Abstract

Sensors measuring the spatial phase of optical waves are widely used in optics. The optical differentiation wavefront sensor (ODWS) reconstructs the wavefront of an optical wave from wavefront slope measurements obtained by inducing linear field-transmission gradients in the far-field. Its dynamic range and sensitivity can be adjusted simply by changing the gradient slope. We numerically and experimentally demonstrate the possibility of implementing the spatially varying transmission gradient using distributions of small pixels that are either transparent or opaque. Binary pixelated filters are achromatic and can be fabricated with high accuracy at relatively low cost using commercial lithography techniques. We study the impact of the noise resulting from pixelation and binarization of the far-field filter for various test wavefronts and sensor parameters. The induced wavefront error is approximately inversely proportional to the pixel size. For an ODWS with dynamic range of 100 rad/mm over a 1-cm pupil, the error is smaller than λ/15 for a wide range of test wavefronts when using 2.5-μm pixels. We experimentally demonstrate the accuracy and consistency of a first-generation ODWS based on binary pixelated filters.

© 2016 Optical Society of America

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Opt. Express 23(4) 5052-5064 (2015)

References

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    [Crossref]

2015 (1)

2013 (1)

2010 (1)

J. W. Cha, J. Ballesta, and P. T. So, “Shack-Hartmann wavefront-sensor-based adaptive optics system for multiphoton microscopy,” J. Biomed. Opt. 15(4), 046022 (2010).
[Crossref] [PubMed]

2009 (2)

P. Martinez, C. Dorrer, M. Kasper, A. Boccaletti, and K. Dohlen, “Design, analysis, and testing of a microdot apodizer for the apodized pupil Lyot coronagraph,” Astron. Astrophys. 495(1), 363–370 (2009).
[Crossref]

A. Sagan, T. J. Antosiewicz, and T. Szoplik, “Three filters for visualization of phase objects with large variations of phase gradients,” Appl. Opt. 48(6), 1143–1152 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (3)

C. Dorrer and J. D. Zuegel, “Design and analysis of binary beam shapers using error diffusion,” J. Opt. Soc. Am. B 24(6), 1268–1275 (2007).
[Crossref]

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007).
[Crossref] [PubMed]

D. Schmidt and O. von der Lühe, “Optical wavefront differentiation: wavefront sensing for solar adaptive optics based on a LCD,” Proc. SPIE 6584, 658408 (2007).
[Crossref]

2006 (3)

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[Crossref]

H. I. Campbell and A. H. Greenaway, “Wavefront sensing: from historical roots to the state-of-the-art,” EAS Publications Series 22, 165–185 (2006).
[Crossref]

W. Zou and J. P. Rolland, “Quantifications of error propagation in slope-based wavefront estimations,” J. Opt. Soc. Am. A 23(10), 2629–2638 (2006).
[Crossref] [PubMed]

2005 (3)

J.-C. Chanteloup, “Multiple-wave lateral shearing interferometry for wave-front sensing,” Appl. Opt. 44(9), 1559–1571 (2005).
[Crossref] [PubMed]

J. E. Oti, V. F. Canales, and M. P. Cagigal, “Improvements on the optical differentiation wavefront sensor,” Mon. Not. R. Astron. Soc. 360(4), 1448–1454 (2005).
[Crossref]

F. Henault, “Wavefront sensor based on varying transmission filters: theory and expected performance,” J. Mod. Opt. 52(14), 1917–1931 (2005).
[Crossref]

2003 (3)

2001 (2)

J. Porter, A. Guirao, I. G. Cox, and D. R. Williams, “Monochromatic aberrations of the human eye in a large population,” J. Opt. Soc. Am. A 18(8), 1793–1803 (2001).
[Crossref] [PubMed]

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001).
[PubMed]

1998 (1)

1988 (2)

O. von der Lühe, “Wavefront error measurement technique using extended, incoherent light sources,” Opt. Eng. 27, 1078–1087 (1988).

