Abstract

A laser beam with circular polarization can be converted into either radial or azimuthal polarization by a microfabricated spiral phase plate and a radial (or azimuthal)-type linear analyzer. The resulting polarization is axially symmetric and is able to produce tightly focused light fields beyond the diffraction limit. We describe in detail the theory behind the technique and the experimental verification of the polarization both in the far field and at the focus of a high numerical aperture lens. Vector properties of the beam under strong focusing conditions were observed by comparing the fluorescence images corresponding to the focal intensity distribution for both radial and azimuthal polarizations. The technique discussed here may easily be implemented to a wide range of optical instruments and devices that require the use of tightly focused light beams.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2007 (2)

2006 (3)

J.-l. Li, K.-i. Ueda, M. Musha, A. Shirakawa, and L.-X. Zhong, "Generation of radially polarized mode in Yb fiber laser by using a dual conical prism," Opt. Lett. 31, 2969-2971 (2006).
[CrossRef] [PubMed]

R. Hongwen, L. Yi-Hsin, and W. Shin-Tson, "Linear to axial or radial polarization conversion using a liquid crystal gel," Appl. Phys. Lett. 89, 051114 (2006).
[CrossRef]

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, "Direct noninterference cylindrical vector beam generation applied in the femtosecond regime," Appl. Phys. Lett. 89, 251114 (2006).
[CrossRef]

2005 (3)

2004 (4)

D. P. Biss and T. G. Brown, "Primary aberrations in focused radially polarized vortex beams," Opt. Express 12, 384-393 (2004).
[CrossRef] [PubMed]

C. J. R. Sheppard and A. Choudhury, "Annular pupils, radial polarization, and superresolution," Appl. Opt. 43, 4322-4327 (2004).
[CrossRef] [PubMed]

N. Hayazawa, Y. Saito, and S. Kawata, "Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy," Appl. Phys. Lett. 85, 6239-6241 (2004).
[CrossRef]

E. Descrovi, L. Vaccaro, W. Nakagawa, L. Aeschimann, U. Staufer, and H. P. Herzig, "Collection of transverse and longitudinal fields by means of apertureless nanoprobes with different metal coating characteristics," Appl. Phys. Lett. 85, 5340-5342 (2004).
[CrossRef]

2003 (3)

2002 (2)

2001 (3)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "The focus of light--theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).

Z. Bomzon, V. Kleiner, and E. Hasman, "Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings," Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

L. E. Helseth, "Roles of polarization, phase and amplitude in solid immersion lens systems," Opt. Commun. 191, 161-172 (2001).
[CrossRef]

2000 (3)

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

K. Youngworth and T. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000).
[CrossRef] [PubMed]

1999 (2)

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D 32, 2871-2875 (1999).
[CrossRef]

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

1996 (1)

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

1993 (1)

1990 (1)

Aeschimann, L.

E. Descrovi, L. Vaccaro, W. Nakagawa, L. Aeschimann, U. Staufer, and H. P. Herzig, "Collection of transverse and longitudinal fields by means of apertureless nanoprobes with different metal coating characteristics," Appl. Phys. Lett. 85, 5340-5342 (2004).
[CrossRef]

Aït-Ameur, K.

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

Biss, D. P.

Blit, S.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Bomzon, Z.

Z. Bomzon, V. Kleiner, and E. Hasman, "Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings," Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Brown, T.

Brown, T. G.

Bu, J.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, "Direct noninterference cylindrical vector beam generation applied in the femtosecond regime," Appl. Phys. Lett. 89, 251114 (2006).
[CrossRef]

Burge, R. E.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, "Direct noninterference cylindrical vector beam generation applied in the femtosecond regime," Appl. Phys. Lett. 89, 251114 (2006).
[CrossRef]

Choudhury, A.

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

Cooper, I. J.

Courjon, D.

T. Grosjean and D. Courjon, "Smallest focal spots," Opt. Commun. 272, 314-319 (2007).
[CrossRef]

Davidson, N.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

de Saint Denis, R.

Descrovi, E.

E. Descrovi, L. Vaccaro, W. Nakagawa, L. Aeschimann, U. Staufer, and H. P. Herzig, "Collection of transverse and longitudinal fields by means of apertureless nanoprobes with different metal coating characteristics," Appl. Phys. Lett. 85, 5340-5342 (2004).
[CrossRef]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "The focus of light--theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "The focus of light--theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Ford, D. H.

Friesem, A. A.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "The focus of light--theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Glur, H.

Grosjean, T.

T. Grosjean and D. Courjon, "Smallest focal spots," Opt. Commun. 272, 314-319 (2007).
[CrossRef]

Hasman, E.

Z. Bomzon, V. Kleiner, and E. Hasman, "Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings," Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Hayazawa, N.

N. Hayazawa, Y. Saito, and S. Kawata, "Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy," Appl. Phys. Lett. 85, 6239-6241 (2004).
[CrossRef]

Helseth, L. E.

