Abstract

We study the optical properties of a two-axis galvanometric optical scanner constituted by a pair of rotating planar mirrors, focusing our attention on the transformation induced on the polarization state of the input beam. We obtain the matrix that defines the transformation of the propagation direction of the beam and the Jones matrix that defines the transformation of the polarization state. Both matrices are expressed in terms of the rotation angles of two mirrors. Finally, we calculate the parameters of the general rotation in the Poincaré sphere that describes the change in the polarization state for each mutual orientation of the mirrors.

© 2010 Optical Society of America

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References

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2009 (1)

C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. 11, 045017 (2009).
[CrossRef]

2008 (1)

2007 (1)

C. Bonato, C. Pernechele, and P. Villoresi, “Influence of all-reflective optical systems in the transmission of polarization-encoded qubits,” J. Opt. Soc. Am. A 9, 899–906 (2007).

2006 (1)

2005 (2)

C. Beck, R. Schlichenmaier, M. Collados, L. Bellot Rubio, and T. Kentischer, “A polarization model for the German Vacuum Tower Telescope from in situ and laboratory measurements,” Astron. Astrophys. 443, 1047–1053 (2005).
[CrossRef]

J. Xie, S. Huang, Z. Duan, Y. Shi, and S. Wen, “Correction of the image distortion for laser galvanometric scanning system,” Opt. Laser Technol. 37, 305–311 (2005).
[CrossRef]

1995 (3)

1980 (1)

1976 (1)

E. F. Borra, “Polarimetry at the Coude focus—Instrumental effects,” Publ. Astron. Soc. Pac. 88, 548–556 (1976).
[CrossRef]

1974 (1)

Aspelmeyer, M.

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Bass, M.

M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, Handbook of Optics: Fundamentals, Techniques, and Design (McGraw-Hill, 1995), Vol. I.

Beck, C.

C. Beck, R. Schlichenmaier, M. Collados, L. Bellot Rubio, and T. Kentischer, “A polarization model for the German Vacuum Tower Telescope from in situ and laboratory measurements,” Astron. Astrophys. 443, 1047–1053 (2005).
[CrossRef]

Bellot Rubio, L.

C. Beck, R. Schlichenmaier, M. Collados, L. Bellot Rubio, and T. Kentischer, “A polarization model for the German Vacuum Tower Telescope from in situ and laboratory measurements,” Astron. Astrophys. 443, 1047–1053 (2005).
[CrossRef]

Blaszczak, Z.

Bonato, C.

C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. 11, 045017 (2009).
[CrossRef]

C. Bonato, C. Pernechele, and P. Villoresi, “Influence of all-reflective optical systems in the transmission of polarization-encoded qubits,” J. Opt. Soc. Am. A 9, 899–906 (2007).

C. Bonato, M. Aspelmeyer, T. Jennewein, C. Pernechele, P. Villoresi, and A. Zeilinger, “Influence of satellite motion on polarization qubits in a Space-Earth quantum communication link,” Opt. Express 14, 10050–10059 (2006).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Borra, E. F.

E. F. Borra, “Polarimetry at the Coude focus—Instrumental effects,” Publ. Astron. Soc. Pac. 88, 548–556 (1976).
[CrossRef]

Chipman, R. A.

R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. (Bellingham) 34, 1636–1645 (1995).
[CrossRef]

Collados, M.

C. Beck, R. Schlichenmaier, M. Collados, L. Bellot Rubio, and T. Kentischer, “A polarization model for the German Vacuum Tower Telescope from in situ and laboratory measurements,” Astron. Astrophys. 443, 1047–1053 (2005).
[CrossRef]

Da Deppo, V.

C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. 11, 045017 (2009).
[CrossRef]

Drexler, W.

W. Drexler and J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications (Springer, 2008).
[CrossRef]

Duan, Z.

J. Xie, S. Huang, Z. Duan, Y. Shi, and S. Wen, “Correction of the image distortion for laser galvanometric scanning system,” Opt. Laser Technol. 37, 305–311 (2005).
[CrossRef]

Fujimoto, J. G.

W. Drexler and J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications (Springer, 2008).
[CrossRef]

Garrison, L. M.

Green, A. E. S.

Hariharan, P.

P. Hariharan, Optical Interferometry, 2nd ed. (Academic, 2003).

Huang, S.

J. Xie, S. Huang, Z. Duan, Y. Shi, and S. Wen, “Correction of the image distortion for laser galvanometric scanning system,” Opt. Laser Technol. 37, 305–311 (2005).
[CrossRef]

Jennewein, T.

Katz, J.

Kentischer, T.