F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27(7), 1223–1225 (1988).
[Crossref] [PubMed]

1984 (1)

1980 (1)

1972 (1)

Antosiewicz, T. J.

Ballesta, J.

J. W. Cha, J. Ballesta, and P. T. So, “Shack-Hartmann wavefront-sensor-based adaptive optics system for multiphoton microscopy,” J. Biomed. Opt. 15(4), 046022 (2010).
[Crossref] [PubMed]

Beau, V.

Boccaletti, A.

P. Martinez, C. Dorrer, M. Kasper, A. Boccaletti, and K. Dohlen, “Design, analysis, and testing of a microdot apodizer for the apodized pupil Lyot coronagraph,” Astron. Astrophys. 495(1), 363–370 (2009).
[Crossref]

Bortz, J. C.

Cagigal, M.

Cagigal, M. P.

M. P. Cagigal and P. J. Valle, “x-y curvature wavefront sensor,” Opt. Lett. 40(8), 1655–1658 (2015).
[Crossref] [PubMed]

J. E. Oti, V. F. Canales, and M. P. Cagigal, “Improvements on the optical differentiation wavefront sensor,” Mon. Not. R. Astron. Soc. 360(4), 1448–1454 (2005).
[Crossref]

Campbell, H. I.

H. I. Campbell and A. H. Greenaway, “Wavefront sensing: from historical roots to the state-of-the-art,” EAS Publications Series 22, 165–185 (2006).
[Crossref]

Canales, V.

Canales, V. F.

J. E. Oti, V. F. Canales, and M. P. Cagigal, “Improvements on the optical differentiation wavefront sensor,” Mon. Not. R. Astron. Soc. 360(4), 1448–1454 (2005).
[Crossref]

Cha, J. W.

J. W. Cha, J. Ballesta, and P. T. So, “Shack-Hartmann wavefront-sensor-based adaptive optics system for multiphoton microscopy,” J. Biomed. Opt. 15(4), 046022 (2010).
[Crossref] [PubMed]

Chanteloup, J.-C.

Climent, V.

Cox, I. G.

Daurios, J.

DeVries, G.

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[Crossref]

Dohlen, K.

P. Martinez, C. Dorrer, M. Kasper, A. Boccaletti, and K. Dohlen, “Design, analysis, and testing of a microdot apodizer for the apodized pupil Lyot coronagraph,” Astron. Astrophys. 495(1), 363–370 (2009).
[Crossref]

Dorrer, C.

C. Dorrer, “Analysis of the chromaticity of near-field binary beam shapers,” Appl. Opt. 52(14), 3368–3380 (2013).
[Crossref] [PubMed]

P. Martinez, C. Dorrer, M. Kasper, A. Boccaletti, and K. Dohlen, “Design, analysis, and testing of a microdot apodizer for the apodized pupil Lyot coronagraph,” Astron. Astrophys. 495(1), 363–370 (2009).
[Crossref]

C. Dorrer and J. D. Zuegel, “Design and analysis of binary beam shapers using error diffusion,” J. Opt. Soc. Am. B 24(6), 1268–1275 (2007).
[Crossref]

Ebbesen, T. W.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007).
[Crossref] [PubMed]

Fernández-Alonso, M.

Fleig, J.

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[Crossref]

Forbes, G.

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[Crossref]

Furuhashi, H.

H. Furuhashi, K. Matsuda, and C. P. Grover, “Visualization of phase objects by use of a differentiation filter,” Appl. Opt. 42(2), 218–226 (2003).
[Crossref] [PubMed]

H. Furuhashi, J. Valle Mayorga, Y. Uchida, and A. Kono, “Phase measurement of optical wavefront by an SLM differentiation filter,” in XIX IMEKO World Congress Fundamental and Applied Metrology (2009), pp. 1–5.

Genet, C.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007).
[Crossref] [PubMed]

Greenaway, A. H.

H. I. Campbell and A. H. Greenaway, “Wavefront sensing: from historical roots to the state-of-the-art,” EAS Publications Series 22, 165–185 (2006).
[Crossref]

Grover, C. P.

