Abstract

We describe the use of spatially incoherent illumination to make quantitative phase imaging of a semi-transparent sample, even out of the paraxial approximation. The image volume electromagnetic field is collected by scanning the image planes with a quadriwave lateral shearing interferometer, while the sample is spatially incoherently illuminated. In comparison to coherent quantitative phase measurements, incoherent illumination enriches the 3D collected spatial frequencies leading to 3D resolution increase (up to a factor 2). The image contrast loss introduced by the incoherent illumination is simulated and used to compensate the measurements. This restores the quantitative value of phase and intensity. Experimental contrast loss compensation and 3D resolution increase is presented using polystyrene and TiO2 micro-beads. Our approach will be useful to make diffraction tomography reconstruction with a simplified setup.

© 2014 Optical Society of America

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2013

2012

2011

2010

2009

2008

2007

2006

2004

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[CrossRef] [PubMed]

G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 29, 2503–2505 (2004).
[CrossRef] [PubMed]

2002

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205, 165–176 (2002).
[CrossRef] [PubMed]

2000

1999

1998

1985

1952

R. Barer, “Interference microscopy and mass determination,” Nature 169, 366–367 (1952).
[CrossRef] [PubMed]

Agranov, G.

Ahn, S.-G.

Aknoun, S.

S. Aknoun, P. Bon, J. Savatier, B. Wattellier, S. Monneret, “Quantitative birefringence imaging of biological samples using quadri-wave interferometry,” Proc. SPIE 8587, 85871D (2013).
[CrossRef]

Arfire, C.

Arnison, M. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[CrossRef] [PubMed]

Babacan, S. D.

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics, advance online publication (2014).
[CrossRef]

Badizadegan, K.

Barbastathis, G.

Barer, R.

R. Barer, “Interference microscopy and mass determination,” Nature 169, 366–367 (1952).
[CrossRef] [PubMed]

Barroca, T.

P. Bon, T. Barroca, S. Lévêque-Fort, E. Fort, “Label-free evanescent microscopy for membrane nano-tomography in living cells,” J. Biophotonics (advance online publication, 2013).
[CrossRef]

Barty, A.

Bergoënd, I.

Bernet, S.

Bhaduri, B.

Bon, P.

Bonod, N.

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge University, 1999), chap. 9, pp. 547–553.

Boss, D.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[CrossRef]

Y. Cotte, F. M. Toy, C. Arfire, S. S. Kou, D. Boss, I. Bergoënd, C. Depeursinge, “Realistic 3d coherent transfer function inverse filtering of complex fields,” Biomed. Opt. Express 2, 2216–2230 (2011).
[CrossRef] [PubMed]

Carney, P. S.

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics, advance online publication (2014).
[CrossRef]

Charrière, F.

Chen, X.

Choi, W.

Choi, W. J.

Choi, Y.

Chowdhury, S.

Chu, K. K.

Cogswell, C. J.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[CrossRef] [PubMed]

Colomb, T.

Cotte, Y.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[CrossRef]

Y. Cotte, F. M. Toy, C. Arfire, S. S. Kou, D. Boss, I. Bergoënd, C. Depeursinge, “Realistic 3d coherent transfer function inverse filtering of complex fields,” Biomed. Opt. Express 2, 2216–2230 (2011).
[CrossRef] [PubMed]

Cuche, E.

Dasari, R. R.

Dayton, A.

Debailleul, M.

Deflores, L. P.

Depeursinge, C.

Ding, H.

Duncan, D. D.

Fang-Yen, C.

Feld, M. S.

Fiolka, R.

Fischer, D. G.

Ford, T. N.

Fort, E.

P. Bon, T. Barroca, S. Lévêque-Fort, E. Fort, “Label-free evanescent microscopy for membrane nano-tomography in living cells,” J. Biophotonics (advance online publication, 2013).
[CrossRef]

Fürhapter, S.

Gao, P.

George, N.

Georges, V.

Gillette, M. U.

Goddard, L. L.

