Abstract

The vectorial shearing interferometer is based on the Mach–Zehnder configuration; it incorporates a displacement shearing system composed of a pair of wedge prisms that modify the optical path difference and the tilt of the sheared wave front with respect to that of the reference wave front. Variable shear and tilt can be implemented along any direction by choice of displacements Δx and Δy. The number of fringes and their orientation can be controlled with the vectorial shear. Knowledge of the prescribed displacements in the x and the y directions permits one to obtain a phase gradient in any direction.

© 2000 Optical Society of America

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  1. J. Flores, G. Paez, M. Strojnik, “Design of a diluted-aperture mirror using the practical cutoff frequency,” Appl. Opt. 38, 6010–6018 (1999).
    [CrossRef]
  2. M. S. Scholl, G. N. Lawrence, “Adaptive optics for in-orbit aberration correction—feasibility study,” Appl. Opt. 34, 7295–7301 (1995).
    [CrossRef] [PubMed]
  3. M. S. Scholl, “Recursive exact ray trace equations through tilted off-axis confocal prolate spheroids,” J. Mod. Opt. 43, 1583–1588 (1996).
    [CrossRef]
  4. M. S. Scholl, “Design parameters for a two-mirror telescope for stray-light sensitive infrared applications,” Infrared Phys. Technol. 37, 251–257 (1996).
    [CrossRef]
  5. G. Páez Padilla and M. Strojnik Scholl, “Recursive relations for ray-tracing through three-dimensional reflective confocal prolate spheroids,” Rev. Mex. Fis. 43, 875–886 (1997).
  6. G. Paez, M. S. Scholl, “Phase-shifted interferometry without phase unwrapping: reconstruction of a decentered wave front,” J. Opt. Soc. Am. A 16, 475–480 (1999).
    [CrossRef]
  7. M. Strojnik, G. Paez, “Testing the aspherical surfaces with the differential rotationally-shearing interferometer,” in Fabrication and Testing of Aspheres, A. Lindquist, M. Piscotty, J. Taylor, eds., Vol. 24 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 119–123.
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    [CrossRef] [PubMed]
  9. M. Strojnik Scholl, J. L. Flores, G. Paez, “Interferometric layout for extrasolar planet detection,” in Infrared Technology and Applications XXV, B. Andresen, M. Strojnik Scholl, eds., Proc. SPIE3698, 857–868 (1999).
  10. M. Strojnik Scholl and G. Paez, “Cancellation of star light generated by a nearby star–planet system upon detection with a rotationally-shearing interferometer,” Infrared Phys. Technol. 40, 357–365 (1999).
    [CrossRef]
  11. G. Paez, M. Strojnik, “Convergent, recursive phase reconstruction from noisy, modulated intensity patterns using synthetic interferograms,” Opt. Lett. 23, 406–408 (1998).
    [CrossRef]
  12. G. Paez and Marija Strojnik, “Fringe analysis and phase reconstruction from modulated intensity patterns,” Opt. Lett. 22, 1669–1971 (1997).
    [CrossRef]
  13. G. Paez, M. Strojnik, “Analysis and minimization of noise effects in phase-shifting interferometry,” in Interferometry 99, Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 295–305 (1999).
    [CrossRef]
  14. G. Paez, M. Strojnik, J. L. Flores, “Phase reconstruction from undersampled intensity pattern(s): underdetection,” in Infrared Spaceborne Remote Sensing VII, M. Strojnik, B. Andresen, eds., Proc. SPIE3759, 29–39 (1999).
    [CrossRef]
  15. B. Sen, D. Sen, “Interference with beams shaped in orthogonal axis,” Opt. Laser Technol. 5, 315–318 (1985).
    [CrossRef]
  16. M. P. Rimmer, J. C. Wyant, “Evaluation of large aberrations using a lateral shear interferometer having variable shear,” Appl. Opt. 14, 142–149 (1977).
    [CrossRef]
  17. G. Paez, M. Strojnik, “Phase reconstruction from undersampled intensity patterns,” J. Opt. Soc. Am. A 17, 46–52 (2000).
    [CrossRef]
  18. G. Paez, M. Strojnik, “Mathematical theory of differential rotational shearing interferometry: asymmetrical aberrations,” in Interferometry 99, Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 335–346 (1999).
    [CrossRef]
  19. G. García Torales, M. Strojnik Scholl, G. Páez, “Controlled wavefront displacement using a thin prism system,” in Infrared Spaceborne Remote Sensing VI, M. Strojnik, B. Andresen, eds., Proc. SPIE3437, 424–428 (1998).
    [CrossRef]
  20. G. Paez, M. S. Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 109–116 (1999).
    [CrossRef]
  21. G. Páez and M. Strojnik Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 261–265 (1999).
    [CrossRef]
  22. M. S. Scholl, G. Paez, “Image-plane incidence for a baffled infrared telescope,” Infrared Phys. Technol. 38, 87–92 (1997).
    [CrossRef]
  23. M. S. Scholl, G. Paez, “Using the y, y-bar diagram to control stray light noise in IR systems,” Infrared Phys. Technol. 38, 25–30 (1997).
    [CrossRef]

