Abstract

We present a hybrid diffractive-refractive optical lens doublet consisting of a varifocal Moiré Fresnel lens and a polymer lens of tunable refractive power. The wide range of focal tunability of each lens and the opposite dispersive characteristics of the diffractive and the refractive element are exploited to obtain an optical system where both the Abbe number and the refractive power can be changed separately. We investigate the performance of the proposed hybrid lens at zero overall refractive power by tuning the Abbe number of a complementary standard lens while maintaining a constant overall focal length for the central wavelength. As an application example, the hybrid lens is used to tune to an optimal operating regime for quantitative phase microscopy based on a two-color transport of intensity (TIE) approach which utilizes chromatic aberrations rather than intensity recordings at several planes to reconstruct the optical path length of a phase object.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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2013 (6)

2012 (1)

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

2011 (1)

2010 (3)

2009 (2)

G. Della Valle, R. Osellame, P. Laporta, “Micromachining of photonic devices by femtosecond laser pulses,” J. Opt. A Pure Appl. Opt. 11, 013001 (2009).
[CrossRef]

G. Zhou, H. M. Leung, H. Yu, A. S. Kumar, F. S. Chau, “Liquid tunable diffractive refractive hybrid lens,” Opt. Lett. 34, 2793–2795 (2009).
[CrossRef] [PubMed]

2008 (1)

2004 (1)

D. Paganin, A. Barty, P. J. McMahon, K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effect of noise,” J. Microsc. 214, 51–61 (2004).
[CrossRef] [PubMed]

2002 (1)

H. J. Caulfield, “The Alvarez-Lohmann lens as a do-nothing machine,” Opt. Laser Technol. 34, 1–5 (2002).
[CrossRef]

2001 (1)

K. A. Nugent, D. P. Paganin, T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
[CrossRef]

2000 (1)

1998 (1)

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

1997 (1)

T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–9346 (1997).
[CrossRef]

1993 (1)

1988 (1)

1987 (1)

1983 (1)

M. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 23, 1434–1441 (1983).
[CrossRef]

1982 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Abrahamsson, S.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Agard, D. A.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Andrés, P.

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

Applegate, B. E.

C. Olsovsky, R. Shelton, O. Carrasco-Zevallos, B. E. Applegate, K. C. Maitland, “Chromatic confocal microscopy for multi-depth imaging of epithelial tissue,” Opt. Express 4, 732–740 (2013).
[CrossRef]

Avicola, K.

Barbastathis, G.

L. Waller, S. S. Kou, C. J. R. Sheppard, G. Barbastathis, “Phase from chromatic aberrations,” Opt. Express 22, 22817–22825 (2010).
[CrossRef]

Bargmann, C. I.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Barton, I. M.

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effect of noise,” J. Microsc. 214, 51–61 (2004).
[CrossRef] [PubMed]

Bernet, S.

Booth, M. J.

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge University, 1999).
[CrossRef]

Carrasco-Zevallos, O.

C. Olsovsky, R. Shelton, O. Carrasco-Zevallos, B. E. Applegate, K. C. Maitland, “Chromatic confocal microscopy for multi-depth imaging of epithelial tissue,” Opt. Express 4, 732–740 (2013).
[CrossRef]

Caulfield, H. J.

H. J. Caulfield, “The Alvarez-Lohmann lens as a do-nothing machine,” Opt. Laser Technol. 34, 1–5 (2002).
[CrossRef]

Chau, F. S.

Chen, J.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Climent, V.

O. Mendoza-Yero, G. Mínguez-Vega, J. Lancis, V. Climent, “Diffractive pulse shaper for arbitrary waveform generation,” Opt. Lett. 35, 535–537 (2010).
[CrossRef] [PubMed]

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

Cu-Nguyen, P.

Dahan, M.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Darzacq, C. D.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Darzacq, X.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Davidson, N.

Della Valle, G.

G. Della Valle, R. Osellame, P. Laporta, “Micromachining of photonic devices by femtosecond laser pulses,” J. Opt. A Pure Appl. Opt. 11, 013001 (2009).
[CrossRef]

Dixit, S. N.

Dodge, M. R.

Fienup, J. R.

Friesem, A.

Fujimoto, J. G.

George, N.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Grewe, A.

Gross, H.

