Abstract

We propose and demonstrate a displacement and angular drift simultaneous measurement technique based on a defocus grating. The displacement and angular drift of the incident beam can be detected by monitoring the movements of ±1 diffraction order spots of the defocus grating. The relationship between drift of the incident beam and movements of ±1 diffraction order spots is studied in detail. Compared with other methods, this technique eliminates the requirement of two or more detecting systems for measuring displacement and angular drift simultaneously. The proof-of-principle experiment shows that the root-mean-square errors of displacement and angular drift measurements are less than 0.5μm and 0.84μrad, respectively.

© 2010 Optical Society of America

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2009 (1)

2007 (1)

2006 (3)

2005 (1)

2004 (3)

2002 (1)

2000 (1)

1999 (2)

1997 (2)

Awwal, A. A. S.

Backus, S.

Blanchard, P. M.

Candy, J. V.

Cao, J. Z.

Cavagnero, G.

Christov, I. P.

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Self-compression of ultrashort pulses through ionization-induced spatiotemporal reshaping,” Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef] [PubMed]

Eiju, T.

Fan, D. Y.

Fekete, P. W.

Ferguson, S. W.

Fisher, D. J.

Gagnon, E.

Gao, Y. Q.

Geng, Y. F.

Gibson, E. A.

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Self-compression of ultrashort pulses through ionization-induced spatiotemporal reshaping,” Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef] [PubMed]

Greenaway, A. H.

Hand, D. P.

Jiang, Z. F.

Kapteyn, H. C.

E. Gagnon, I. Thomann, A. Paul, A. L. Lytle, S. Backus, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, “Long-term carrier-envelope phase stability from a grating-based, chirped pulse amplifier,” Opt. Lett. 31, 1866–1868 (2006).
[CrossRef] [PubMed]

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Self-compression of ultrashort pulses through ionization-induced spatiotemporal reshaping,” Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef] [PubMed]

Lambert, R. W.

Lin, Z. Q.

Liu, D. Z.

Liu, X. F.

Lv, F. N.

Lytle, A. L.

Mana, G.

Mannoni, A.

Margheri, G.

Martínez, R. C.

Massa, E.

Matsuda, K.

McClay, W. A.

Murnane, M. M.

E. Gagnon, I. Thomann, A. Paul, A. L. Lytle, S. Backus, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, “Long-term carrier-envelope phase stability from a grating-based, chirped pulse amplifier,” Opt. Lett. 31, 1866–1868 (2006).
[CrossRef] [PubMed]

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Self-compression of ultrashort pulses through ionization-induced spatiotemporal reshaping,” Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef] [PubMed]

O’Byrne, J. W.

Palma, C.

Paul, A.

Popmintchev, T.

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Self-compression of ultrashort pulses through ionization-induced spatiotemporal reshaping,” Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef] [PubMed]

Quercioli, F.

Roy, M.

Sandhu, A. S.

Shephard, J. D.

Sheppard, C. J. R.

Taghizadeh, M. R.

Thomann, I.

Waddie, A. J.

Wagner, N. L.

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Self-compression of ultrashort pulses through ionization-induced spatiotemporal reshaping,” Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef] [PubMed]

Woods, S. C.

Xi, F. J.

Xiao, J. B.

Xu, R. F.

Xu, X. J.

Zhou, X. J.

Zhu, B. Q.

Zhu, J. Q.

Appl. Opt. (8)

C. Palma, “Decentered Gaussian beams, ray bundles, and Bessel–Gauss beams,” Appl. Opt. 36, 1116–1120 (1997).
[CrossRef] [PubMed]

G. Margheri, A. Mannoni, and F. Quercioli, “High-resolution angular and displacement sensing based on the excitation of surface plasma waves,” Appl. Opt. 36, 4521–4525(1997).
[CrossRef] [PubMed]

P. M. Blanchard and A. H. Greenaway, “Simultaneous multiplane imaging with a distorted diffraction grating,” Appl. Opt. 38, 6692–6699 (1999).
[CrossRef]

K. Matsuda, M. Roy, J. W. O’Byrne, P. W. Fekete, T. Eiju, and C. J. R. Sheppard, “Straightness measurements by use of a reflection confocal optical system,” Appl. Opt. 38, 5310–5318(1999).
[CrossRef]