2000

1999

G. Paez, M. S. Scholl, “Phase-shifted interferometry without phase unwrapping: reconstruction of a decentered wave front,” J. Opt. Soc. Am. A 16, 475–480 (1999).
[CrossRef]

J. Flores, G. Paez, M. Strojnik, “Design of a diluted-aperture mirror using the practical cutoff frequency,” Appl. Opt. 38, 6010–6018 (1999).
[CrossRef]

M. Strojnik Scholl and G. Paez, “Cancellation of star light generated by a nearby star–planet system upon detection with a rotationally-shearing interferometer,” Infrared Phys. Technol. 40, 357–365 (1999).
[CrossRef]

G. Paez, M. S. Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 109–116 (1999).
[CrossRef]

G. Páez and M. Strojnik Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 261–265 (1999).
[CrossRef]

1998

1997

G. Paez and Marija Strojnik, “Fringe analysis and phase reconstruction from modulated intensity patterns,” Opt. Lett. 22, 1669–1971 (1997).
[CrossRef]

G. Páez Padilla and M. Strojnik Scholl, “Recursive relations for ray-tracing through three-dimensional reflective confocal prolate spheroids,” Rev. Mex. Fis. 43, 875–886 (1997).

M. S. Scholl, G. Paez, “Image-plane incidence for a baffled infrared telescope,” Infrared Phys. Technol. 38, 87–92 (1997).
[CrossRef]

M. S. Scholl, G. Paez, “Using the y, y-bar diagram to control stray light noise in IR systems,” Infrared Phys. Technol. 38, 25–30 (1997).
[CrossRef]

1996

M. S. Scholl, “Recursive exact ray trace equations through tilted off-axis confocal prolate spheroids,” J. Mod. Opt. 43, 1583–1588 (1996).
[CrossRef]

M. S. Scholl, “Design parameters for a two-mirror telescope for stray-light sensitive infrared applications,” Infrared Phys. Technol. 37, 251–257 (1996).
[CrossRef]

1995

1985

B. Sen, D. Sen, “Interference with beams shaped in orthogonal axis,” Opt. Laser Technol. 5, 315–318 (1985).
[CrossRef]

1977

1970

Flores, J.

Flores, J. L.

G. Paez, M. Strojnik, J. L. Flores, “Phase reconstruction from undersampled intensity pattern(s): underdetection,” in Infrared Spaceborne Remote Sensing VII, M. Strojnik, B. Andresen, eds., Proc. SPIE3759, 29–39 (1999).
[CrossRef]

M. Strojnik Scholl, J. L. Flores, G. Paez, “Interferometric layout for extrasolar planet detection,” in Infrared Technology and Applications XXV, B. Andresen, M. Strojnik Scholl, eds., Proc. SPIE3698, 857–868 (1999).

García Torales, G.

G. García Torales, M. Strojnik Scholl, G. Páez, “Controlled wavefront displacement using a thin prism system,” in Infrared Spaceborne Remote Sensing VI, M. Strojnik, B. Andresen, eds., Proc. SPIE3437, 424–428 (1998).
[CrossRef]

Lawrence, G. N.

Murty, M. V. R. K.

Paez, G.