L. E. Helseth, "Roles of polarization, phase and amplitude in solid immersion lens systems," Opt. Commun. 191, 161-172 (2001).
[CrossRef]

Herzig, H. P.

E. Descrovi, L. Vaccaro, W. Nakagawa, L. Aeschimann, U. Staufer, and H. P. Herzig, "Collection of transverse and longitudinal fields by means of apertureless nanoprobes with different metal coating characteristics," Appl. Phys. Lett. 85, 5340-5342 (2004).
[CrossRef]

Hierle, R.

Hong, M. H.

Hongwen, R.

R. Hongwen, L. Yi-Hsin, and W. Shin-Tson, "Linear to axial or radial polarization conversion using a liquid crystal gel," Appl. Phys. Lett. 89, 051114 (2006).
[CrossRef]

Jackel, S.

Kawata, S.

N. Hayazawa, Y. Saito, and S. Kawata, "Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy," Appl. Phys. Lett. 85, 6239-6241 (2004).
[CrossRef]

Kim, G. H.

Kimura, W. D.

Kleiner, V.

Z. Bomzon, V. Kleiner, and E. Hasman, "Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings," Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

Lai, W. J.

Leger, J.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "The focus of light--theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Li, J.-l.

Lim, B. C.

Lim, Y. L.

Low, D. K. Y.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, "Direct noninterference cylindrical vector beam generation applied in the femtosecond regime," Appl. Phys. Lett. 89, 251114 (2006).
[CrossRef]

Meir, A.

Moh, K. J.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, "Direct noninterference cylindrical vector beam generation applied in the femtosecond regime," Appl. Phys. Lett. 89, 251114 (2006).
[CrossRef]

Moshe, I.

Musha, M.

Nakagawa, W.

E. Descrovi, L. Vaccaro, W. Nakagawa, L. Aeschimann, U. Staufer, and H. P. Herzig, "Collection of transverse and longitudinal fields by means of apertureless nanoprobes with different metal coating characteristics," Appl. Phys. Lett. 85, 5340-5342 (2004).
[CrossRef]

Nesterov, A. V.

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D 32, 2871-2875 (1999).
[CrossRef]

Niziev, V. G.

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D 32, 2871-2875 (1999).
[CrossRef]

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

Oron, R.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Passilly, N.

Phua, P. B.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "The focus of light--theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Roch, J. F.

Roth, M. S.

Roy, M.

Saito, Y.

N. Hayazawa, Y. Saito, and S. Kawata, "Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy," Appl. Phys. Lett. 85, 6239-6241 (2004).
[CrossRef]

Schadt, M.

Sheppard, C. J. R.

Shin-Tson, W.

R. Hongwen, L. Yi-Hsin, and W. Shin-Tson, "Linear to axial or radial polarization conversion using a liquid crystal gel," Appl. Phys. Lett. 89, 051114 (2006).
[CrossRef]

Shirakawa, A.

Stalder, M.

Staufer, U.

E. Descrovi, L. Vaccaro, W. Nakagawa, L. Aeschimann, U. Staufer, and H. P. Herzig, "Collection of transverse and longitudinal fields by means of apertureless nanoprobes with different metal coating characteristics," Appl. Phys. Lett. 85, 5340-5342 (2004).
[CrossRef]

Teo, H. H.

Tiaw, K. S.

Tidwell, S. C.

Treussart, F.

Ueda, K.-i.

Vaccaro, L.

E. Descrovi, L. Vaccaro, W. Nakagawa, L. Aeschimann, U. Staufer, and H. P. Herzig, "Collection of transverse and longitudinal fields by means of apertureless nanoprobes with different metal coating characteristics," Appl. Phys. Lett. 85, 5340-5342 (2004).
[CrossRef]

Weber, H. P.

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

Wyss, E. W.

Yakunin, V. P.

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D 32, 2871-2875 (1999).
[CrossRef]

Yi-Hsin, L.

R. Hongwen, L. Yi-Hsin, and W. Shin-Tson, "Linear to axial or radial polarization conversion using a liquid crystal gel," Appl. Phys. Lett. 89, 051114 (2006).
[CrossRef]

Youngworth, K.

Yuan, X. C.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, "Direct noninterference cylindrical vector beam generation applied in the femtosecond regime," Appl. Phys. Lett. 89, 251114 (2006).
[CrossRef]

Zhan, Q.

Zhong, L.-X.

Appl. Opt. (4)

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "The focus of light--theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).

