Spatial filtering is a commonly deployed technique to improve the quality of laser beams by optically filtering the noise. In the “textbook” example, the noise is usually assumed to be high frequency and the laser beam, Gaussian. In this case, the filtering is achieved by a simple pinhole placed at the common focal plane of two lenses. Here, we explain how to generalize the concept of spatial filtering to arbitrary beam profiles: spatial filtering of structured light. We show how to construct the spatial filters using a range of structured light examples and highlight under what conditions spatial filtering works. In the process, we address some misconceptions in the community as to how and when spatial filters can be applied, extend the concept of spatial filtering to arbitrary beam types, and provide a theoretical and experimental framework for further study at both the undergraduate and graduate level.

1.
A. E.
Siegman
,
Lasers
(
University Science Books
,
Mill Valley, CA
,
1986
).
2.
B. D.
Van Veen
and
K. M.
Buckley
, “
Beamforming: A versatile approach to spatial filtering
,”
IEEE ASSP Mag.
5
,
4
24
(
1988
).
3.
A.
Vander Lugt
, “
Signal detection by complex spatial filtering
,”
IEEE Trans. Inf. Theory
10
,
139
145
(
1964
).
4.
B. R.
Brown
and
A. W.
Lohmann
, “
Complex spatial filtering with binary masks
,”
Appl. Opt.
5
,
967
969
(
1966
).
5.
A. W.
Lohmann
and
D. P.
Paris
, “
Computer generated spatial filters for coherent optical data processing
,”
Appl. Opt.
7
,
651
655
(
1968
).
6.
S.
Fürhapter
,
A.
Jesacher
,
S.
Bernet
, and
M.
Ritsch-Marte
, “
Spiral phase contrast imaging in microscopy
,”
Opt. Express
13
,
689
694
(
2005
).
7.
S. J.
Luck
and
S. A.
Hillyard
, “
Spatial filtering during visual search: Evidence from human electrophysiology
,”
J. Exp. Psychol. Human
20
,
1000
1014
(
1994
).
8.
E.
O'Neill
, “
Spatial filtering in optics
,”
IEEE Trans. Inf. Theory
2
,
56
65
(
1956
).
9.
J. W.
Goodman
,
Introduction to Fourier Optics
(
Roberts and Company Publishers
,
CO
,
2005
).
10.
L.
Maigyte
and
K.
Staliunas
, “
Spatial filtering with photonic crystals
,”
Appl. Phys. Rev.
2
,
011102
(
2015
).
11.
S.
Szatmári
,
Z.
Bakonyi
, and
P.
Simon
, “
Active spatial filtering of laser beams
,”
Opt. Commun.
134
,
199
204
(
1997
).
12.
B.
Mahieu
,
D.
Gauthier
,
M.
Perdrix
 et al, “
Spatial quality improvement of a Ti: Sapphire laser beam by modal filtering
,”
Appl. Phys. B
118
,
47
60
(
2015
).
13.
J. E.
Murray
,
D.
Milam
,
C. D.
Boley
 et al, “
Spatial filter pinhole development for the national ignition facility
,”
Appl. Opt.
39
,
1405
1420
(
2000
).
14.
H.
Rubinsztein-Dunlop
,
A.
Forbes
,
M. V.
Berry
 et al, “
Roadmap on structured light
,”
J. Opt.
19
,
013001
(
2017
).
15.
A.
Forbes
, “
Structured light: Tailored for purpose
,”
Opt. Photonics News
31
,
24
31
(
2020
).
16.
A.
Forbes
and
I.
Nape
, “
Quantum mechanics with patterns of light: Progress in high dimensional and multidimensional entanglement with structured light
,”
AVS Quantum Sci.
1
,
011701
(
2019
).
17.
A.
Forbes
,
Laser Beam Propagation: Generation and Propagation of Customized Light
(
CRC Press
,
FL
,
2014
).
18.
C.
Rosales-Guzmán
,
B.
Ndagano
, and
A.
Forbes
, “
A review of complex vector light fields and their applications
,”
J. Opt.
20
,
123001
(
2018
).
19.
J.
Chen
,
C.
Wan
, and
Q.
Zhan
, “
Vectorial optical fields: Recent advances and future prospects
,”
Sci. Bull.
63
,
54
74
(
2018
).
20.
A.
Forbes
,
Laser Beam Propagation: Generation and Propagation of Customized Light
(
CRC Press LLC
,
FL
,
2014
).
21.
A.
Forbes
, “
Structured light from lasers
,”
Laser Photonics Rev.
13
,
1900140
(
2019
).
22.
G.
Lazarev
,
P.-J.
Chen
,
J.
