Abstract

A modal wavefront sensor provides a direct way of measuring the presence of a given type of aberration in the incident beam and is less computationally intensive compared to a zonal wavefront sensor. Such a sensor is particularly useful when the incident beam is aberrated with a limited number of lower order aberrations. The modal wavefront sensor has found applications in diverse areas and research is on to further improve the performance of the sensor. However, one major issue of the modal wavefront sensor is the inter-modal cross talk, that is the effect on the sensor output due to the presence of other aberration modes present in the beam. Although inter-modal cross talk is an inherent phenomenon in a modal wavefront sensor, so far there has not been any comprehensive study on the same in general. Besides there does not exist a model for quick numerical evaluation of the inter-modal cross talk in a modal wavefront sensor. In this paper we develop the theoretical expressions to define an inter-modal cross talk co-efficient and further expand it to a polynomial form. The polynomial form provides a quick way to calculate the effect of inter-modal cross talk. The correctness of the polynomial form thus developed is demonstrated by comparing the results with those obtained using the basic theory and the experimental implementation of a modal wavefront sensor. The polynomial form of the cross talk co-efficient is further used to investigate the effect of inter-modal cross talk in the sensing of a few important low order aberrations.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

F. Loosen, J. Stehr, L. Alber, I. Harder, and N. Lindlein, “A holography-based modal wavefront sensor for the precise positioning of a light emitter using a high-resolution computer-generated hologram,” IEEE Photon. J. 10, 1–11 (2018).
[Crossref]

2017 (3)

M. J. Wilby, C. U. Keller, F. Snik, V. Korkiakoski, and A. G. M. Pietrow, “The coronagraphic modal wavefront sensor: a hybrid focal-plane sensor for the high-contrast imaging of circumstellar environments,” Astron. Astrophys. 597, A112 (2017).
[Crossref]

S. Konwar and B. R. Boruah, “Note: Current induced fluctuations in the orientation of the beam diffracted by a liquid crystal spatial light modulator,” Rev. Sci. Instrum. 88, 066104 (2017).
[Crossref] [PubMed]

K. Yao, J. Wang, X. Liu, X. Lin, and L. Chen, “Analysis of a holographic laser adaptive optics system using a deformable mirror,” Appl. Opt. 56, 6639–6648 (2017).
[Crossref] [PubMed]

2016 (1)

2014 (2)

2013 (1)

C. Liu, Y. Yang, S. Guo, R. Xu, T. Men, and C. Wen, “Modal wavefront sensor employing stratified computer-generated holographic elements,” Opt. Lasers Eng. 51, 1265–1271 (2013).
[Crossref]

2012 (1)

2010 (1)

2009 (1)

B. R. Boruah, “Dynamic manipulation of a laser beam using a liquid crystal spatial light modulator,” Am. J. Phys. 77, 331–336 (2009).
[Crossref]

2007 (1)

2003 (1)

M. J. Booth, “Direct measurement of zernike aberration modes with a modal wavefront sensor,” Proc. SPIE 5162, 79–91 (2003).
[Crossref]

2002 (1)

2000 (1)

1998 (1)

1995 (1)

1983 (1)

W. Southwell, “What’s wrong with cross coupling in modal wave-front estimation?” Proc. SPIE 365, 97–105 (1983).
[Crossref]

1981 (1)

1979 (1)

1976 (1)

Alber, L.

F. Loosen, J. Stehr, L. Alber, I. Harder, and N. Lindlein, “A holography-based modal wavefront sensor for the precise positioning of a light emitter using a high-resolution computer-generated hologram,” IEEE Photon. J. 10, 1–11 (2018).
[Crossref]

Bernet, S.

Booth, M.

Booth, M. J.

Boruah, B. R.

S. Konwar and B. R. Boruah, “Note: Current induced fluctuations in the orientation of the beam diffracted by a liquid crystal spatial light modulator,” Rev. Sci. Instrum. 88, 066104 (2017).
[Crossref] [PubMed]

B. R. Boruah, “Dynamic manipulation of a laser beam using a liquid crystal spatial light modulator,” Am. J. Phys. 77, 331–336 (2009).
[Crossref]

Chen, L.