C. Beck, R. Schlichenmaier, M. Collados, L. Bellot Rubio, and T. Kentischer, “A polarization model for the German Vacuum Tower Telescope from in situ and laboratory measurements,” Astron. Astrophys. 443, 1047–1053 (2005).
[CrossRef]

Li, Y.

Marshall, G. F.

G. F. Marshall, Handbook of Optical and Laser Scanning (CRC, 2004).
[CrossRef]

Naletto, G.

C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. 11, 045017 (2009).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1991).

Pernechele, C.

C. Bonato, C. Pernechele, and P. Villoresi, “Influence of all-reflective optical systems in the transmission of polarization-encoded qubits,” J. Opt. Soc. Am. A 9, 899–906 (2007).

C. Bonato, M. Aspelmeyer, T. Jennewein, C. Pernechele, P. Villoresi, and A. Zeilinger, “Influence of satellite motion on polarization qubits in a Space-Earth quantum communication link,” Opt. Express 14, 10050–10059 (2006).
[CrossRef] [PubMed]

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics (Addison Wesley, 1985).

Schlichenmaier, R.

C. Beck, R. Schlichenmaier, M. Collados, L. Bellot Rubio, and T. Kentischer, “A polarization model for the German Vacuum Tower Telescope from in situ and laboratory measurements,” Astron. Astrophys. 443, 1047–1053 (2005).
[CrossRef]

Shi, Y.

J. Xie, S. Huang, Z. Duan, Y. Shi, and S. Wen, “Correction of the image distortion for laser galvanometric scanning system,” Opt. Laser Technol. 37, 305–311 (2005).
[CrossRef]

Stryland, E. W. V.

M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, Handbook of Optics: Fundamentals, Techniques, and Design (McGraw-Hill, 1995), Vol. I.

Tomaello, A.

C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. 11, 045017 (2009).
[CrossRef]

Villoresi, P.

C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. 11, 045017 (2009).
[CrossRef]

C. Bonato, C. Pernechele, and P. Villoresi, “Influence of all-reflective optical systems in the transmission of polarization-encoded qubits,” J. Opt. Soc. Am. A 9, 899–906 (2007).

C. Bonato, M. Aspelmeyer, T. Jennewein, C. Pernechele, P. Villoresi, and A. Zeilinger, “Influence of satellite motion on polarization qubits in a Space-Earth quantum communication link,” Opt. Express 14, 10050–10059 (2006).
[CrossRef] [PubMed]

Wen, S.

J. Xie, S. Huang, Z. Duan, Y. Shi, and S. Wen, “Correction of the image distortion for laser galvanometric scanning system,” Opt. Laser Technol. 37, 305–311 (2005).
[CrossRef]

Williams, D. R.

M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, Handbook of Optics: Fundamentals, Techniques, and Design (McGraw-Hill, 1995), Vol. I.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Wolfe, W. L.

M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, Handbook of Optics: Fundamentals, Techniques, and Design (McGraw-Hill, 1995), Vol. I.

Xie, J.

J. Xie, S. Huang, Z. Duan, Y. Shi, and S. Wen, “Correction of the image distortion for laser galvanometric scanning system,” Opt. Laser Technol. 37, 305–311 (2005).
[CrossRef]

Zeilinger, A.

Zook, J. D.

Appl. Opt. (5)

Astron. Astrophys. (1)

C. Beck, R. Schlichenmaier, M. Collados, L. Bellot Rubio, and T. Kentischer, “A polarization model for the German Vacuum Tower Telescope from in situ and laboratory measurements,” Astron. Astrophys. 443, 1047–1053 (2005).
[CrossRef]

J. Opt. Soc. Am. A (1)

C. Bonato, C. Pernechele, and P. Villoresi, “Influence of all-reflective optical systems in the transmission of polarization-encoded qubits,” J. Opt. Soc. Am. A 9, 899–906 (2007).

New J. Phys. (1)

C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. 11, 045017 (2009).
[CrossRef]

Opt. Eng. (Bellingham) (1)

R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. (Bellingham) 34, 1636–1645 (1995).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

J. Xie, S. Huang, Z. Duan, Y. Shi, and S. Wen, “Correction of the image distortion for laser galvanometric scanning system,” Opt. Laser Technol. 37, 305–311 (2005).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

E. F. Borra, “Polarimetry at the Coude focus—Instrumental effects,” Publ. Astron. Soc. Pac. 88, 548–556 (1976).
[CrossRef]

Other (8)

M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, Handbook of Optics: Fundamentals, Techniques, and Design (McGraw-Hill, 1995), Vol. I.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1991).

J. J. Sakurai, Modern Quantum Mechanics (Addison Wesley, 1985).

G. F. Marshall, Handbook of Optical and Laser Scanning (CRC, 2004).
[CrossRef]

P. Hariharan, Optical Interferometry, 2nd ed. (Academic, 2003).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

W. Drexler and J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications (Springer, 2008).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

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Figures (5)

Fig. 1
Fig. 1

Schematic view of the galvo scanning system considered in this work, together with the general coordinate system and the unit vectors used in the calculations. An example of the optical path of a beam is indicated by the thick (red online) line.

