Abstract

Wavefront sensors and more general phase retrieval methods have recently attracted a lot of attention in a host of application domains, ranging from astronomy to scientific imaging and microscopy. In this paper, we introduce a new class of sensor, the Coded Wavefront Sensor, which provides high spatio-temporal resolution using a simple masked sensor under white light illumination. Specifically, we demonstrate megapixel spatial resolution and phase accuracy better than 0.1 wavelengths at reconstruction rates of 50 Hz or more, thus opening up many new applications from high-resolution adaptive optics to real-time phase retrieval in microscopy.

© 2017 Optical Society of America

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References

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  1. B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” Journal of Refractive Surgery 17, S573–S577 (2001).
    [PubMed]
  2. L. K. Saddlemyer, G. Herriot, J.-P. Véran, and J. M. Fletcher, “Design aspects of the reconstructor for the Gemini adaptive optics system (Altair),” Proc. SPIE 3353, 150–159 (1998).
    [Crossref]
  3. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
    [Crossref]
  4. O. Fauvarque, B. Neichel, T. Fusco, S. Thierry, and J.-F. Sauvage, “Variation around a pyramid theme: optical recombination and optimal use of photons,” Opt. Lett. 40, 3528–3531 (2015).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  7. L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18, 12552–12561 (2010).
    [Crossref] [PubMed]
  8. Z. Jingshan, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22, 10661–10674 (2014).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  11. R. Tumbar, R. A. Stack, and D. J. Brady, “Wave-front sensing with a sampling field sensor,” Appl. Opt. 39, 72–84 (2000).
    [Crossref]
  12. J. C. Chanteloup, “Multiple-wave lateral shearing interferometry for wave-front sensing,” Appl. Opt. 44, 1559–1571 (2005).
    [Crossref] [PubMed]
  13. B. K. Horn and B. G. Schunck, “Determining optical flow,” Artificial intelligence 17, 185–203 (1981).
    [Crossref]
  14. T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in ECCV’04, 8th European Conference on Computer Vision (2004) pp. 25–36.
  15. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning 3, 1–122 (2011).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  19. R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Computer Science Technical Report CSTR 2, 1–11 (2005).
  20. A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: Mask enhanced cameras for heterodyned light fields and coded aperture refocusing,” ACM Trans. Graph. 26, 69 (2007).
    [Crossref]
  21. K. Marwah, G. Wetzstein, Y. Bando, and R. Raskar, “Compressive light field photography using overcomplete dictionaries and optimized projections,” ACM Trans. Graph. 32, 46 (2013).
    [Crossref]
  22. H. Richard and M. Raffel, “Principle and applications of the background oriented schlieren (bos) method,” Measurement Science and Technology 12, 1576 (2001).
    [Crossref]
  23. B. Atcheson, W. Heidrich, and I. Ihrke, “An evaluation of optical flow algorithms for background oriented schlieren imaging,” Experiments in Fluids 46, 467–476 (2009).
    [Crossref]

2015 (1)

2014 (1)

2013 (1)

K. Marwah, G. Wetzstein, Y. Bando, and R. Raskar, “Compressive light field photography using overcomplete dictionaries and optimized projections,” ACM Trans. Graph. 32, 46 (2013).
[Crossref]

2011 (1)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning 3, 1–122 (2011).
[Crossref]

2010 (1)

2009 (2)

K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17, 19662–19673 (2009).
[Crossref] [PubMed]

B. Atcheson, W. Heidrich, and I. Ihrke, “An evaluation of optical flow algorithms for background oriented schlieren imaging,” Experiments in Fluids 46, 467–476 (2009).
[Crossref]

2007 (2)

A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: Mask enhanced cameras for heterodyned light fields and coded aperture refocusing,” ACM Trans. Graph. 26, 69 (2007).
[Crossref]

A. F. Brooks, T.-L. Kelly, P. J. Veitch, and J. Munch, “Ultra-sensitive wavefront measurement using a Hartmann sensor,” Opt. Express 15, 10370–10375 (2007).
[Crossref] [PubMed]

2005 (2)

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

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Computer Science Technical Report CSTR 2, 1–11 (2005).

2004 (1)

T. E. Gureyev, A. Pogany, D. M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

2001 (2)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” Journal of Refractive Surgery 17, S573–S577 (2001).
[PubMed]

H. Richard and M. Raffel, “Principle and applications of the background oriented schlieren (bos) method,” Measurement Science and Technology 12, 1576 (2001).
[Crossref]

2000 (1)

1998 (1)

L. K. Saddlemyer, G. Herriot, J.-P. Véran, and J. M. Fletcher, “Design aspects of the reconstructor for the Gemini adaptive optics system (Altair),” Proc. SPIE 3353, 150–159 (1998).
[Crossref]

1996 (1)

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[Crossref]

1992 (1)

R.G. Lane, A. Glindemann, and J.C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves in random media 2, 3, 209–224 (1992).
[Crossref]

1988 (1)

1983 (1)

1981 (1)

B. K. Horn and B. G. Schunck, “Determining optical flow,” Artificial intelligence 17, 185–203 (1981).
[Crossref]

Agrawal, A.