Corbett, A. D.

Cubalchini, R.

Diaz-Santana, L.

Dong, S.

Feng, F.

Guo, S.

C. Liu, Y. Yang, S. Guo, R. Xu, T. Men, and C. Wen, “Modal wavefront sensor employing stratified computer-generated holographic elements,” Opt. Lasers Eng. 51, 1265–1271 (2013).
[Crossref]

Haist, T.

Harder, I.

F. Loosen, J. Stehr, L. Alber, I. Harder, and N. Lindlein, “A holography-based modal wavefront sensor for the precise positioning of a light emitter using a high-resolution computer-generated hologram,” IEEE Photon. J. 10, 1–11 (2018).
[Crossref]

Herrmann, J.

Jesacher, A.

Keller, C. U.

M. J. Wilby, C. U. Keller, F. Snik, V. Korkiakoski, and A. G. M. Pietrow, “The coronagraphic modal wavefront sensor: a hybrid focal-plane sensor for the high-contrast imaging of circumstellar environments,” Astron. Astrophys. 597, A112 (2017).
[Crossref]

Kong, F.

Konwar, S.

S. Konwar and B. R. Boruah, “Note: Current induced fluctuations in the orientation of the beam diffracted by a liquid crystal spatial light modulator,” Rev. Sci. Instrum. 88, 066104 (2017).
[Crossref] [PubMed]

Korkiakoski, V.

M. J. Wilby, C. U. Keller, F. Snik, V. Korkiakoski, and A. G. M. Pietrow, “The coronagraphic modal wavefront sensor: a hybrid focal-plane sensor for the high-contrast imaging of circumstellar environments,” Astron. Astrophys. 597, A112 (2017).
[Crossref]

Lambert, A.

Lin, X.

Lindlein, N.

F. Loosen, J. Stehr, L. Alber, I. Harder, and N. Lindlein, “A holography-based modal wavefront sensor for the precise positioning of a light emitter using a high-resolution computer-generated hologram,” IEEE Photon. J. 10, 1–11 (2018).
[Crossref]

Liu, C.

C. Liu, Y. Yang, S. Guo, R. Xu, T. Men, and C. Wen, “Modal wavefront sensor employing stratified computer-generated holographic elements,” Opt. Lasers Eng. 51, 1265–1271 (2013).
[Crossref]

Liu, X.

Loosen, F.

F. Loosen, J. Stehr, L. Alber, I. Harder, and N. Lindlein, “A holography-based modal wavefront sensor for the precise positioning of a light emitter using a high-resolution computer-generated hologram,” IEEE Photon. J. 10, 1–11 (2018).
[Crossref]

Marshall, G. D.

Men, T.

C. Liu, Y. Yang, S. Guo, R. Xu, T. Men, and C. Wen, “Modal wavefront sensor employing stratified computer-generated holographic elements,” Opt. Lasers Eng. 51, 1265–1271 (2013).
[Crossref]

Neil, M. A.

Neil, M. A. A.

Németh, G.

Noll, R. J.

Osten, W.

Pietrow, A. G. M.

M. J. Wilby, C. U. Keller, F. Snik, V. Korkiakoski, and A. G. M. Pietrow, “The coronagraphic modal wavefront sensor: a hybrid focal-plane sensor for the high-contrast imaging of circumstellar environments,” Astron. Astrophys. 597, A112 (2017).
[Crossref]

Ritsch-Marte, M.

Roider, C.

Ruppel, T.

Sawodny, O.

Snik, F.

M. J. Wilby, C. U. Keller, F. Snik, V. Korkiakoski, and A. G. M. Pietrow, “The coronagraphic modal wavefront sensor: a hybrid focal-plane sensor for the high-contrast imaging of circumstellar environments,” Astron. Astrophys. 597, A112 (2017).
[Crossref]

Southwell, W.

W. Southwell, “What’s wrong with cross coupling in modal wave-front estimation?” Proc. SPIE 365, 97–105 (1983).
[Crossref]

Stehr, J.