Fig. 2
Fig. 2

Azimuth and ellipticity of output polarization states from a galvo scanner. (a),(b) β = 20 ° ; (c),(d) β = 0 ° ; (e),(f) β = + 20 ° . Dark lines refer to the output states calculated using the Jones matrix of Eq. (27), while light lines have been obtained assuming lossless mirrors. The polarization states of the input beam are H (solid lines), V (dashed lines), + 45 (dashed-dotted lines), and −45 (dotted lines). In the right panels, dashed lines appear superimposed on solid lines, while dotted lines appear superimposed on dashed-dotted lines.

Fig. 3
Fig. 3

Azimuth and ellipticity of output polarization states from a galvo scanner. (a),(b) β = 20 ° ; (c),(d) β = 0 ° ; (e),(f) β = + 20 ° . Dark lines refer to the output states calculated using the Jones matrix of Eq. (27), while light lines have been obtained assuming lossless mirrors. The polarization states of the input beam are R (solid lines) and L (dashed lines). In the right panels, all lines appear superimposed.

Fig. 4
Fig. 4

Traces of the direction of the rotation axis associated with the Jones matrix of a galvo scanner as a function of the α and β angles. The lines are traced on the surface of the Poincaré sphere, whose bottom part is depicted with a gray grid. Radial lines correspond to meridians drawn at 15° intervals, while circles correspond to parallels drawn at 10° intervals starting from the south pole. Dark (blue online) lines refer to a fixed β ( β = 20 ° , dashed line; β = 0 ° , solid line; β = 20 ° , dashed-dotted line) and 20 ° α 20 ° , while light (red online) lines refer to a fixed α ( α = 20 ° , dashed line; α = 0 ° , solid line; α = 20 ° , dashed-dotted line) and 20 ° β 20 ° .

Fig. 5
Fig. 5

Rotation angle δ associated with the Jones matrix of a galvo scanner as a function of the α and β angles. Dark (blue online) lines refer to a fixed β ( β = 20 ° , dashed line; β = 0 ° , solid line; β = 20 ° , dashed-dotted line) and 20 ° α 20 ° , while light (red online) lines refer to a fixed α ( α = 20 ° , dashed line; α = 0 ° , solid line; α = 20 ° , dashed-dotted line) and 20 ° β 20 ° .

Equations (39)

