Phase retrieval, i.e., the reconstruction of phase information from intensity information, is a central problem in many optical systems. Imaging the emission from a point source such as a single molecule is one example. Here, we demonstrate that a deep residual neural net is able to quickly and accurately extract the hidden phase for general point spread functions (PSFs) formed by Zernike-type phase modulations. Five slices of the 3D PSF at different focal positions within a two micrometer range around the focus are sufficient to retrieve the first six orders of Zernike coefficients.

1.
Y.
Shechtman
,
Y. C.
Eldar
,
O.
Cohen
,
H. N.
Chapman
,
J. W.
Miao
, and
M.
Segev
,
IEEE Signal Proc. Mag.
32
,
87
109
(
2015
).
2.
S.
Marchesini
,
H.
He
,
H. N.
Chapman
,
S. P.
Hau-Riege
,
A.
Noy
,
M. R.
Howells
,
U.
Weierstall
, and
J. C. H.
Spence
,
Phys. Rev. B
68
,
140101
(
2003
).
3.
K.
Jaganathan
,
Y.
Eldar
, and
B.
Hassibi
, “
Phase retrieval: An overview of recent developments
,” in
Optical Compressive Imaging
, edited by
A.
Stern
(
CRC Press
,
2016
).
4.
R. W.
Gerchberg
and
W. O.
Saxton
,
Optik
35
,
237
246
(
1972
).
5.
C.
Guo
,
C.
Wei
,
J. B.
Tan
,
K.
Chen
,
S. T.
Liu
,
Q.
Wu
, and
Z. J.
Liu
,
Opt. Laser Eng.
89
,
2
12
(
2017
).
6.
B. M.
Hanser
,
M. G.
Gustafsson
,
D. A.
Agard
, and
J. W.
Sedat
,
J. Microsc.
216
,
32
48
(
2004
).
7.
S.
Quirin
,
S. R.
Pavani
, and
R.
Piestun
,
Proc. Natl. Acad. Sci. U. S. A.
109
,
675
679
(
2012
).
8.
P. N.
Petrov
,
Y.
Shechtman
, and
W. E.
Moerner
,
Opt. Express
25
,
7945
7959
(
2017
).
9.
M.
Siemons
,
C. N.
Hulleman
,
R. O.
Thorsen
,
C. S.
Smith
, and
S.
Stallinga
,
Opt. Express
26
,
8397
8416
(
2018
).
10.
W. X.
Wang
,
F.
Ye
,
H.
Shen
,
N. A.
Moringo
,
C.
Dutta
,
J. T.
Robinson
, and
C. F.
Landes
,
Opt. Express
27
,
3799
3816
(
2019
).
11.
D.
Debarre
,
M. J.
Booth
, and
T.
Wilson
,
Opt. Express
15
,
8176
8190
(
2007
).
12.
A.
Facomprez
,
E.
Beaurepaire
, and
D.
Debarre
,
Opt. Express
20
,
2598
2612
(
2012
).
13.
E.
Nehme
,
L. E.
Weiss
,
T.
Michaeli
, and
Y.
Shechtman
,
Optica
5
,
458
464
(
2018
).
14.
W.
Ouyang
,
A.
Aristov
,
M.
Lelek
,
X.
Hao
, and
C.
Zimmer
,
Nat. Biotechnol.
36
,
460
468
(
2018
).
15.
K. M.
He
,
X. Y.
Zhang
,
S. Q.
Ren
, and
J.
Sun
, in
Proceedings of the IEEE Conference on CVPR
(
2016
), pp.
770
778
.
16.
P.
Zhang
,
S.
Liu
,
A.
Chaurasia
,
D.
Ma
,
M. J.
Mlodzianoski
,
E.
Culurciello
, and
F.
Huang
,
Nat. Methods
15
,
913
916
(
2018
).
17.
P.
Zelger
,
K.
Kaser
,
B.
Rossboth
,
L.
Velas
,
G. J.
Schutz
, and
A.
Jesacher
,
Opt. Express
26
,
33166
33179
(
2018
).
18.
A.
Goy
,
K.
Arthur
,
S.
Li
, and
G.
Barbastathis
,
Phys. Rev. Lett.
121
,
243902
(
2018
).
19.
H.
Guo
,
N.
Korablinova
,
Q.
Ren
, and
J.
Bille
,
Opt. Express
14
,
6456
6462
(
2006
).
20.
S. W.
Paine
and
J. R.
Fienup
,
Opt. Lett.
43
,
1235
1238
(
2018
).
21.
Y.
Nishizaki
,
M.
Valdivia
,
R.
Horisaki
,
K.
Kitaguchi
,
M.
Saito
,
J.
Tanida
, and
E.
Vera
,
Opt. Express
27
,
240
251
(
2019
).
22.
M. D.
Lew
and
W. E.
Moerner
,
Nano Lett.
14
,
6407
6413
(
2014
).
23.
K. I.
Mortensen
,
L. S.
Churchman
,
J. A.
Spudich
, and
H.
Flyvbjerg
,
Nat. Methods
7
,
377
381
(
2010
).
24.
A. S.
Backer
and
W. E.
Moerner
,
J. Phys. Chem. B
118
,
8313
8329
(
2014
).
25.
B.
Moomaw
,
Method Cell Biol.
114
,
243
283
(
2013
).
26.
W. E.
Moerner
and
D. P.
Fromm
,
Rev. Sci. Instrum.
74
,
3597
3619
(
2003
).
27.
M. C.
Won
,
D.
Mnyama
, and
R. H. T.
Bates
,
Opt. Acta
32
,
377
396
(
1985
).
28.
A.
von Diezmann
,
M. Y.
Lee
,
M. D.
Lew
, and
W. E.
Moerner
,
Optica
2
,
985
993
(
2015
).
29.
T.
Yan
,
C. J.
Richardson
,
M.
Zhang
, and
A.
Gahlmann
,
Opt. Express
27
,
12582
12599
(
2019
).
30.
N.
Srivastava
,
G.
Hinton
,
A.
Krizhevsky
,
I.
Sutskever
, and
R.
Salakhutdinov
,
J. Mach. Learn. Res.
15
,
1929
1958
(
2014
).
31.
Y.
Shechtman
,
S. J.
Sahl
,
A. S.
Backer
, and
W. E.
Moerner
,
Phys. Rev. Lett.
113
,
133902
(
2014
).
32.
A.
von Diezmann
,
Y.
Shechtman
, and
W. E.
Moerner
,
Chem. Rev.
117
,
7244
7275
(
2017
).

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