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics, advance online publication (2014).
[CrossRef]

Goncharsky, A.

A. Tikhonov, A. Goncharsky, V. Stepanov, A. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Mathematics and Its Applications (Springer, 1995).
[CrossRef]

Gravelle, B.

Guérineau, N.

Haeberlé, O.

Heintzmann, R.

Herminjard, S.

Iglesias, I.

Iwai, H.

Izatt, J.

Jacques, S. L.

K. G. Phillips, S. L. Jacques, O. J. T. McCarty, “Measurement of single cell refractive index, dry mass, volume, and density using a transillumination microscope,” Phys. Rev. Lett. 109, 118105 (2012).
[CrossRef] [PubMed]

Jeon, D. I.

Jesacher, A.

Jourdain, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[CrossRef]

Kemper, B.

B. Kemper, A. Vollmer, C. E. Rommel, J. Schnekenburger, G. von Bally, “Simplified approach for quantitative digital holographic phase contrast imaging of living cells,” J. Biomed. Opt. 16, 026014 (2011).
[CrossRef] [PubMed]

Kim, S.

Kim, T.

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics, advance online publication (2014).
[CrossRef]

Kou, S. S.

Kühn, J.

Larkin, K. G.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[CrossRef] [PubMed]

Lauer, V.

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205, 165–176 (2002).
[CrossRef] [PubMed]

Lee, B. H.

Lee, K. J.

Lévêque-Fort, S.

P. Bon, T. Barroca, S. Lévêque-Fort, E. Fort, “Label-free evanescent microscopy for membrane nano-tomography in living cells,” J. Biophotonics (advance online publication, 2013).
[CrossRef]

Liu, C.

Magistretti, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[CrossRef]

Marquet, P.

Maucort, G.

Maurer, C.

McCarty, O. J. T.

K. G. Phillips, S. L. Jacques, O. J. T. McCarty, “Measurement of single cell refractive index, dry mass, volume, and density using a transillumination microscope,” Phys. Rev. Lett. 109, 118105 (2012).
[CrossRef] [PubMed]

Mertz, J.

Millet, L.

Mir, M.

Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, G. Popescu, “Spatial light interference microscopy (slim),” Opt. Express 19, 1016–1026 (2011).
[CrossRef] [PubMed]

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics, advance online publication (2014).
[CrossRef]

Monneret, S.

Montfort, F.

Morin, R.

Nugent, K. A.

Osten, W.

Paganin, D.

Parthasarathy, A. B.

Pavillon, N.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[CrossRef]

Pedrini, G.

Phillips, K. G.

K. G. Phillips, S. L. Jacques, O. J. T. McCarty, “Measurement of single cell refractive index, dry mass, volume, and density using a transillumination microscope,” Phys. Rev. Lett. 109, 118105 (2012).
[CrossRef] [PubMed]

Popescu, G.

Prahl, S. A.

Primot, J.

Rigneault, H.

Ritsch-Marte, M.

Roberts, A.

Rogers, J.

Rolly, B.

Rommel, C. E.

B. Kemper, A. Vollmer, C. E. Rommel, J. Schnekenburger, G. von Bally, “Simplified approach for quantitative digital holographic phase contrast imaging of living cells,” J. Biomed. Opt. 16, 026014 (2011).
[CrossRef] [PubMed]

Savatier, J.

S. Aknoun, P. Bon, J. Savatier, B. Wattellier, S. Monneret, “Quantitative birefringence imaging of biological samples using quadri-wave interferometry,” Proc. SPIE 8587, 85871D (2013).
[CrossRef]

Schnekenburger, J.

B. Kemper, A. Vollmer, C. E. Rommel, J. Schnekenburger, G. von Bally, “Simplified approach for quantitative digital holographic phase contrast imaging of living cells,” J. Biomed. Opt. 16, 026014 (2011).
[CrossRef] [PubMed]

Shaked, N. T.

Sheppard, C. J.

Sheppard, C. J. R.