Guirao, A.

Henault, F.

F. Henault, “Wavefront sensor based on varying transmission filters: theory and expected performance,” J. Mod. Opt. 52(14), 1917–1931 (2005).
[Crossref]

Kasper, M.

P. Martinez, C. Dorrer, M. Kasper, A. Boccaletti, and K. Dohlen, “Design, analysis, and testing of a microdot apodizer for the apodized pupil Lyot coronagraph,” Astron. Astrophys. 495(1), 363–370 (2009).
[Crossref]

Kono, A.

H. Furuhashi, J. Valle Mayorga, Y. Uchida, and A. Kono, “Phase measurement of optical wavefront by an SLM differentiation filter,” in XIX IMEKO World Congress Fundamental and Applied Metrology (2009), pp. 1–5.

Lancis, J.

Lavergne, M.

Martinez, P.

P. Martinez, C. Dorrer, M. Kasper, A. Boccaletti, and K. Dohlen, “Design, analysis, and testing of a microdot apodizer for the apodized pupil Lyot coronagraph,” Astron. Astrophys. 495(1), 363–370 (2009).
[Crossref]

Matsuda, K.

Miladinovic, D.

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[Crossref]

Murphy, P.

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[Crossref]

Néauport, J.

O’Donohue, S.

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[Crossref]

Oti, J.

Oti, J. E.

J. E. Oti, V. F. Canales, and M. P. Cagigal, “Improvements on the optical differentiation wavefront sensor,” Mon. Not. R. Astron. Soc. 360(4), 1448–1454 (2005).
[Crossref]

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001).
[PubMed]

Porter, J.

Ribeyre, X.

Roddier, F.

Rolland, J. P.

Sagan, A.

Schmidt, D.

D. Schmidt and O. von der Lühe, “Optical wavefront differentiation: wavefront sensing for solar adaptive optics based on a LCD,” Proc. SPIE 6584, 658408 (2007).
[Crossref]

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001).
[PubMed]

So, P. T.

J. W. Cha, J. Ballesta, and P. T. So, “Shack-Hartmann wavefront-sensor-based adaptive optics system for multiphoton microscopy,” J. Biomed. Opt. 15(4), 046022 (2010).
[Crossref] [PubMed]

Southwell, W. H.

Sprague, R. A.

Szoplik, T.

Tajahuerce, E.

Thompson, B. J.

Thompson, K. P.

Uchida, Y.

H. Furuhashi, J. Valle Mayorga, Y. Uchida, and A. Kono, “Phase measurement of optical wavefront by an SLM differentiation filter,” in XIX IMEKO World Congress Fundamental and Applied Metrology (2009), pp. 1–5.

Valla, D.

Valle, P. J.

Valle Mayorga, J.

H. Furuhashi, J. Valle Mayorga, Y. Uchida, and A. Kono, “Phase measurement of optical wavefront by an SLM differentiation filter,” in XIX IMEKO World Congress Fundamental and Applied Metrology (2009), pp. 1–5.

Videau, L.

von der Lühe, O.

D. Schmidt and O. von der Lühe, “Optical wavefront differentiation: wavefront sensing for solar adaptive optics based on a LCD,” Proc. SPIE 6584, 658408 (2007).
[Crossref]

O. von der Lühe, “Wavefront error measurement technique using extended, incoherent light sources,” Opt. Eng. 27, 1078–1087 (1988).

Williams, D. R.

Zou, W.

Zuegel, J. D.