A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: Mask enhanced cameras for heterodyned light fields and coded aperture refocusing,” ACM Trans. Graph. 26, 69 (2007).
[Crossref]

Atcheson, B.

B. Atcheson, W. Heidrich, and I. Ihrke, “An evaluation of optical flow algorithms for background oriented schlieren imaging,” Experiments in Fluids 46, 467–476 (2009).
[Crossref]

Bando, Y.

K. Marwah, G. Wetzstein, Y. Bando, and R. Raskar, “Compressive light field photography using overcomplete dictionaries and optimized projections,” ACM Trans. Graph. 32, 46 (2013).
[Crossref]

Barbastathis, G.

Boyd, S.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning 3, 1–122 (2011).
[Crossref]

Brady, D. J.

Brédif, M.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Computer Science Technical Report CSTR 2, 1–11 (2005).

Brooks, A. F.

Brox, T.

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in ECCV’04, 8th European Conference on Computer Vision (2004) pp. 25–36.

Bruhn, A.

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in ECCV’04, 8th European Conference on Computer Vision (2004) pp. 25–36.

Chanteloup, J. C.

Chu, E.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning 3, 1–122 (2011).
[Crossref]

Claus, R. A.

Dainty, J.C.

R.G. Lane, A. Glindemann, and J.C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves in random media 2, 3, 209–224 (1992).
[Crossref]

Dauwels, J.

Duval, G.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Computer Science Technical Report CSTR 2, 1–11 (2005).

Eckstein, J.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning 3, 1–122 (2011).
[Crossref]

Fauvarque, O.

Fletcher, J. M.

L. K. Saddlemyer, G. Herriot, J.-P. Véran, and J. M. Fletcher, “Design aspects of the reconstructor for the Gemini adaptive optics system (Altair),” Proc. SPIE 3353, 150–159 (1998).
[Crossref]

Fusco, T.

Glindemann, A.

R.G. Lane, A. Glindemann, and J.C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves in random media 2, 3, 209–224 (1992).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).

Gureyev, T. E.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

Hanrahan, P.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Computer Science Technical Report CSTR 2, 1–11 (2005).

Heidrich, W.

B. Atcheson, W. Heidrich, and I. Ihrke, “An evaluation of optical flow algorithms for background oriented schlieren imaging,” Experiments in Fluids 46, 467–476 (2009).
[Crossref]

Herriot, G.

L. K. Saddlemyer, G. Herriot, J.-P. Véran, and J. M. Fletcher, “Design aspects of the reconstructor for the Gemini adaptive optics system (Altair),” Proc. SPIE 3353, 150–159 (1998).
[Crossref]

Horn, B. K.

B. K. Horn and B. G. Schunck, “Determining optical flow,” Artificial intelligence 17, 185–203 (1981).
[Crossref]

Horowitz, M.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Computer Science Technical Report CSTR 2, 1–11 (2005).

Ihrke, I.

B. Atcheson, W. Heidrich, and I. Ihrke, “An evaluation of optical flow algorithms for background oriented schlieren imaging,” Experiments in Fluids 46, 467–476 (2009).
[Crossref]

Jingshan, Z.

Kelly, T.-L.

Lane, R.G.

R.G. Lane, A. Glindemann, and J.C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves in random media 2, 3, 209–224 (1992).
[Crossref]

Levoy, M.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Computer Science Technical Report CSTR 2, 1–11 (2005).

Marwah, K.

K. Marwah, G. Wetzstein, Y. Bando, and R. Raskar, “Compressive light field photography using overcomplete dictionaries and optimized projections,” ACM Trans. Graph. 32, 46 (2013).
[Crossref]

Matsushima, K.

Mohan, A.

A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: Mask enhanced cameras for heterodyned light fields and coded aperture refocusing,” ACM Trans. Graph. 26, 69 (2007).
[Crossref]

Munch, J.

Neichel, B.

Ng, R.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Computer Science Technical Report CSTR 2, 1–11 (2005).

Paganin, D. M.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

Papenberg, N.

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in ECCV’04, 8th European Conference on Computer Vision (2004) pp. 25–36.