Simon, B.

Smith, N. I.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[CrossRef] [PubMed]

Stemmer, A.

Stepanov, V.

A. Tikhonov, A. Goncharsky, V. Stepanov, A. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Mathematics and Its Applications (Springer, 1995).
[CrossRef]

Stout, B.

Streibl, N.

Sung, Y.

Tangella, K.

Tikhonov, A.

A. Tikhonov, A. Goncharsky, V. Stepanov, A. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Mathematics and Its Applications (Springer, 1995).
[CrossRef]

Toy, F.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[CrossRef]

Toy, F. M.

Unarunotai, S.

Vaughan, J. C.

Vollmer, A.

B. Kemper, A. Vollmer, C. E. Rommel, J. Schnekenburger, G. von Bally, “Simplified approach for quantitative digital holographic phase contrast imaging of living cells,” J. Biomed. Opt. 16, 026014 (2011).
[CrossRef] [PubMed]

von Bally, G.

B. Kemper, A. Vollmer, C. E. Rommel, J. Schnekenburger, G. von Bally, “Simplified approach for quantitative digital holographic phase contrast imaging of living cells,” J. Biomed. Opt. 16, 026014 (2011).
[CrossRef] [PubMed]

Waller, L.

Wang, Z.

Wattellier, B.

Wenger, J.

Wicker, K.

Wiener, N.

N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series (The MIT Press, 1964).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge University, 1999), chap. 9, pp. 547–553.

Yagola, A.

A. Tikhonov, A. Goncharsky, V. Stepanov, A. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Mathematics and Its Applications (Springer, 1995).
[CrossRef]

Yang, T. D.

Yoon, J.-H.

Zhou, R.

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics, advance online publication (2014).
[CrossRef]

Appl. Opt.

Biomed. Opt. Express

J. Biomed. Opt.

B. Kemper, A. Vollmer, C. E. Rommel, J. Schnekenburger, G. von Bally, “Simplified approach for quantitative digital holographic phase contrast imaging of living cells,” J. Biomed. Opt. 16, 026014 (2011).
[CrossRef] [PubMed]

J. Microsc.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[CrossRef] [PubMed]

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205, 165–176 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Nat. Photonics

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[CrossRef]

Nature

R. Barer, “Interference microscopy and mass determination,” Nature 169, 366–367 (1952).
[CrossRef] [PubMed]

Opt. Express

S. S. Kou, C. J. Sheppard, “Imaging in digital holographic microscopy,” Opt. Express 15, 13640–13648 (2007).
[CrossRef] [PubMed]

X. Chen, N. George, G. Agranov, C. Liu, B. Gravelle, “Sensor modulation transfer function measurement using band-limited laser speckle,” Opt. Express 16, 20047–20059 (2008).
[CrossRef] [PubMed]

W. J. Choi, D. I. Jeon, S.-G. Ahn, J.-H. Yoon, S. Kim, B. H. Lee, “Full-field optical coherence microscopy for identifying live cancer cells by quantitative measurement of refractive index distribution,” Opt. Express 18, 23285–23295 (2010).
[CrossRef] [PubMed]

Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, G. Popescu, “Spatial light interference microscopy (slim),” Opt. Express 19, 1016–1026 (2011).
[CrossRef] [PubMed]

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17, 266–277 (2009).
[CrossRef] [PubMed]

R. Fiolka, K. Wicker, R. Heintzmann, A. Stemmer, “Simplified approach to diffraction tomography in optical microscopy,” Opt. Express 17, 12407–12417 (2009).
[CrossRef] [PubMed]

P. Bon, G. Maucort, B. Wattellier, S. Monneret, “Quadriwave lateral shearing interferometry for quantitative phase microscopy of living cells,” Opt. Express 17, 13080–13094 (2009).
[CrossRef] [PubMed]

S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006).
[CrossRef] [PubMed]

Opt. Lett.