F. Loosen, J. Stehr, L. Alber, I. Harder, and N. Lindlein, “A holography-based modal wavefront sensor for the precise positioning of a light emitter using a high-resolution computer-generated hologram,” IEEE Photon. J. 10, 1–11 (2018).
[Crossref]

Török, P.

Varga, P.

Wang, J.

Wen, C.

C. Liu, Y. Yang, S. Guo, R. Xu, T. Men, and C. Wen, “Modal wavefront sensor employing stratified computer-generated holographic elements,” Opt. Lasers Eng. 51, 1265–1271 (2013).
[Crossref]

White, I. H.

Wilby, M. J.

M. J. Wilby, C. U. Keller, F. Snik, V. Korkiakoski, and A. G. M. Pietrow, “The coronagraphic modal wavefront sensor: a hybrid focal-plane sensor for the high-contrast imaging of circumstellar environments,” Astron. Astrophys. 597, A112 (2017).
[Crossref]

Wilkinson, T. D.

Wilson, T.

Xu, R.

C. Liu, Y. Yang, S. Guo, R. Xu, T. Men, and C. Wen, “Modal wavefront sensor employing stratified computer-generated holographic elements,” Opt. Lasers Eng. 51, 1265–1271 (2013).
[Crossref]

Yang, Y.

C. Liu, Y. Yang, S. Guo, R. Xu, T. Men, and C. Wen, “Modal wavefront sensor employing stratified computer-generated holographic elements,” Opt. Lasers Eng. 51, 1265–1271 (2013).
[Crossref]

Yao, K.

Zhong, J. J.

Am. J. Phys. (1)

B. R. Boruah, “Dynamic manipulation of a laser beam using a liquid crystal spatial light modulator,” Am. J. Phys. 77, 331–336 (2009).
[Crossref]

Appl. Opt. (1)

Astron. Astrophys. (1)

M. J. Wilby, C. U. Keller, F. Snik, V. Korkiakoski, and A. G. M. Pietrow, “The coronagraphic modal wavefront sensor: a hybrid focal-plane sensor for the high-contrast imaging of circumstellar environments,” Astron. Astrophys. 597, A112 (2017).
[Crossref]

IEEE Photon. J. (1)

F. Loosen, J. Stehr, L. Alber, I. Harder, and N. Lindlein, “A holography-based modal wavefront sensor for the precise positioning of a light emitter using a high-resolution computer-generated hologram,” IEEE Photon. J. 10, 1–11 (2018).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Opt. Express (2)

Opt. Lasers Eng. (1)

C. Liu, Y. Yang, S. Guo, R. Xu, T. Men, and C. Wen, “Modal wavefront sensor employing stratified computer-generated holographic elements,” Opt. Lasers Eng. 51, 1265–1271 (2013).
[Crossref]

Opt. Lett. (1)

Proc. SPIE (2)

M. J. Booth, “Direct measurement of zernike aberration modes with a modal wavefront sensor,” Proc. SPIE 5162, 79–91 (2003).
[Crossref]

W. Southwell, “What’s wrong with cross coupling in modal wave-front estimation?” Proc. SPIE 365, 97–105 (1983).
[Crossref]

Rev. Sci. Instrum. (1)

S. Konwar and B. R. Boruah, “Note: Current induced fluctuations in the orientation of the beam diffracted by a liquid crystal spatial light modulator,” Rev. Sci. Instrum. 88, 066104 (2017).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1 Working principle of a modal wavefront sensor.
Fig. 2
Fig. 2 The plots of Fa, c for (a) the sensor mode Z4 in the presence of input modes Z5, Z7, Z9 and Z11 and (b) the sensor mode Z5 in the presence of input modes Z4, Z7, Z9 and Z11. The data obtained using the integral form of the sensor output for the sensor mode Zs and the input mode Zc is indicated as Zs Zct and the data obtained using the polynomial form of Fa, c for the sensor mode Zs and the input mode Zc is indicated as Zs Zcp.
Fig. 3
Fig. 3 Diagram of the experimental arrangement.
Fig. 4
Fig. 4 Plots of Fa, c vs c for the sensor mode Z4 obtained (a) using the polynomial and (b) experimentally. Similarly plots of Fa, c vs c for the sensor mode Z5 obtained (c) using the polynomial and (d) experimentally. The cross talk co-efficient for the sensor mode Zs and the input mode Zc is indicated as Zs Zc.