H. Gross, Handbook of Optical Systems (John Wiley, 2005).
[CrossRef]

Gureyev, T. E.

K. A. Nugent, D. P. Paganin, T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
[CrossRef]

T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–9346 (1997).
[CrossRef]

Gustafsson, M. G. L.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Hajj, B.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Harm, W.

Hartl, I.

Hasman, E.

Hillenbrand, M.

Ippen, E. P.

Jesacher, A.

Katsov, A. Y.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Kou, S. S.

L. Waller, S. S. Kou, C. J. R. Sheppard, G. Barbastathis, “Phase from chromatic aberrations,” Opt. Express 22, 22817–22825 (2010).
[CrossRef]

Kowalevicz, A. M.

Kumar, A. S.

Lancis, J.

O. Mendoza-Yero, G. Mínguez-Vega, J. Lancis, V. Climent, “Diffractive pulse shaper for arbitrary waveform generation,” Opt. Lett. 35, 535–537 (2010).
[CrossRef] [PubMed]

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

Laporta, P.

G. Della Valle, R. Osellame, P. Laporta, “Micromachining of photonic devices by femtosecond laser pulses,” J. Opt. A Pure Appl. Opt. 11, 013001 (2009).
[CrossRef]

Leung, H. M.

Liebetraut, P.

Mader, D.

Maitland, K. C.

C. Olsovsky, R. Shelton, O. Carrasco-Zevallos, B. E. Applegate, K. C. Maitland, “Chromatic confocal microscopy for multi-depth imaging of epithelial tissue,” Opt. Express 4, 732–740 (2013).
[CrossRef]

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effect of noise,” J. Microsc. 214, 51–61 (2004).
[CrossRef] [PubMed]

Mendoza-Yero, O.

Mínguez-Vega, G.

Minoshima, K.

Mizuguchi, G.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Mueller, F.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effect of noise,” J. Microsc. 214, 51–61 (2004).
[CrossRef] [PubMed]

K. A. Nugent, D. P. Paganin, T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
[CrossRef]

T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–9346 (1997).
[CrossRef]

Olsovsky, C.

C. Olsovsky, R. Shelton, O. Carrasco-Zevallos, B. E. Applegate, K. C. Maitland, “Chromatic confocal microscopy for multi-depth imaging of epithelial tissue,” Opt. Express 4, 732–740 (2013).
[CrossRef]

Osellame, R.

G. Della Valle, R. Osellame, P. Laporta, “Micromachining of photonic devices by femtosecond laser pulses,” J. Opt. A Pure Appl. Opt. 11, 013001 (2009).
[CrossRef]

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effect of noise,” J. Microsc. 214, 51–61 (2004).
[CrossRef] [PubMed]

Paganin, D. P.

K. A. Nugent, D. P. Paganin, T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
[CrossRef]

Peyghambarian, N.

Peyman, G.

Ritsch-Marte, M.

Roider, C.

Salter, P. S.

Savidis, N.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schwiegerling, J.

Seifert, A.

Shelton, R.

C. Olsovsky, R. Shelton, O. Carrasco-Zevallos, B. E. Applegate, K. C. Maitland, “Chromatic confocal microscopy for multi-depth imaging of epithelial tissue,” Opt. Express 4, 732–740 (2013).
[CrossRef]

Sheppard, C. J. R.

L. Waller, S. S. Kou, C. J. R. Sheppard, G. Barbastathis, “Phase from chromatic aberrations,” Opt. Express 22, 22817–22825 (2010).
[CrossRef]

Sinzinger, S.

Soule, P.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Stallinga, S.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Stone, T.

Summers, L. J.

Tajahuerce, E.

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

Teague, M.

M. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 23, 1434–1441 (1983).
[CrossRef]

Thompson, C. A.

Valley, P.

Waibel, P.

Waller, L.

L. Waller, S. S. Kou, C. J. R. Sheppard, G. Barbastathis, “Phase from chromatic aberrations,” Opt. Express 22, 22817–22825 (2010).
[CrossRef]

Wilhelmsen, J.

Wisniewski, J.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge University, 1999).
[CrossRef]

Wu, C.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Yu, H.

Zappe, H.

Zhou, G.

