lion, Y.

V. Rosso, L. Zhang, F. Michel, Y. Renotte, Y. Lion, and A.-M. Habraken, “Out-of-plane displacement derivative measurement: comparison of results obtained by a shearographic interferometer using the separation of the polarization states and the finite element method,” Proc. SPIE 6343, 634327 (2006).
[CrossRef]

Lu, P. P.

Mansell, J.

Mason, K.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, “Thermal compensation system description,” LIGO-T050064-00-R (2005).

Michel, F.

V. Rosso, L. Zhang, F. Michel, Y. Renotte, Y. Lion, and A.-M. Habraken, “Out-of-plane displacement derivative measurement: comparison of results obtained by a shearographic interferometer using the separation of the polarization states and the finite element method,” Proc. SPIE 6343, 634327 (2006).
[CrossRef]

Moreno, G.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, “Thermal compensation system description,” LIGO-T050064-00-R (2005).

Murukeshan, V. M.

V. M. Murukeshan, O. Lin Seng, and A. Asundi, “Polarization phase shifting shearography for optical metrological applications,” Opt. Laser Technol. 30, 527–531 (1998).
[CrossRef]

Nikogosyan, D. N.

D. N. Nikogosyan, “Beta barium borate (BBO) a review of its properties and applications,” Appl. Phys. A 52, 359–368 (1991).
[CrossRef]

Ottaway, D.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, “Thermal compensation system description,” LIGO-T050064-00-R (2005).

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998), MATLAB code posted by Bruce Spottiswoode.

Raskar, R.

A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” in Proceedings of Tenth IEEE International Conference on Computer Vision (IEEE, 2005), 174–181.

Renotte, Y.

V. Rosso, L. Zhang, F. Michel, Y. Renotte, Y. Lion, and A.-M. Habraken, “Out-of-plane displacement derivative measurement: comparison of results obtained by a shearographic interferometer using the separation of the polarization states and the finite element method,” Proc. SPIE 6343, 634327 (2006).
[CrossRef]

Rosso, V.

V. Rosso, L. Zhang, F. Michel, Y. Renotte, Y. Lion, and A.-M. Habraken, “Out-of-plane displacement derivative measurement: comparison of results obtained by a shearographic interferometer using the separation of the polarization states and the finite element method,” Proc. SPIE 6343, 634327 (2006).
[CrossRef]

Rudiger, A.

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1991).
[CrossRef]

Schilling, R.

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1991).
[CrossRef]

Servin, M.

Smith, M.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, “Thermal compensation system description,” LIGO-T050064-00-R (2005).

Traeger, S.

Vinet, J. Y.

P. Hello and J. Y. Vinet, “Analytical models of thermal aberrations in massive mirrors heated by high power laser beams,” French J. Phys. 51, 1267–1282 (1990).
[CrossRef]

P. Hello and J. Y. Vinet, “Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams,” French J. Phys. 51, 2243–2261 (1990).
[CrossRef]

Vorvick, C.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, “Thermal compensation system description,” LIGO-T050064-00-R (2005).

Willems, P.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, “Thermal compensation system description,” LIGO-T050064-00-R (2005).

Winkler, W.

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1991).
[CrossRef]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals, Propagation and Control of Laser Radiation (Wiley, 2003).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals, Propagation and Control of Laser Radiation (Wiley, 2003).

Zhang, L.

V. Rosso, L. Zhang, F. Michel, Y. Renotte, Y. Lion, and A.-M. Habraken, “Out-of-plane displacement derivative measurement: comparison of results obtained by a shearographic interferometer using the separation of the polarization states and the finite element method,” Proc. SPIE 6343, 634327 (2006).
[CrossRef]

Zhu Tuan, L.

Zucker, M.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, “Thermal compensation system description,” LIGO-T050064-00-R (2005).

Appl. Opt. (1)

Appl. Phys. A (1)

D. N. Nikogosyan, “Beta barium borate (BBO) a review of its properties and applications,” Appl. Phys. A 52, 359–368 (1991).
[CrossRef]

French J. Phys. (2)

P. Hello and J. Y. Vinet, “Analytical models of thermal aberrations in massive mirrors heated by high power laser beams,” French J. Phys. 51, 1267–1282 (1990).
[CrossRef]

P. Hello and J. Y. Vinet, “Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams,” French J. Phys. 51, 2243–2261 (1990).
[CrossRef]

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

J. Opt. Technol. (1)

Opt. Express (1)

Opt. Laser Technol. (1)

V. M. Murukeshan, O. Lin Seng, and A. Asundi, “Polarization phase shifting shearography for optical metrological applications,” Opt. Laser Technol. 30, 527–531 (1998).
[CrossRef]

Phys. Rev. A (1)

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1991).
[CrossRef]

Proc. SPIE (1)

V. Rosso, L. Zhang, F. Michel, Y. Renotte, Y. Lion, and A.-M. Habraken, “Out-of-plane displacement derivative measurement: comparison of results obtained by a shearographic interferometer using the separation of the polarization states and the finite element method,” Proc. SPIE 6343, 634327 (2006).
[CrossRef]

Other (9)

G. R. Fowels, Introduction to Modern Optics, 2nd ed. (Dover, 1975).

A. Yariv and P. Yeh, Optical Waves in Crystals, Propagation and Control of Laser Radiation (Wiley, 2003).

M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover, 1970).