Parikh, N.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning 3, 1–122 (2011).
[Crossref]

Peleato, B.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning 3, 1–122 (2011).
[Crossref]

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” Journal of Refractive Surgery 17, S573–S577 (2001).
[PubMed]

Pogany, A.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

Raffel, M.

H. Richard and M. Raffel, “Principle and applications of the background oriented schlieren (bos) method,” Measurement Science and Technology 12, 1576 (2001).
[Crossref]

Ragazzoni, R.

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[Crossref]

Raskar, R.

K. Marwah, G. Wetzstein, Y. Bando, and R. Raskar, “Compressive light field photography using overcomplete dictionaries and optimized projections,” ACM Trans. Graph. 32, 46 (2013).
[Crossref]

A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: Mask enhanced cameras for heterodyned light fields and coded aperture refocusing,” ACM Trans. Graph. 26, 69 (2007).
[Crossref]

Richard, H.

H. Richard and M. Raffel, “Principle and applications of the background oriented schlieren (bos) method,” Measurement Science and Technology 12, 1576 (2001).
[Crossref]

Roddier, F.

Saddlemyer, L. K.

L. K. Saddlemyer, G. Herriot, J.-P. Véran, and J. M. Fletcher, “Design aspects of the reconstructor for the Gemini adaptive optics system (Altair),” Proc. SPIE 3353, 150–159 (1998).
[Crossref]

Sauvage, J.-F.

Schunck, B. G.

B. K. Horn and B. G. Schunck, “Determining optical flow,” Artificial intelligence 17, 185–203 (1981).
[Crossref]

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” Journal of Refractive Surgery 17, S573–S577 (2001).
[PubMed]

Shimobaba, T.

Stack, R. A.

Teague, M. R.

Thierry, S.

Tian, L.

Tumbar, R.

Tumblin, J.

A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: Mask enhanced cameras for heterodyned light fields and coded aperture refocusing,” ACM Trans. Graph. 26, 69 (2007).
[Crossref]

Veeraraghavan, A.

A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: Mask enhanced cameras for heterodyned light fields and coded aperture refocusing,” ACM Trans. Graph. 26, 69 (2007).
[Crossref]

Veitch, P. J.

Véran, J.-P.

L. K. Saddlemyer, G. Herriot, J.-P. Véran, and J. M. Fletcher, “Design aspects of the reconstructor for the Gemini adaptive optics system (Altair),” Proc. SPIE 3353, 150–159 (1998).
[Crossref]

Waller, L.

Weickert, J.

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in ECCV’04, 8th European Conference on Computer Vision (2004) pp. 25–36.

Wetzstein, G.

K. Marwah, G. Wetzstein, Y. Bando, and R. Raskar, “Compressive light field photography using overcomplete dictionaries and optimized projections,” ACM Trans. Graph. 32, 46 (2013).
[Crossref]

Wilkins, S.W.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

ACM Trans. Graph. (2)

A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: Mask enhanced cameras for heterodyned light fields and coded aperture refocusing,” ACM Trans. Graph. 26, 69 (2007).
[Crossref]

K. Marwah, G. Wetzstein, Y. Bando, and R. Raskar, “Compressive light field photography using overcomplete dictionaries and optimized projections,” ACM Trans. Graph. 32, 46 (2013).
[Crossref]

Appl. Opt. (3)

Artificial intelligence (1)

B. K. Horn and B. G. Schunck, “Determining optical flow,” Artificial intelligence 17, 185–203 (1981).
[Crossref]

Computer Science Technical Report CSTR (1)

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Computer Science Technical Report CSTR 2, 1–11 (2005).

Experiments in Fluids (1)

B. Atcheson, W. Heidrich, and I. Ihrke, “An evaluation of optical flow algorithms for background oriented schlieren imaging,” Experiments in Fluids 46, 467–476 (2009).
[Crossref]

Foundations and Trends in Machine Learning (1)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning 3, 1–122 (2011).
[Crossref]

J. Mod. Opt. (1)

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[Crossref]

J. Opt. Soc. Am. (1)

Journal of Refractive Surgery (1)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” Journal of Refractive Surgery 17, S573–S577 (2001).
[PubMed]

Measurement Science and Technology (1)

H. Richard and M. Raffel, “Principle and applications of the background oriented schlieren (bos) method,” Measurement Science and Technology 12, 1576 (2001).
[Crossref]

Opt. Commun. (1)

T. E. Gureyev, A. Pogany, D. M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Proc. SPIE (1)

L. K. Saddlemyer, G. Herriot, J.-P. Véran, and J. M. Fletcher, “Design aspects of the reconstructor for the Gemini adaptive optics system (Altair),” Proc. SPIE 3353, 150–159 (1998).
[Crossref]

Waves in random media (1)

R.G. Lane, A. Glindemann, and J.C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves in random media 2, 3, 209–224 (1992).
[Crossref]

Other (2)

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in ECCV’04, 8th European Conference on Computer Vision (2004) pp. 25–36.