G. Paez, M. Strojnik, “Phase reconstruction from undersampled intensity patterns,” J. Opt. Soc. Am. A 17, 46–52 (2000).
[CrossRef]

G. Paez, M. S. Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 109–116 (1999).
[CrossRef]

G. Paez, M. S. Scholl, “Phase-shifted interferometry without phase unwrapping: reconstruction of a decentered wave front,” J. Opt. Soc. Am. A 16, 475–480 (1999).
[CrossRef]

J. Flores, G. Paez, M. Strojnik, “Design of a diluted-aperture mirror using the practical cutoff frequency,” Appl. Opt. 38, 6010–6018 (1999).
[CrossRef]

G. Paez, M. Strojnik, “Convergent, recursive phase reconstruction from noisy, modulated intensity patterns using synthetic interferograms,” Opt. Lett. 23, 406–408 (1998).
[CrossRef]

M. S. Scholl, G. Paez, “Image-plane incidence for a baffled infrared telescope,” Infrared Phys. Technol. 38, 87–92 (1997).
[CrossRef]

M. S. Scholl, G. Paez, “Using the y, y-bar diagram to control stray light noise in IR systems,” Infrared Phys. Technol. 38, 25–30 (1997).
[CrossRef]

M. Strojnik Scholl, J. L. Flores, G. Paez, “Interferometric layout for extrasolar planet detection,” in Infrared Technology and Applications XXV, B. Andresen, M. Strojnik Scholl, eds., Proc. SPIE3698, 857–868 (1999).

G. Paez, M. Strojnik, “Analysis and minimization of noise effects in phase-shifting interferometry,” in Interferometry 99, Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 295–305 (1999).
[CrossRef]

M. Strojnik, G. Paez, “Testing the aspherical surfaces with the differential rotationally-shearing interferometer,” in Fabrication and Testing of Aspheres, A. Lindquist, M. Piscotty, J. Taylor, eds., Vol. 24 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 119–123.

G. Paez, M. Strojnik, “Mathematical theory of differential rotational shearing interferometry: asymmetrical aberrations,” in Interferometry 99, Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 335–346 (1999).
[CrossRef]

G. Paez, M. Strojnik, J. L. Flores, “Phase reconstruction from undersampled intensity pattern(s): underdetection,” in Infrared Spaceborne Remote Sensing VII, M. Strojnik, B. Andresen, eds., Proc. SPIE3759, 29–39 (1999).
[CrossRef]

Páez, G.

G. García Torales, M. Strojnik Scholl, G. Páez, “Controlled wavefront displacement using a thin prism system,” in Infrared Spaceborne Remote Sensing VI, M. Strojnik, B. Andresen, eds., Proc. SPIE3437, 424–428 (1998).
[CrossRef]

Páez and M. Strojnik Scholl, G.

G. Páez and M. Strojnik Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 261–265 (1999).
[CrossRef]

Paez and Marija Strojnik, G.

Páez Padilla and M. Strojnik Scholl, G.

G. Páez Padilla and M. Strojnik Scholl, “Recursive relations for ray-tracing through three-dimensional reflective confocal prolate spheroids,” Rev. Mex. Fis. 43, 875–886 (1997).

Rimmer, M. P.

Scholl, M. S.

G. Paez, M. S. Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 109–116 (1999).
[CrossRef]

G. Paez, M. S. Scholl, “Phase-shifted interferometry without phase unwrapping: reconstruction of a decentered wave front,” J. Opt. Soc. Am. A 16, 475–480 (1999).
[CrossRef]

M. S. Scholl, G. Paez, “Using the y, y-bar diagram to control stray light noise in IR systems,” Infrared Phys. Technol. 38, 25–30 (1997).
[CrossRef]

M. S. Scholl, G. Paez, “Image-plane incidence for a baffled infrared telescope,” Infrared Phys. Technol. 38, 87–92 (1997).
[CrossRef]

M. S. Scholl, “Recursive exact ray trace equations through tilted off-axis confocal prolate spheroids,” J. Mod. Opt. 43, 1583–1588 (1996).
[CrossRef]

M. S. Scholl, “Design parameters for a two-mirror telescope for stray-light sensitive infrared applications,” Infrared Phys. Technol. 37, 251–257 (1996).
[CrossRef]

M. S. Scholl, G. N. Lawrence, “Adaptive optics for in-orbit aberration correction—feasibility study,” Appl. Opt. 34, 7295–7301 (1995).
[CrossRef] [PubMed]

Scholl, M. Strojnik

M. Strojnik Scholl, J. L. Flores, G. Paez, “Interferometric layout for extrasolar planet detection,” in Infrared Technology and Applications XXV, B. Andresen, M. Strojnik Scholl, eds., Proc. SPIE3698, 857–868 (1999).

Sen, B.