Appl. Phys. Lett. (6)

E. Descrovi, L. Vaccaro, W. Nakagawa, L. Aeschimann, U. Staufer, and H. P. Herzig, "Collection of transverse and longitudinal fields by means of apertureless nanoprobes with different metal coating characteristics," Appl. Phys. Lett. 85, 5340-5342 (2004).
[CrossRef]

R. Hongwen, L. Yi-Hsin, and W. Shin-Tson, "Linear to axial or radial polarization conversion using a liquid crystal gel," Appl. Phys. Lett. 89, 051114 (2006).
[CrossRef]

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, "Direct noninterference cylindrical vector beam generation applied in the femtosecond regime," Appl. Phys. Lett. 89, 251114 (2006).
[CrossRef]

Z. Bomzon, V. Kleiner, and E. Hasman, "Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings," Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

N. Hayazawa, Y. Saito, and S. Kawata, "Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy," Appl. Phys. Lett. 85, 6239-6241 (2004).
[CrossRef]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

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

J. Phys. D (2)

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D 32, 2871-2875 (1999).
[CrossRef]

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

Opt. Commun. (4)

L. E. Helseth, "Roles of polarization, phase and amplitude in solid immersion lens systems," Opt. Commun. 191, 161-172 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

T. Grosjean and D. Courjon, "Smallest focal spots," Opt. Commun. 272, 314-319 (2007).
[CrossRef]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

Opt. Express (4)

Opt. Lett. (6)

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

(a) Single wave train of circularly polarized light. (b) Field distribution with observer looking into the light as it propagates.

Fig. 2
Fig. 2

(Color online) Vector distribution of circular polarized vortex beams at (a) Φ = 0 or 2π, (b) Φ = π / 4 , (c) Φ = π / 2 , (d) Φ = 3 π / 4 , and (e) Φ = π .

Fig. 3
Fig. 3

(a) Design schematic and (b) actual prototype of the SPP element, where t s is the step thickness and t is the thickness at a given point is proportional to the azimuthal angle θ. (c) Concentric transmission axes of an azimuthal-type analyzer.

Fig. 4
Fig. 4

Conceptual diagram of generating (a) radial polarization via radial-type analyzer and (b) azimuth polarization via an azimuth-type analyzer.

Fig. 5
Fig. 5

(a) Layout for the analyzer-SPP technique and (b) experiment layout for detecting the longitudinal electric field at the focus. Emission observed through a bandpass (BP) filter.

Fig. 6
Fig. 6

Observed intensity distributions for (i) azimuthal, (ii) radial, and when a mismatched SPP is used (iii) and (iv). The arrows indicate the direction of the linear polarizer used to sample the beam.

Fig. 7
Fig. 7

(Color online) Snapshots of the polarization distribution for both a mismatched radial-type beam [i(a)–i(d)] and a mismatched azimuth-type beam [ii(a)–ii(d)].

Fig. 8
Fig. 8

Experiment results showing (a) the presence of the longitudinal component in focused radial polarization and (b) its absence in azimuthal polarization. Right insets show the expected beam profile expected in theory.

Equations (13)

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[ cos ( θ ) sin ( θ ) ]
[ sin ( θ ) cos ( θ ) ] .
e i θ [ 1 i ] = [ e i θ i e i θ ] .
e i θ [ a b c d ] [ 1 i ] = [ sin ( θ ) cos ( θ ) ] ,
[ a b c d ]
[ 1 2   sin ( 2 θ ) sin 2 ( θ ) cos 2 ( θ ) 1 2   sin ( 2 θ ) ] .
[ cos 2 ( θ ) 1 2   sin ( 2 θ ) 1 2   sin ( 2 θ ) sin 2 ( θ ) ] .
t = t s θ 2 π + t 0 ,
[ 1 2   sin ( 2 θ ) sin 2 ( θ ) cos 2 ( θ ) 1 2   sin ( 2 θ ) ] [ 1 0 ] = [ 1 2   sin ( 2 θ ) cos 2 ( θ ) ] ,
[ cos 2 ( θ ) 1 2   sin ( 2 θ ) 1 2   sin ( 2 θ ) sin 2 ( θ ) ] [ 1 i ] = [ cos 2 ( θ ) i 1 2   sin ( 2 θ ) 1 2   sin ( 2 θ ) i sin 2 ( θ ) ]
[ 1 2   sin ( 2 θ ) sin 2 ( θ ) cos 2 ( θ ) 1 2   sin ( 2 θ ) ] [ 1 i ] = [ 1 2   sin ( 2 θ ) + i sin 2 ( θ ) cos 2 ( θ ) i 1 2   sin ( 2 θ ) ]
e i θ [ cos 2 ( θ ) 1 2   sin ( 2 θ ) 1 2   sin ( 2 θ ) sin 2 ( θ ) ] [ 1 i ] = [ cos   θ   cos   2 θ 2 i   sin   θ cos 2 θ sin   θ   cos   2 θ 2 i sin 2 θ   cos   θ ]
e i θ [ 1 2   sin ( 2 θ ) sin 2 ( θ ) cos 2 ( θ ) 1 2   sin ( 2 θ ) ] [ 1 i ] = [ sin   θ   cos   2 θ + 2 i sin 2 θ   cos   θ cos   θ   cos   2 θ 2 i   sin   θ cos 2 θ ]

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