Y. Choi, T. D. Yang, K. J. Lee, W. Choi, “Full-field and single-shot quantitative phase microscopy using dynamic speckle illumination,” Opt. Lett. 36, 2465–2467 (2011).
[CrossRef] [PubMed]

G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 29, 2503–2505 (2004).
[CrossRef] [PubMed]

S. S. Kou, L. Waller, G. Barbastathis, C. J. R. Sheppard, “Transport-of-intensity approach to differential interference contrast (ti-dic) microscopy for quantitative phase imaging,” Opt. Lett. 35, 447–449 (2010).
[CrossRef] [PubMed]

I. Iglesias, “Pyramid phase microscopy,” Opt. Lett. 36, 3636–3638 (2011).
[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

Schematic of the optical setup used in this paper.

Fig. 2
Fig. 2

5 μm polystyrene microbead immersed in a n = 1.556 medium, imaged at 250×, NAcoll = 1.3. (a) QWLSI interferogram with NAillum ≈ 0.08. (b) QWLSI interferogram with NAillum = 1.3. (c) Profile plots of (a) (dashed red line) and (b) (solid black line). (d) OPD image retrieved from (a). (e) OPD image retrieved from (b). (f) Profile plots of (d) (dashed red line) and (e) (solid black line).

Fig. 3
Fig. 3

Measurable object frequencies in trans-illumination when illuminated by a plane wave propagating: (a) along the (Koz) axis. (b) along the optical axis (red curve), with an angle respect to the optical axis off π/4 (pink curve) and with an angle of −π/2 (brown). The dashed dot regions indicate the frequencies filtrated by the collection microscope objective of numerical aperture NA coll = 0.75 λ m λ 0.

Fig. 4
Fig. 4

Measurable object frequencies with an objective of NAcoll = 0.75λm0, transilluminated by a incoherent source. (a) Polychromatic spatially coherent illumination with λmax = 2λmin. In pink: coherent contribution at λm = λmax, in red: at λm = 〈λ〉, in brown: at λm = λmin. In green: complete accessible frequencies. (b) Monochromatic (λm = 〈λ〉) spatially incoherent illumination with NAill = NAcoll. In red: coherent contribution at θill = 0, in pink: at θill = θobj/2, in brown: at θill = θobj. In green: complete accessible frequencies.

Fig. 5
Fig. 5

(a) OPD images under coherent illumination of a 5 μm polystyrene bead immersed in nmed = 1.542 with the setup described in the part 2.1. Top: Simulated image using G-POC. Bottom: Experimental image. (b) Same as (a) but with SI illumination. (c) Profile plots of the OPD images. Black dots: experiment with coherent illumination (a,top). Gray line: simulation (a,bottom). Red dots: experiment with SII (a,top). Burgundy line: simulation (a,bottom). (d) Fourier transfrom (logarithmic scale) of a the 3D Rytov EM field. Top: Simulation using G-POC. Bottom: Experiment.

Fig. 6
Fig. 6

(a) 2D radial target (N = 60 and Δn0 = 0.02). (b) Simulated coherent OPD simulated from the object (a) with a plane-wave illumination along the optical axis. (c) Same as (b) with the maximum illumination angle θx,ill = asin (NAill/nim). (d) Simulated SII OPD obtained from the object (a) with NAcoll = NAill = 1.3. (e) MTF with NAcoll = NAill = 1.3. In red: projective MTF for the intensity obtained from the simulation. In black: projective MTF for the OPD obtained from the simulation. In dashed gray: Theoretical MTF under paraxial approximation. (f) Same as (e) with NAcoll = 1.3 and NAill = 1.0.

Fig. 7
Fig. 7

(a) Imaged of a 3D radial target (N = 4, νz = 4% of 2NA/λ) observed in the z = 0 plane. Top: simulated SII OPD. Bottom: simulated Rytov combined coherent EM fields. The contrast enhancement compared to the Rytov fields is visible on the SII OPD. (b) 3D OPD MTF visualized in the (νr, νz) plane (logarithmic scale) considering NAcoll = NAill = 1.3. (c) Same as (b) for the 3D intensity MTF. (d) Profile plots from (c), modulation amplitude as a function of the lateral frequency normalized at 2NAcoll. Black: νz = 0. Short dashed red: νz = 4% of 2NAcoll. Medium dashed brown: νz = −4%. Dashed dot orange: νz = 20%. Large dashed yellow: νz = 30%. (e) Zoom on (d).