B. Sen, D. Sen, “Interference with beams shaped in orthogonal axis,” Opt. Laser Technol. 5, 315–318 (1985).
[CrossRef]

Sen, D.

B. Sen, D. Sen, “Interference with beams shaped in orthogonal axis,” Opt. Laser Technol. 5, 315–318 (1985).
[CrossRef]

Strojnik, M.

G. Paez, M. Strojnik, “Phase reconstruction from undersampled intensity patterns,” J. Opt. Soc. Am. A 17, 46–52 (2000).
[CrossRef]

J. Flores, G. Paez, M. Strojnik, “Design of a diluted-aperture mirror using the practical cutoff frequency,” Appl. Opt. 38, 6010–6018 (1999).
[CrossRef]

G. Paez, M. Strojnik, “Convergent, recursive phase reconstruction from noisy, modulated intensity patterns using synthetic interferograms,” Opt. Lett. 23, 406–408 (1998).
[CrossRef]

G. Paez, M. Strojnik, “Mathematical theory of differential rotational shearing interferometry: asymmetrical aberrations,” in Interferometry 99, Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 335–346 (1999).
[CrossRef]

G. Paez, M. Strojnik, J. L. Flores, “Phase reconstruction from undersampled intensity pattern(s): underdetection,” in Infrared Spaceborne Remote Sensing VII, M. Strojnik, B. Andresen, eds., Proc. SPIE3759, 29–39 (1999).
[CrossRef]

M. Strojnik, G. Paez, “Testing the aspherical surfaces with the differential rotationally-shearing interferometer,” in Fabrication and Testing of Aspheres, A. Lindquist, M. Piscotty, J. Taylor, eds., Vol. 24 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 119–123.

G. Paez, M. Strojnik, “Analysis and minimization of noise effects in phase-shifting interferometry,” in Interferometry 99, Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 295–305 (1999).
[CrossRef]

Strojnik Scholl, M.

G. García Torales, M. Strojnik Scholl, G. Páez, “Controlled wavefront displacement using a thin prism system,” in Infrared Spaceborne Remote Sensing VI, M. Strojnik, B. Andresen, eds., Proc. SPIE3437, 424–428 (1998).
[CrossRef]

Strojnik Scholl and G. Paez, M.

M. Strojnik Scholl and G. Paez, “Cancellation of star light generated by a nearby star–planet system upon detection with a rotationally-shearing interferometer,” Infrared Phys. Technol. 40, 357–365 (1999).
[CrossRef]

Wyant, J. C.

Appl. Opt.

Infrared Phys. Technol.

G. Paez, M. S. Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 109–116 (1999).
[CrossRef]

G. Páez and M. Strojnik Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 261–265 (1999).
[CrossRef]

M. S. Scholl, G. Paez, “Image-plane incidence for a baffled infrared telescope,” Infrared Phys. Technol. 38, 87–92 (1997).
[CrossRef]

M. S. Scholl, G. Paez, “Using the y, y-bar diagram to control stray light noise in IR systems,” Infrared Phys. Technol. 38, 25–30 (1997).
[CrossRef]

M. S. Scholl, “Design parameters for a two-mirror telescope for stray-light sensitive infrared applications,” Infrared Phys. Technol. 37, 251–257 (1996).
[CrossRef]

M. Strojnik Scholl and G. Paez, “Cancellation of star light generated by a nearby star–planet system upon detection with a rotationally-shearing interferometer,” Infrared Phys. Technol. 40, 357–365 (1999).
[CrossRef]

J. Mod. Opt.

M. S. Scholl, “Recursive exact ray trace equations through tilted off-axis confocal prolate spheroids,” J. Mod. Opt. 43, 1583–1588 (1996).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Laser Technol.

B. Sen, D. Sen, “Interference with beams shaped in orthogonal axis,” Opt. Laser Technol. 5, 315–318 (1985).
[CrossRef]

Opt. Lett.

Rev. Mex. Fis.

G. Páez Padilla and M. Strojnik Scholl, “Recursive relations for ray-tracing through three-dimensional reflective confocal prolate spheroids,” Rev. Mex. Fis. 43, 875–886 (1997).

Other

M. Strojnik, G. Paez, “Testing the aspherical surfaces with the differential rotationally-shearing interferometer,” in Fabrication and Testing of Aspheres, A. Lindquist, M. Piscotty, J. Taylor, eds., Vol. 24 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 119–123.