Tables (2)

Tables Icon

Table 1 Table of Co-Efficients in the Fa, c Polynomial

Tables Icon

Table 2 Table of Fa, c for Some Low Order Zernike Modes

Equations (29)

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F [ E ( r , θ ) ] = 0 0 2 π E ( r , θ ) exp { i 2 π ρ r cos ( ϕ θ ) } r d r d θ ,
Z e v e n j = n + 1 R n m ( r ) 2 cos ( m θ ) Z o d d j = n + 1 R n m ( r ) 2 sin ( m θ ) } for m 0 , Z j = n + 1 R n m ( r ) for m 0 ,
R n m ( r ) = q = 0 ( n m ) 2 ( 1 ) q ( n q ) ! q ! [ n + m 2 q ] ! [ n m 2 q ] ! r n 2 q .
| 0 2 π exp { i j = 1 a j Z j ( r , θ ) } exp { i 2 π ρ r cos ( ϕ θ ) } r d r d θ | 2 .
I a , c ( 0 ) = | 0 0 2 π exp { i a Z s ( r , θ ) + i c Z c ( r , θ ) } r d r d θ | 2 .
S o = I a b , c ( 0 ) I a + b , c ( 0 ) I a b , c ( 0 ) + I a + b , c ( 0 ) = δ I a , b , c σ I a , b , c .
I a , c ( 0 ) = | 0 0 2 π p = 0 [ i a Z s ] p p ! q = 0 [ i c Z c ] q q ! r d r d θ | 2 ,
= | 0 0 2 π p , q = 0 i ( p + q ) a p Z s   p c q Z c   q p ! q ! r d r d θ | 2 .
I a , c ( 0 ) = | 0 0 2 π p + q = e v e n ( 1 ) p + q 2 a p Z s   p c q Z c   q p ! q ! r d r d θ + i 0 0 2 π p + q = o d d ( 1 ) p + q 1 2 a p Z s   p c q Z c   q p ! q ! r d r d θ | 2 .
I a , c ( 0 ) = [ u , v = 0 f e ( u , v ) + u , v = 0 f e ( u + 1 2 , v + 1 2 ) ] 2 + [ u , v = 0 f o ( u , v + 1 2 ) + u , v = 0 f o ( u + 1 2 , v ) ] 2 ,
f e ( u , v ) = 0 0 2 π ( 1 ) 2 u + 2 v 2 a 2 u Z s   2 u c 2 v Z c   2 v ( 2 u ) ! ( 2 v ) ! r d r d θ f o ( u , v ) = 0 0 2 π ( 1 ) 2 u + 2 v 1 2 a 2 u Z s   2 u c 2 v Z c   2 v ( 2 u ) ! ( 2 v ) ! r d r d θ .
I a , c ( 0 ) = [ u , v = 0 f e ( u , v ) ] 2 + [ u , v = 0 f e ( u + 1 2 , v + 1 2 ) ] 2 + 2 u , v = 0 f e ( u , v ) u , v = 0 f e ( u + 1 2 , v + 1 2 ) + [ u , v = 0 f o ( u , v + 1 2 ) ] 2 + [ u , v = 0 f o ( u + 1 2 , v ) ] 2 + 2 u , v = 0 f o ( u , v + 1 2 ) u , v = 0 f o ( u + 1 2 , v ) .
I a , c ( 0 ) = [ u = 0 f e ( u , 0 ) + u = 0 , v = 0 f e ( u , v ) ] 2 + [ u , v = 0 f e ( u + 1 2 , v + 1 2 ) ] 2 + 2 u , v = 0 f e ( u , v ) u , v = 0 f e ( u + 1 2 , v + 1 2 ) + [ u , v = 0 f o ( u , v + 1 2 ) ] 2 + [ u = 0 f o ( u + 1 2 , 0 ) + u = 0 , v = 1 f o ( u + 1 2 , v ) ] 2 + 2 u , v = 0 f o ( u , v + 1 2 ) u , v = 0 f o ( u + 1 2 , v ) .