Strauss
 et al, “
Beyond the display: Phase-only liquid crystal on silicon devices and their applications in photonics
,”
Opt. Express
27
,
16206
16249
(
2019
).
23.
A.
Forbes
,
A.
Dudley
, and
M.
McLaren
, “
Creation and detection of optical modes with spatial light modulators
,”
Adv. Opt. Photonics
8
,
200
227
(
2016
).
24.
C.
Rosales-Guzmán
and
A.
Forbes
,
How to Shape Light with Spatial Light Modulators
(
SPIE Press
,
WA
,
2017
), p.
57
.
25.
S.
Turtaev
,
I. T.
Leite
,
K. J.
Mitchell
 et al, “
Comparison of nematic liquid-crystal and dmd based spatial light modulation in complex photonics
,”
Opt. Express
25
,
29874
29884
(
2017
).
26.
S.
Scholes
,
R.
Kara
,
J.
Pinnell
 et al, “
Structured light with digital micromirror devices: A guide to best practice
,”
Opt. Eng.
59
,
041202
(
2019
).
27.
D.
Gossman
,
B.
Perez-Garcia
,
R. I.
Hernandez-Aranda
, and
A.
Forbes
, “
Optical interference with digital holograms
,”
Am. J. Phys.
84
,
508
516
(
2016
).
28.
B. R.
Boruah
, “
Dynamic manipulation of a laser beam using a liquid crystal spatial light modulator
,”
Am. J. Phys.
77
,
331
336
(
2009
).
29.
A.
Dudley
,
N.
Majola
,
N.
Chetty
, and
A.
Forbes
, “
Implementing digital holograms to create and measure complex-plane optical fields
,”
Am. J. Phys.
84
,
106
112
(
2016
).
30.
D.
Huang
,
H.
Timmers
,
A.
Roberts
 et al, “
A low-cost spatial light modulator for use in undergraduate and graduate optics labs
,”
Am. J. Phys.
80
,
211
215
(
2012
).
31.
A. V.
Carpentier
,
H.
Michinel
,
J. R.
Salgueiro
, and
D.
Olivieri
, “
Making optical vortices with computer-generated holograms
,”
Am. J. Phys.
76
,
916
921
(
2008
).
32.
S.
Panarin
,
J.
Müller
,
S.
Prabhakar
, and
R.
Fickler
, “
Spatial structuring of light for undergraduate laboratory
,” preprint arXiv:2003.12713 (
2020
).
33.
M. A.
Golub
,
I. N.
Sisakian
,
V. A.
Soifer
, and
G. V.
Uvarov
, “
New measurement techniques for modal power distribution in fibers
,” in
San Jose-DL Tentative
(
International Society for Optics and Photonics
,
WA
,
1991
), pp.
273
282
.
34.
E.
Bolduc
,
N.
Bent
,
E.
Santamato
,
E.
Karimi
, and
R. W.
Boyd
, “
Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram
,”
Opt. Lett.
38
,
3546
3549
(
2013
).
35.
See <https://github.com/JPinnell/Spatial-filtering-of-structured-light> for associated MATLAB scripts and functions.
36.
A.
Jesacher
,
A.
Schwaighofer
,
S.
Fürhapter
 et al, “
Wavefront correction of spatial light modulators using an optical vortex image
,”
Opt. Express
15
,
5801
5808
(
2007
).
37.
L.
Burger
,
I. A.
Litvin
, and
A.
Forbes
, “
Simulating atmospheric turbulence using a phase-only spatial light modulator
,”
S. Afr. J. Sci.
104
,
129
134
(
2008
).
38.
R. J.
Noll
, “
Zernike polynomials and atmospheric turbulence
,”
J. Opt. Soc. Am.
66
,
207
211
(
1976
).
39.
M. A.
Cox
,
N.
Mphuthi
,
I.
Nape
 et al, “
Structured light in turbulence
,” preprint arXiv:2005.14586 (
2020
).
40.
F.
Roddier
,
Adaptive Optics in Astronomy
(
Cambridge U. P.
,
Cambridge
,
1999
).
41.
A. P.
Mosk
,
A.
Lagendijk
,
G.
Lerosey
, and
M.
Fink
, “
Controlling waves in space and time for imaging and focusing in complex media
,”
Nat. Photon.
6
,
283
292
(
2012
).
42.
M.
Plöschner
,
T.
Tyc
, and
T.
Čižmár
, “
Seeing through chaos in multimode fibres
,”
Nat. Photon.
9
,
529
535
(
2015
).
43.
A. E.
Siegman
, “
Binary phase plates cannot improve laser beam quality
,”
Opt. Lett.
18
,
675
677
(
1993
).
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