G. Paez, M. Strojnik, “Analysis and minimization of noise effects in phase-shifting interferometry,” in Interferometry 99, Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 295–305 (1999).
[CrossRef]

G. Paez, M. Strojnik, J. L. Flores, “Phase reconstruction from undersampled intensity pattern(s): underdetection,” in Infrared Spaceborne Remote Sensing VII, M. Strojnik, B. Andresen, eds., Proc. SPIE3759, 29–39 (1999).
[CrossRef]

M. Strojnik Scholl, J. L. Flores, G. Paez, “Interferometric layout for extrasolar planet detection,” in Infrared Technology and Applications XXV, B. Andresen, M. Strojnik Scholl, eds., Proc. SPIE3698, 857–868 (1999).

G. Paez, M. Strojnik, “Mathematical theory of differential rotational shearing interferometry: asymmetrical aberrations,” in Interferometry 99, Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 335–346 (1999).
[CrossRef]

G. García Torales, M. Strojnik Scholl, G. Páez, “Controlled wavefront displacement using a thin prism system,” in Infrared Spaceborne Remote Sensing VI, M. Strojnik, B. Andresen, eds., Proc. SPIE3437, 424–428 (1998).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Original and modified wave fronts in the vectorial shearing interferometer, with the displacement or shear occurring along an arbitrary direction.

Fig. 2
Fig. 2

Experimental setup with the vectorial shearing interferometer for testing a positive lens in transmission.

Fig. 3
Fig. 3

Simulated intensity patterns of the spherical aberration (2.5 waves) and defocus (±4 waves) for the Mach–Zehnder interferometer (top row); for the linear shearing interferometer (next row); and twice in a vectorial shearing interferometer, first with a larger radius vector pointing to the first quadrant (first row from bottom), and second with a smaller radius vector pointing to the fourth quadrant (bottom row).

Fig. 4
Fig. 4

Simulated intensity patterns of the astigmatism (3 waves) and defocus (±4 waves) for the Mach–Zehnder interferometer; the linear shearing interferometer; and twice in a vectorial shearing interferometer, first with a larger radius vector pointing to the first quadrant and second with a smaller radius vector pointing to the fourth quadrant (in the same order as for Fig. 3).

Fig. 5
Fig. 5

Simulated intensity patterns of the coma (7 waves) and defocus (±4 waves) for the Mach–Zehnder interferometer; the linear shearing interferometer; and twice in a vectorial shearing interferometer, first with a larger radius vector pointing to the first quadrant and second with a smaller radius vector pointing to the fourth quadrant (in the same order as for Fig. 3).

Fig. 6
Fig. 6

Simulated intensity patterns of the coma (7 waves), a small amount of spherical aberration (0.5 waves), and defocus (±4 waves) for the Mach–Zehnder interferometer; the linear shearing interferometer; and twice in a vectorial shearing interferometer, first with a larger radius vector pointing to the first quadrant and second with a smaller radius vector pointing to the fourth quadrant (in the same order as for Fig. 3).

Fig. 7
Fig. 7

Experimental setup with the vectorial shearing interferometer for testing an off-axis parabolic mirror in reflection.

Fig. 8
Fig. 8

Experimentally obtained interferometric patterns of the lens for (a) a small and (b) a large shearing vector obtained in the vectorial shearing interferometer used in transmission.

Fig. 9
Fig. 9

Experimentally obtained intensity patterns of the parabolic mirror for (a) a small and (b) a large shearing vector obtained in the vectorial shearing interferometer used in reflection.

Equations (12)

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

OPD=WSHEAREDx, y-WORIGINALx, y
OPD=WSHEAREDρ, θ-WORIGINALρ, θ.
Wρ, ϕ=Fpiston term+Eρsin ϕ+cos ϕtiltsx, y+Dρ2defocus+Cρ21+2 cos2 ϕastigmatism+Bρ3 cos ϕcoma+Aρ4spherical.
α=δOPDδy=TAf,
OPDLAT=Wx+Δx, y-Wx, y
OPDLAT=Wx, y+Δy-Wx, y.
δ=2δi cosβ/2,
b cos βy1+y2+Ls2π=nλ.
δWx, yδx S+δWx, yδy T=nλ.
OPDVEC=Wx+Δx, y+Δy-Wx, y,
OPDVEC=Wρ+Δρ, θ+Δθ-Wρ, θ.
Δρ=Δx2+Δy21/2,  θ=arctanΔy/Δx.

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