I a , c ( 0 ) = [ u = 0 f e ( u , 0 ) ] 2 + [ u = 0 f o ( u + 1 2 , 0 ) ] 2 + [ u , v = 0 f e ( u + 1 2 , v + 1 2 ) ] 2 + [ v , u = 0 f o ( u , v 1 2 ) ] 2 + [ u = 0 , v = 0 f o ( u + 1 2 , v ) ] 2 + [ u = 0 , v = 1 f e ( u , v ) ] 2 + 2 [ u = 0 f e ( u , 0 ) + u = 0 , v = 1 f e ( u , v ) + u , v = 0 f e ( u , v ) u , v = 0 f e ( u + 1 2 , v + 1 2 ) + u = 0 f o ( u + 1 2 , 0 ) u = 0 , v = 1 f o ( u + 1 2 , v ) + u , v = 0 f o ( u , v + 1 2 ) u , v = 0 f o ( u + 1 2 , v ) ] ,
I a , c ( 0 ) = I a N C ( 0 ) + I a , c C ( 0 ) ,
I a N C ( 0 ) = [ u = 0 f e ( u , 0 ) ] 2 + [ u = 0 f o ( u + 1 2 , 0 ) ] 2 ,
I a , c C ( 0 ) = [ u , v = 0 f e ( u + 1 2 , v + 1 2 ) ] 2 + [ u , v = 0 f o ( u , v + 1 2 ) ] 2 + [ u = 0 , v = 1 f o ( u + 1 2 , v ) ] 2 + [ u = 0 , v = 1 f e ( u , v ) ] 2 + 2 [ u = 0 f e ( u , 0 ) u = 0 , v = 0 f e ( u , v ) + u , v = 0 f e ( u , v ) u , v = 0 f e ( u + 1 2 , v + 1 2 ) + u = 0 f o ( u + 1 2 , 0 ) u = 0 , v = 1 f o ( u + 1 2 , v ) + u , v = 0 f o ( u , v + 1 2 ) u , v = 0 f o ( u + 1 2 , v ) ] .
S o = δ I a , b N C + δ I a , b , c C σ I a , b N C + σ I a , b , c C ,
δ I a , b N C = I a b N C ( 0 ) I a + b N C ( 0 ) δ I a , b , c N C = I a b , c C ( 0 ) I a + b , c C ( 0 ) δ I a , b , c C = I a b , c C ( 0 ) + I a + b N C ( 0 ) σ I a , b , c C = I a b , c C ( 0 ) + I a + b , c C ( 0 ) .
S o = S o N C × F a , c ,
S o N C = δ I a , b N C σ I a , b N C = Ideal sensor output ,
F a , c = 1 + δ I a , b , c C δ I a , b N C 1 + σ I a , b , c C σ I a , b N C = ( 1 + δ I a , b , c C δ I a , b N C ) ( 1 + σ I a , b , c C σ I a , b N C ) 1 .
ϕ R M S ( F a , c ) = λ 2 π a ( 1 F a , c ) ,
F a , c = n = 0 ( 1 ) n [ ( σ I a , b , c C σ I a , b N C ) n + δ I a , b , c C δ I a , b N C ( σ I a , b , c C σ I a , b N C ) n ] .
F a , c = 1 + c A 1 + c 2 A 2 + c 3 A 3 + c 4 A 4 + c 5 A 5