Appl. Opt. (8)

Astron. Astrophys. (1)

P. Martinez, C. Dorrer, M. Kasper, A. Boccaletti, and K. Dohlen, “Design, analysis, and testing of a microdot apodizer for the apodized pupil Lyot coronagraph,” Astron. Astrophys. 495(1), 363–370 (2009).
[Crossref]

EAS Publications Series (1)

H. I. Campbell and A. H. Greenaway, “Wavefront sensing: from historical roots to the state-of-the-art,” EAS Publications Series 22, 165–185 (2006).
[Crossref]

J. Biomed. Opt. (1)

J. W. Cha, J. Ballesta, and P. T. So, “Shack-Hartmann wavefront-sensor-based adaptive optics system for multiphoton microscopy,” J. Biomed. Opt. 15(4), 046022 (2010).
[Crossref] [PubMed]

J. Mod. Opt. (1)

F. Henault, “Wavefront sensor based on varying transmission filters: theory and expected performance,” J. Mod. Opt. 52(14), 1917–1931 (2005).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

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

J. Refract. Surg. (1)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001).
[PubMed]

Mon. Not. R. Astron. Soc. (1)

J. E. Oti, V. F. Canales, and M. P. Cagigal, “Improvements on the optical differentiation wavefront sensor,” Mon. Not. R. Astron. Soc. 360(4), 1448–1454 (2005).
[Crossref]

Nature (1)

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007).
[Crossref] [PubMed]

Opt. Eng. (1)

O. von der Lühe, “Wavefront error measurement technique using extended, incoherent light sources,” Opt. Eng. 27, 1078–1087 (1988).

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (2)

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[Crossref]

D. Schmidt and O. von der Lühe, “Optical wavefront differentiation: wavefront sensing for solar adaptive optics based on a LCD,” Proc. SPIE 6584, 658408 (2007).
[Crossref]

Other (7)

H. Furuhashi, J. Valle Mayorga, Y. Uchida, and A. Kono, “Phase measurement of optical wavefront by an SLM differentiation filter,” in XIX IMEKO World Congress Fundamental and Applied Metrology (2009), pp. 1–5.

E. Gendron, M. Brangier, G. Chenegros, F. Vidal, Z. Hubert, G. Rousset, and F. Pouplard, “A new sensor for laser tomography on ELTs,” in 1st AO4ELT Conference (2010), paper 05003.
[Crossref]

R. Ulichney, Digital Halftoning (MIT Press, 1987).

C.-P. Huang and Y.-Y. Zhu, “Plasmonics: manipulating light at the subwavelength scale,” in Active and Passive Electronic Components 2007 (2007), paper 30946.

J.-P. Zou and B. Wattellier, “Adaptive optics for high-peak-power lasers – an optical adaptive closed-loop used for high-energy short-pulse laser facilities: laser wave-front correction and focal-spot shaping,” in Topics in Adaptive Optics, Bob Tyson ed. (Intech, 2012).

D. Malacara, Optical Shop Testing, 3rd ed. (John Wiley and Sons, 2007).

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. Z. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Annals - Manufacturing Technol. 62, 823–846 (2013).

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Figures (13)