Appl. Opt. (4)

J. Microsc. (1)

D. Paganin, A. Barty, P. J. McMahon, K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effect of noise,” J. Microsc. 214, 51–61 (2004).
[CrossRef] [PubMed]

J. Opt. A Pure Appl. Opt. (1)

G. Della Valle, R. Osellame, P. Laporta, “Micromachining of photonic devices by femtosecond laser pulses,” J. Opt. A Pure Appl. Opt. 11, 013001 (2009).
[CrossRef]

J. Opt. Soc. Am. A (2)

Nat. Methods (1)

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[CrossRef] [PubMed]

Opt. Commun. (2)

T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–9346 (1997).
[CrossRef]

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

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H. J. Caulfield, “The Alvarez-Lohmann lens as a do-nothing machine,” Opt. Laser Technol. 34, 1–5 (2002).
[CrossRef]

Opt. Lett. (4)

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Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Phys. Today (1)

K. A. Nugent, D. P. Paganin, T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

(a) Simulation of the refractive power of the Moiré lens for the design wavelength (black) and the wavelengths used for the definition of the Abbe number (blue, green and red), see inset. (b) Simulation of the effective Abbe number of the DRD over the entire tuning range of both elements. Note the logarithmic color scale.

Fig. 2
Fig. 2

Tuning the dispersive characteristics of a spherical lens with a tunable diffractive-refractive doublet (DRD). (a) Schematic of the DRD, consisting itself of a varifocal Moiré diffractive lens and a varifocal polymere membrane refractive lens, used in front of the spherical lens (SL). Illustrated is a negative Abbe number of the spherical lens with fdiff > 0, fref < 0 and |fdiff| ≅ |fref|. (b) Photographies of axial focus profiles captured on a screen as illustrated in (a). The corresponding effective Abbe numbers Veff were calculated with Eq. (7) and range from negative (top) to postive (bottom) values.

Fig. 3
Fig. 3

Experimental results for tuning the Abbe number of a standard spherical lens with a focal length of fL = 0.15 m with the DRD. (a) Measured shift in focus for a red wavelength (656.3 nm) and a blue wavelength (488.1 nm) with respect to the refractive power at the design wavelength (632.8 nm) of the Moiré lens while maintaining zero effective refractive power of the DRD for the central wavelength (587.6 nm). The red and blue curves correspond to Eq. (8) and the green curve indicates the constant focal length for the central wavelength. (b) Experimentally determined effective Abbe number of the compound system compared to its theoretical values (within the thin lens approximation) given by Eq. (7) (black curve).

Fig. 4
Fig. 4

Experimental set-up of an inverted microscope for the determination of refractive index profiles of direct laser written waveguides in fused silica. The DRD to adjust system chromatic aberrations to an optimal amount of defocus between the green and the red color channel is placed in a plane conjugated to the back focal plane of the objective lens, which is relayed by lenses L1 and L2. The intermediate image is imaged onto a color camera by lens L3.

Fig. 5
Fig. 5

Phase microscopy of DLW-waveguides in fused silica. (a) Comparison of peak optical path length (OPL) acquired for different amounts of defocus with single-shot TIE from system chromatic aberrations (square), single-shot TIE with tuned chromatic aberrations (circles) and two-shot TIE (asterisks). Error bars represent standard deviation for 10 measurements. (b) Transverse OPL profile with reduced standard deviation at the optimal amount of defocus δz = 2.3 μm.

Equations (13)

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ϕ ( λ ) = λ λ 0 ϕ 0 ,
ϕ eff V eff = ϕ R V R + ϕ D V D ,
ϕ eff ( λ ) = ϕ L + ϕ DRD ( λ ) ,
ϕ DRD ( λ ) = ϕ D ( λ d ) λ λ d λ d
d ϕ eff ( λ ) d λ = 1 λ d ϕ D ( λ ) .
d ϕ L d λ = ϕ L V L ( λ F λ C ) ,
V eff = 3.452 ϕ L ϕ D .
ϕ ( λ ) = ϕ L + ϕ R ( λ d ) + ϕ D ( λ ) ,
Δ f F / C = n F / C n d n F / C 1 f L .
V = n d 1 n F n C .
2 π λ I ( x , y ) z = [ I ( x , y ) ϕ ( x , y ) ] ,
d I ( x ) d z = λ 2 π d d x [ I ( x ) d d x ϕ ( x ) ] ,
ϕ ( x ) = 2 π λ ( 1 I ( x ) d I ( x ) d z d x ) d x .

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