Fig. 8
Fig. 8

OPD Z-stacks of a 5 μm polystyrene bead. Up: z = 0 plane. Down: y = 0 plane. (a) OPD measured with a quasi-plane-wave illumination. The Z-stack is obtained by numerical propagation. (b) OPD stack measured under SII with axial scanning of the objective. (c) Same as (b) plus deconvolution using the 3D MTFOPD. (d) Same as (b) plus deconvolution of the z = 0 image (b).up using the projective MTFOPD. (e) OPD line-outs along x by the center of the bead. Black curve: coherent illumination. Red dot curve: SI illumination. Yellow dashed dot curve: SII plus projective MTFOPD. Orange curve: SII plus 3D MTFOPD.

Fig. 9
Fig. 9

OPD Z-stacks of a 100 nm TiO2 bead. Up: z = 0 plane. Down: y = 0 plane. (a) OPD measured with a quasi-plane-wave illumination. The Z-stack is obtained with axial scanning of the objective. (b) OPD stack measured under SII with axial scanning of the objective. (c) Same as (b) plus deconvolution using the 3D MTFOPD. (d) Same as (b) plus deconvolution of the z = 0 image (b).up using the projective MTFOPD. (e) OPD profile plots in the z = 0 plane of coherent OPD (black line), SII OPD (red dot line), 2D deconvolved SII OPD (orange line) and 3D deconvolved SII OPD (yellow dashed dot line). The resolution gain using SII is clearly visible.

Tables (1)

Tables Icon

Table 1 Comparison between theoretical and experimental PSFs.

Equations (16)

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OPD sample Δ n × d z cos ( | θ ill | ) ,
i ( θ ill , x , z ) I ( θ ill , x , z ) [ 1 + cos ( 2 π Λ [ x z p OPD x ( θ ill , x , z ) z p τ ( θ ill ) ] ) ] ,
{ I = FT 1 [ FT [ i ] δ ( k ) ] OPD x = 1 α Arg { FT 1 [ FT [ i ] δ ( k Λ 2 π ) ] } ,
i S I ( x , z ) = θ ill i ( θ ill , x , z ) d θ ill .
τ ( θ ill ) = tan ( | θ ill | ) M ,
r x y = λ 0 NA ill + NA coll ,
k d = k i + K o ,
k i 2 = ( k i x + K o x ) 2 + ( k i y + K o y ) 2 + ( k i z + K o z ) 2 .
{ K o z = k i z + k i z 2 ( K o x 2 + K o y 2 + 2 K o x k i x + 2 K o y k i y ) [ 1 ] K o z = k i z k i z 2 ( K o x 2 + K o y 2 + 2 K o x k i x + 2 K o y k i y ) [ 2 ]
I ˜ Deconv = I ˜ MTF ,
{ 1 MTF reg 1 when MTF 0 1 MTF reg 1 MTF els e .
1 MTF reg Wiener ( ν ) = 1 MTF ( ν ) [ | MTF ( ν ) | 2 | MTF ( ν ) | 2 + 1 SNR ( ν ) ] .
Δ n ( r , ψ ) = Δ n 0 cos ( ψ N ) ,
OPD D T 3 D ˜ ( K o x , K o y , K o z ) 2 π λ = Re ( θ i [ E Rytov i ˜ ( k d x , k d y ; z = 0 ) δ ( k d 2 k i 2 ) ] j k d z π d θ i ) ,
Δ n ( r , ψ , z ) = Δ n 0 cos ( ψ N ) e j 2 π ν z 0 z ,
r PSF = 1.22 λ NA coll + NA ill ,

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