Supplementary Material (1)

NameDescription
» Visualization 1: MP4 (5216 KB)      Video of system in action

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Figures (6)

Fig. 1
Fig. 1

Schematic of the Coded Wavefront Sensor. (a) A simple calibration is performed by capturing the diffraction pattern of a planar wavefront, namely the reference image. (b) By just recording the diffraction pattern of a distorted wavefront (i.e. the measurement image) and compare it to the reference image captured in (a), the distorted wavefront can be reconstructed. Small arrows indicate local distortion directions.

Fig. 2
Fig. 2

Accuracy experiments. Top left shows the reconstruction error in RMS for different wavefront range (Peak-to-Valley) for different z. Specifically, results of three wavefront range spherical waves are shown when z = 1 mm. To visualize the difference between the measurement and the reference, the logarithm of their substraction are shown as inset.

Fig. 3
Fig. 3

Synthetic atmospheric turbulence. Most left shows the reconstruction RMS versus the turbulence RMS. Specifically, one scale of the turbulence is shown on the right.

Fig. 4
Fig. 4

Experimental setup for accuracy validation. Under collimated incoherent illumination from a white light source, the SLM generates a known distorted wavefront, which is then captured by our Coded Wavefront Sensor in the conjugate plane.

Fig. 5
Fig. 5

Selected experimental results. The ground truth wavefronts, our reconstructed wavefronts, and the wavefront errors are shown. The wavefronts are shown in interference fringes where one fringe maps to wavefront difference of λ = 632.8 nm. Scale bar is 1 mm.

Fig. 6
Fig. 6

Wavefront visualization of the heat flow and the defocusing. The setup diagrams are simplified versions of the real situations. See Visualization 1 for full experimental details. Scale bar is 2 mm.

Tables (2)

Tables Icon

Table 1 Reconstruction error for all experimental wavefronts, where λ = 632.8 nm.

Tables Icon

Table 2 Timing performance of our GPU implementation for solving Eq. (3).

Equations (13)

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

u z ( r ) exp [ j ϕ ( r ) ] p z ( r ( z / k ) ϕ ( r ) ) ,
I ( r ) = I 0 ( r ( z / k ) ϕ ( r ) ) .
z k ϕ ( r ) I 0 ( r ) + I ( r ) I 0 ( r ) = 0 ,
ϕ = a 2 π d s λ z ,
z k 2 ϕ ( r ) = z R 0 ,
minimize ϕ GM ϕ + g t 2 2 + α ϕ 2 2 ,
u 0 ( r ) = f 0 ( r ) p 0 ( r ) ,
u z ( r ) = exp [ j k z ( 1 + 2 k 2 ) 1 / 2 ] u 0 ( r ) = exp ( j 2 π r ρ ) exp [ j k z ( 1 λ 2 ρ 2 2 ) 1 / 2 ] × P 0 ( ρ ) F 0 ( ρ ρ ) d ρ d ρ exp ( j k z ) exp ( j 2 π r ρ ) exp [ j k z ( 1 λ 2 ρ 2 2 ) 1 / 2 ] × exp ( j 2 π ( r λ z ρ ) ρ ) exp [ j k z ( 1 λ 2 ρ 2 2 ) 1 / 2 ] F 0 ( ρ ) d ρ P 0 ( ρ ) d ρ = exp ( j k z ) exp ( j 2 π r ρ ) exp [ j k z ( 1 λ 2 ρ 2 2 ) 1 / 2 ] × P 0 ( ρ ) exp [ j k z ( 1 + 2 k 2 ) 1 / 2 ] f 0 ( r λ z ρ ) d ρ ,
( 1 λ 2 ρ 2 2 ) 1 / 2 ( 1 λ 2 ρ 2 2 ) 1 / 2 + ( 1 λ 2 ρ 2 2 ) 1 / 2 λ 2 ρ ρ 1 ,
ϕ ( r λ z ρ ) ϕ ( r ) λ z ρ ϕ ( r ) + 1 2 λ z ρ ( λ z 2 ϕ ( r ) ) ρ = ϕ ( r ) λ z ρ ϕ ( r ) + π λ z z R ρ 2 2 .
d m d m * = π λ R z 2 .
exp [ j k z ( 1 + 2 k 2 ) 1 / 2 ] f 0 ( r λ z ρ ) exp ( j k z ) exp [ j ϕ ( r ) ] exp [ j λ z ρ ϕ ( r ) ] .
u z ( r ) exp [ j ϕ ( r ) ] p z ( r ( z / k ) ϕ ( r ) ) ,

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