A 1 = δ C 1 σ C 1 A 2 = δ C 2 σ C 2 + σ C 1 2 δ C 1 σ C 1 A 3 = δ C 3 σ C 3 + 2 σ C 1 σ C 2 σ C 1 3 δ C 2 σ C 1 δ C 1 σ C 2 + δ C 1 σ C 1 2 A 4 = δ C 4 σ C 4 + σ C 2 2 + 2 σ C 1 σ C 3 3 σ C 1 2 σ C 2 + σ C 1 4 δ C 3 σ C 1 + δ C 2 σ C 1 2 + 2 δ C 1 σ C 1 σ C 2 δ C 2 σ C 2 δ C 1 σ C 3 δ C 1 σ C 1 3 A 5 = δ C 5 σ C 5 + 2 σ C 1 σ C 4 + 2 σ C 2 σ C 3 3 σ C 1 2 σ C 3 3 σ C 2 2 σ C 1 + 4 σ C 1 3 σ C 2 σ C 1 5 δ C 4 σ C 1 δ C 3 σ C 2 δ C 2 σ C 3 δ C 1 σ C 4 + δ C 3 σ C 1 2 + 2 δ C 2 σ C 1 σ C 2 + δ C 1 σ C 2 2 + 2 δ C 1 σ C 1 σ C 3 δ C 2 σ C 1 3 3 δ C 1 σ C 1 2 σ C 2 + δ C 1 σ C 1 4 .
σ C 1 , 2 , 3 , 4 , 5 = ( 1 σ I a , b N C ) { C 1 , 2 , 3 , 4 , 5 ( a + b ) } δ C 1 , 2 , 3 , 4 , 5 = ( 1 δ I a , b N C ) { C 1 , 2 , 3 , 4 , 5 ( a b ) C 1 , 2 , 3 , 4 , 5 ( a + b ) } ,
C 1 ( a ) = 1 c { 2 u = 0 f o ( u , 1 2 ) u = 0 f o ( u + 1 2 , 0 ) + 2 u = 0 f e ( u , 0 ) u = 0 f e ( u + 1 2 , 1 2 ) } C 2 ( a ) = 1 c 2 { | u = 0 f e ( u + 1 2 , 1 2 ) | 2 + | u = 0 f o ( u , 1 2 ) | 2 + 2 u = 0 f e ( u , 0 ) u = 0 f e ( u , 1 ) + 2 u = 0 f o ( u + 1 2 , 0 ) u = 0 f o ( u + 1 2 , 1 ) } C 3 ( a ) = 1 c 3 { 2 u = 0 f e ( u , 0 ) u = 0 f e ( u + 1 2 , 3 2 ) + 2 u = 0 f e ( u , 1 ) u = 0 f e ( u + 1 2 , 1 2 ) + 2 u = 0 f o ( u , 1 2 ) u = 0 f o ( u + 1 2 , 1 ) + 2 u = 0 f o ( u , 3 2 ) u = 0 f o ( u + 1 2 , 0 ) } , C 4 ( a ) = 1 c 4 { | u = 0 f o ( u + 1 2 , 1 ) | 2 + | u = 0 f e ( u , 1 ) | 2 + 2 u = 0 f e ( u + 1 2 , 1 2 ) u = 0 f e ( u + 1 2 , 3 2 ) + 2 u = 0 f o ( u , 1 2 ) u = 0 f o ( u , 3 2 ) + 2 u = 0 f e ( u , 0 ) u = 0 f e ( u , 2 ) + 2 u = 0 f o ( u + 1 2 , 0 ) u = 0 f o ( u + 1 2 , 2 ) } , C 5 ( a ) = 1 c 5 { 2 u = 0 f e ( u , 0 ) u = 0 f e ( u + 1 2 , 5 2 ) + 2 u = 0 f e ( u , 1 ) u = 0 f e ( u + 1 2 , 3 2 ) + 2 u = 0 f e ( u , 2 ) u = 0 f e ( u + 1 2 , 1 2 ) + 2 u = 0 f o ( u , 1 2 ) u = 0 f o ( u + 1 2 , 2 ) + 2 u = 0 f o ( u , 3 2 ) u = 0 f o ( u + 1 2 , 1 ) + 2 u = 0 f o ( u , 5 2 ) u = 0 f o ( u + 1 2 , 0 ) } .
F a , c = ( δ I a , b , c σ I a , b , c ) ( δ I a , b , c = 0 σ I a , b , c = 0 ) 1 .

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