Fig. 1
Fig. 1 Principle of the optical differentiation wavefront sensor. A filter with linear field transmission gradient is located at the Fourier plane of a 4f line.
Fig. 2
Fig. 2 Fluence in the detection plane plotted on a logarithmic scale for (a) a continuous filter, (b), (c), and (d) binary pixelated filters with 10-μm, 5-μm, and 2.5-μm pixels, respectively. The input pupil (1-cm diameter) is reimaged to the detection plane and appears as a disk with 25% of the fluence of the input beam.
Fig. 3
Fig. 3 (a) Transmission of a continuous linear field-transmission filter (red line), and binary pixelated filters (red bars) for pixel sizes equal to (b) 10 μm, (c) 5 μm, and (d) 2.5 μm. The lineout of the far-field fluence for a flat wavefront over the 1-cm pupil is plotted with a continuous blue line.
Fig. 4
Fig. 4 Reconstructed wavefront for a flat input wavefront across a 1-cm pupil for BPF-ODWS’s with filter pixel size equal to (a) 10 μm, (b) 5 μm, and (c) 2.5 μm.
Fig. 5
Fig. 5 (a) rms and (b) peak errors vs the amplitude of a phase modulation φ0 for an ideal ODWS and BPF-ODWS’s with pixel size equal to 10 μm, 5 μm, and 2.5 μm. (c) shows the fluence distribution in the plane of the ODWS filter for φ0 = 8 rad, with the white lines at u = ± 5 mm indicating the edge of the filter when the gradient is in the horizontal direction.
Fig. 6
Fig. 6 (a) rms and (b) peak error for sinewaves oriented at α = 0, α = 30°, and α = 45° measured with an ODWS implemented with a continuous filter (continuous lines) and a pixelated filter with 2.5-μm pixels (dashed lines).
Fig. 7
Fig. 7 rms (1st line) and peak (2nd line) errors for Zernike polynomials of Noll order j from 4 to 15 and amplitude aj from 0 to 10π radians reconstructed by an ideal ODWS (1st column) and BPF-ODWS’s based on 10-μm, 5-μm, and 2.5-μm pixels (2nd, 3rd, and 4th column, respectively). White lines have been used to separate the groups of polynomials with different orders.
Fig. 8
Fig. 8 Ten random wavefronts, expressed in radians, generated by linear combination of Zernike polynomials of order 2, 3, and 4 with random amplitudes uniformly between −10 and + 10 radians.
Fig. 9
Fig. 9 (a) rms and (b) peak errors vs φ0 for an ideal ODWS and various BPF-ODWS’s for a 2-cm-diameter round pupil and f = 1 m. These plots can be compared to the plots shown in Figs. 5(a) and 5(b).
Fig. 10
Fig. 10 (a) rms and (b) peak error vs φ0 for an ideal ODWS and various BPF-ODWS’s for a 1-cm-diameter round pupil and f = 0.5 m. These plots can be compared to the plots shown in Figs. 5(a) and 5(b).
Fig. 11
Fig. 11 (a) Spatially resolved field transmission of the mask having four gradients with different orientations, two zones with constant transmission, and alignment features. (b) Lineout of the horizontal gradient integrated along the vertical direction.
Fig. 12
Fig. 12 Wavefront introduced by the deformable mirror reconstructed with (a) the two gradients Fx and Fy, and (b) the two gradients F+ and F-. (c) Difference between these two wavefronts. All wavefronts are expressed in waves.
Fig. 13
Fig. 13 Wavefront of a lens with focal length equal to 2 m (a) calculated, (b) measured with the optical differentiation wavefront sensor and (c) measured with a Shack-Hartmann wavefront sensor. All wavefronts are expressed in waves.

Tables (5)

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Table 1 Averages of the peak and rms errors over the range of φ0 from 0 to 2π

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Table 2 Statistics on the rms and peak errors, expressed in radians, for an ideal ODWS and various BPF-ODWS’s measuring Zernike polynomials of Noll index between 4 and 15 and amplitude between 0 and 25 radians.

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Table 3 Statistics on the rms and peak errors, expressed in radians, for an ideal ODWS and various BPF-ODWS’s measuring random wavefronts obtained as linear combinations of Zernike polynomials of Noll index between 4 and 15 and random amplitude between −10 and 10 radians.

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Table 4 Averages of the peak and rms errors over the range of φ0 from 0 to 2π for a 2-cm input pupil. This table can be compared to Tables 1 and 5.

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Table 5 Averages of the peak and rms errors over the range of φ0 from 0 to 2π for f = 0.5 m. This table can be compared to Tables 1 and 4.

Equations (9)

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t ( u , v ) = 1 2 + u W ,
ϕ x = π W λ f [ 2 F x F 0 1 ] .
ϕ y = π W λ f [ 2 F y F 0 1 ] .
t ( u , v ) = 1 2 + u W for W / 2 < u < W / 2
t ( u , v ) = 0 for u < W / 2
t ( u , v ) = 1 for u > W / 2
φ ( x , y ) = φ 0 cos [ 2 π ( x cos ( α ) + y sin ( α ) ) / p ) ] ,
j ( r , q ) = a j Z j ( r , q ) .
φ ( r , θ ) = j = 4 15 a j Z j ( r , θ ) .

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