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998), MATLAB code posted by Bruce Spottiswoode.

A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” in Proceedings of Tenth IEEE International Conference on Computer Vision (IEEE, 2005), 174–181.

A. F. Brooks, “Hartmann wavefront sensors for advanced gravitational wave interferometers,” Ph.D. thesis (University of Adelaide, 2007).

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, “Thermal compensation system description,” LIGO-T050064-00-R (2005).

G. Harry, “Advanced LIGO test masses and core optics,” LIGO-G1000098 (2010).

M. Flanigan and G. Billingsley, “Advanced LIGO input test mass (ITM),” LIGO-E080511-V3-D (2008).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic diagram of the shearing interferometer. A helium neon laser polarized at 45° by a linear polarizer (LP) passes through a lens and a 5 μm pinhole to clean the spatial mode of the beam. The laser then reflects off the test optic. A beta barium borate (BBO) crystal shears the beam in x or y depending on its (adjustable) orientation. A half-wave plate placed after the BBO crystal rotates the polarization so each component of the sheared wavefront has a polarization component along the transmission axis of the polarizing beam splitter (PBS). The interference pattern is recorded by a digital camera and subsequently analyzed.

Fig. 2.
Fig. 2.

Interferogram from the shearing interferometer showing a horizontal shear, as noted by the displacement of one of the two shadows of a pin. The wavefront being measured has a roughly uniform curvature (responsible for the parallel fringes) and an additional small bump from a thermally deformed optic. The effect of the bump is visually evident as a deviation in the straightness of the fringes.

Fig. 3.
Fig. 3.

Red (upper dashed curve) and green (lower dashed curve) values for a representative pixel as a function of time. The composite value (solid curve) is formed by stitching their derivatives together after appropriate scaling, then integrating.

Fig. 4.
Fig. 4.

Measured spectral noise floor of our shearing interferometer. The spatial frequency units are normalized to the amount of shear so that one spatial frequency unit is dk=2πs/x where s is the magnitude of the shear and x is the width of the region of the surface being measured. For our experiment with s=860μm and x=6.3mm, dk=0.86rad.

Fig. 5.
Fig. 5.

Normalized radial gradient of the longitudinal thermally induced deformation from partial absorption of a Gaussian beam. The deformation u¯z=uz/uc is normalized to the characteristic thermal deformation uc given in Eq. (18) for the left axis, and normalized to the maximum deformation umax given in Eq. (19) on the right axis. The transverse coordinate r¯=r/w, is normalized to w, the Gaussian beam radius of the heating beam.

Fig. 6.
Fig. 6.

Percent reduction in the measured peak surface deformation due to spatial filtering from the shearing interferometer over a spatial length scale s. This is calculated according to expression (26) and multiplied by 100% to express as a percentage.

Fig. 7.
Fig. 7.

Thermal deformation measurement (in nanometers) from the shearing interferometer. A peak height of 72 nm was observed prior to accounting for the spatial filtering of the shearing interferometer, corresponding to a height of 80 nm when the filtering is taken into account.

Equations (26)

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

I=I1+I2+2I1I2cos[(ϕ·s⃗)+Γ].
[DxDyDz]=ϵ0[no2000no2000ne2][ExEyEz].
D[cosθsinθ]=ϵ0[no200ne2][ExEz].
D⃗·E⃗=|D||E|cosρ
ρ=cos1([cos2θno2+sin2θne2]cos2θno4+sin2θne4).
Γ=2πλ(neno)Lπλ(ne3r33no3r13)LdV,
2I1I2cos(ϕ·s⃗+Γ0+mcos(ωt)).
cos(ϕ·s⃗+Γ0+mcos(ωt))=cos(ϕ·s⃗+Γ0)cos(mcos(ωt))sin(ϕ·s⃗+Γ0)sin(mcos(ωt))
cos(ϕ·s⃗+Γ0+mcos(ωt))=cos(ϕ·s⃗+Γ0)[J0(m)+2n=1(1)nJ2n(m)cos(2nωt)]sin(ϕ·s⃗+Γ0)[2n=1(1)nJ2n1(m)cos((2n1)ωt)].
A1=2sin(ϕ·s⃗+Γ0)J1(m),
A2=2cos(ϕ·s⃗+Γ0)J2(m),
A3=2sin(ϕ·s⃗+Γ0)J3(m),
ϕ·s⃗=tan1(J2(m)A1J1(m)A2)Γ0.
A1A3=J1(m)J3(m).
tan1(zd)tan1(zr)=tan1(zdzr1+zdzr).
2h=ϕdϕr
uz(r)=uc8[E1(2r2/w2)+γ+ln(2r2/w2)],
uc=2αϵPπκ(1+ν)
umaxuz(0)uz(w),
umax=5.5×107wxs.
ϕ·s⃗tan1(2n)π/212n.
Δhmin=λ4π2n.
ϕ(r)·s⃗ϕ(r⃗+s⃗/2)ϕ(r⃗s⃗/2).
ϕ(r0+s)ϕ(r0)+ϕ(r0)s+12ϕ(r0)s2,
s2ϕϕ,
hmodel(0)hmeasured(0)hmodel(0)11ϕ(0)w0(ϕ(r+s/2)ϕ(rs/2))sdr,

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