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N M 1 = [ cos ( α + 45 ° ) sin ( α + 45 ° ) sin   γ 1 sin ( α + 45 ° ) cos   γ 1 ] .
N M 2 = [ 0 cos ( γ 2 + β ) sin ( γ 2 + β ) ] .
R = [ 1 2 N x 2 2 N x N y 2 N x N z 2 N x N y 1 2 N y 2 2 N y N z 2 N x N z 2 N y N z 1 2 N z 2 ] .
R M 1 = [ sin ( 2 α ) cos ( 2 α ) sin   γ 1 cos ( 2 α ) cos   γ 1 cos ( 2 α ) sin   γ 1 cos 2 γ 1 sin ( 2 α ) sin 2 γ 1 [ 1 + sin ( 2 α ) ] cos   γ 1   sin   γ 1 cos ( 2 α ) cos   γ 1 [ 1 + sin ( 2 α ) ] cos   γ 1   sin   γ 1 sin 2 γ 1 sin ( 2 α ) cos 2 γ 1 ] .
R M 2 = [ 1 0 0 0 cos [ 2 ( γ 2 + β ) ] sin [ 2 ( γ 2 + β ) ] 0 sin [ 2 ( γ 2 + β ) ] cos [ 2 ( γ 2 + β ) ] ] = [ 1 0 0 0 sin ( γ 1 + 2 β ) cos ( γ 1 + 2 β ) 0 cos ( γ 1 + 2 β ) sin ( γ 1 + 2 β ) ] .
G = [ sin ( 2 α ) cos ( 2 α ) sin   γ 1 cos ( 2 α ) cos   γ 1 cos ( 2 α ) cos ( 2 β ) sin ( 2 α ) cos ( 2 β ) sin   γ 1 sin ( 2 β ) cos   γ 1 sin ( 2 α ) cos ( 2 β ) cos   γ 1 sin ( 2 β ) sin   γ 1 cos ( 2 α ) sin ( 2 β ) sin ( 2 α ) sin ( 2 β ) sin   γ 1 + cos ( 2 β ) cos   γ 1 sin ( 2 α ) sin ( 2 β ) cos   γ 1 + cos ( 2 β ) sin   γ 1 ] .
k 3 = [ sin ( 2 α ) cos ( 2 α ) cos ( 2 β ) cos ( 2 α ) sin ( 2 β ) ] .
r s ( n ̃ , θ i ) = sin ( θ i θ t ) sin ( θ i + θ t ) ,
r p ( n ̃ , θ i ) = tan ( θ i θ t ) tan ( θ i + θ t ) ,
cos   θ 1 = k 1 N M 1
k 2 = R M 1 k 1 .
s 1 = k 1 × k 2 | k 1 × k 2 | ,     p 1 = s 1 × k 1 | s 1 × k 1 | .
[ E 1 p E 1 s ] = R ( η 0 ) [ E 1 y E 1 z ] ,
R ( ϑ ) = [ cos   ϑ sin   ϑ sin   ϑ sin   ϑ ] ,
c = a × b | a × b | ,
ϕ = ang ( a , b ) = u   arccos ( a b ) .
s 2 = s 1 ,     p 2 = s 2 × k 2 | s 2 × k 2 | .
[ E 2 p E 2 s ] = [ r 1 p ( n ̃ 1 , θ 1 ) 0 0 r 1 s ( n ̃ 1 , θ 1 ) ] [ E 1 p E 1 s ] ,
cos   θ 2 = k 2 N M 2 ,
k 3 = R M 2 k 2 = G k 1 .
s 1 = k 2 × k 3 | k 2 × k 3 | ,     p 1 = s 1 × k 2 | s 1 × k 2 | ,
s 2 = s 1 ,     p 2 = s 2 × k 2 | s 2 × k 2 | ,
[ E 2 p E 2 s ] = R ( η 1 ) [ E 2 p E 2 s ] ,
[ E 3 p E 3 s ] = [ r 2 p ( n ̃ 2 , θ 2 ) 0 0 r 2 s ( n ̃ 2 , θ 2 ) ] [ E 2 p E 2 s ] ,
h = z × k 3 | z × k 3 | ,     v = k 3 × h | k 3 × h | .
[ E 3 h E 3 v ] = R ( η 2 ) [ E 3 p E 3 s ] ,
M = [ cos   η 2 sin   η 2 sin   η 2 cos   η 2 ] [ r 2 p 0 0 r 2 s ] [ cos   η 1 sin   η 1 sin   η 1 cos   η 1 ] [ r 1 p 0 0 r 1 s ] [ cos   η 0 sin   η 0 sin   η 0 cos   η 0 ] ,
A = [ r p 0 0 r s ] = [ ρ p   exp ( i ϕ p ) 0 0 ρ s   exp ( i ϕ s ) ] ,
A = ρ p ρ s   exp ( i ϕ p + ϕ s 2 ) [ A   exp ( i Φ ) 0 0 A 1   exp ( i Φ ) ] ,
A = ρ p ρ s   exp ( i ϕ p + ϕ s 2 ) [ exp ( i Φ ) 0 0 exp ( i Φ ) ] .
M [ cos   η 2 sin   η 2 sin   η 2 cos   η 2 ] [ exp ( i Φ 2 ) 0 0 exp ( i Φ 2 ) ] [ cos   η 1 sin   η 1 sin   η 1 cos   η 1 ] [ exp ( i Φ 1 ) 0 0 exp ( i Φ 1 ) ] [ cos   η 0 sin   η 0 sin   η 0 cos   η 0 ] ,
M exp ( i σ y η 2 ) exp ( i σ z Φ 2 ) exp ( i σ y η 1 ) exp ( i σ z Φ 1 ) exp ( i σ y η 0 ) ,
σ x = [ 0 1 1 0 ] ,     σ y = [ 0 i i 0 ] ,     σ z = [ 1 0 0 1 ] .
U = exp ( i n σ δ 2 ) ,
U = a 0 1 + i a σ = [ a 0 + i a z i a x + a y i a x a y a 0 i a z ] ,
a 0 = sin   Φ 1   sin   Φ 2   cos ( η 0 η 1 + η 2 ) + cos   Φ 1   cos   Φ 2   cos ( η 0 + η 1 + η 2 ) ,
a x = sin   Φ 1   cos   Φ 2   sin ( η 0 η 1 η 2 ) cos   Φ 1   sin   Φ 2   sin ( η 0 + η 1 η 2 ) ,
a y = cos   Φ 1   cos   Φ 2   sin ( η 0 + η 1 + η 2 ) + sin   Φ 1   sin   Φ 2   sin ( η 0 η 1 + η 2 ) ,
a z = sin   Φ 1   cos   Φ 2   cos ( η 0 η 1 η 2 ) + cos   Φ 1   sin   Φ 2   cos ( η 0 + η 1 η 2 ) .

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