P. M. Blanchard, D. J. Fisher, S. C. Woods, and A. H. Greenaway, “Phase-diversity wave-front sensing with a distorted diffraction grating,” Appl. Opt. 39, 6649–6655 (2000).
[CrossRef]

K. Matsuda, M. Roy, T. Eiju, J. W. O’Byrne, and C. J. R. Sheppard, “Straightness measurements with a reflection confocal optical system—an experimental study,” Appl. Opt. 41, 3966–3970(2002).
[CrossRef] [PubMed]

R. W. Lambert, R. C. Martínez, A. J. Waddie, J. D. Shephard, M. R. Taghizadeh, A. H. Greenaway, and D. P. Hand, “Compact optical system for pulse-to-pulse laser beam quality measurement and applications in laser machining,” Appl. Opt. 43, 5037–5046 (2004).
[CrossRef] [PubMed]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. 48, 1591–1597 (2009).
[CrossRef] [PubMed]

Chin. Opt. Lett. (2)

J. Opt. Soc. Am. A (3)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Self-compression of ultrashort pulses through ionization-induced spatiotemporal reshaping,” Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Movements of ± 1 diffraction order spots with decentered and tilted factor: (a) with decentered factor ( a x , a y ) and (b) with tilted factor ( c x , c y ) .

Fig. 2
Fig. 2

Schematic of the experimental setup used for measuring the displacement and angular drift: (a) experimental setup and (b) principle of displacement and angular drift generation.

Fig. 3
Fig. 3

Experimental results of displacement drift measurement in the (a) x and (b) y directions.

Fig. 4
Fig. 4

Experimental results of angular drift measurement in the (a) x and (b) y directions.

Fig. 5
Fig. 5

Movements of ± 1 diffraction order spots with decentered factor ( a x , a y ) , depending on the detection distance z: (a) f < z < f g f f g f and (b) z > f g f f g f .

Equations (21)

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t ( x , y ) = m = + a m e i π m ( x 2 + y 2 ) / λ f g e i 2 π m x o x / λ f g e i π m x o 2 / λ f g ,
E ( x , y , z 1 ) = exp [ ( x a x ) 2 + ( y a y ) 2 w 2 ( z 1 ) ] exp { i k [ ( x a x ) 2 + ( y a y ) 2 2 R ( z 1 ) + z 1 ] ψ } exp [ i k ( c x x + c y y ) ] ,
E 2 ( x 2 , y 2 , z ) = i exp ( i k L ) λ B exp [ i k D ( x 2 2 + y 2 2 ) 2 B ] E ( x 1 , y 1 , z 1 ) t ( x 1 , y 1 ) exp { i k 2 B [ A ( x 1 2 + y 1 2 ) 2 ( x 2 x 1 + y 2 y 1 ) ] } d x 1 d y 1 ,
E d m ( x 2 , y 2 , z ) = F E d m exp { [ k x 2 i B 2 w 2 ( z 1 ) a x k i R ( z 1 ) a x + k m x 0 i f g ] 2 4 w 2 ( z 1 ) + 2 k i R ( z 1 ) 2 k A i B 2 k m i f g } exp { [ k y 2 i B 2 w 2 ( z 1 ) a y k i R ( z 1 ) a y ] 2 4 w 2 ( z 1 ) + 2 k i R ( z 1 ) 2 k A i B 2 k m i f g } ,
I d m ( x 2 , y 2 , z ) = E d m ( x 2 , y 2 , z ) E d m * ( x 2 , y 2 , z ) = F I d m exp { [ x 2 ( A a x + B m f g a x B m x 0 f g ) ] 2 2 B 2 w 2 ( z 1 ) k 2 + 2 B 2 w 2 ( z 1 ) [ 1 2 R ( z 1 ) A 2 B m 2 f g ] 2 } exp { ( y 2 A a y B m f g a y ) 2 2 B 2 w ( z 1 ) 2 k 2 + 2 B 2 w ( z 1 ) 2 [ 1 2 R ( z 1 ) A 2 B m 2 f g ] 2 } ,
F E d m = 2 π 2 sin ( m π 2 ) sinc ( m 2 ) λ B [ 1 w ( z 1 ) 2 + i ( k 2 R ( z 1 ) k A 2 B k m 2 f g ) ] exp [ a x 2 + a y 2 w 2 ( z 1 ) ψ + i k D ( x 2 2 + y 2 2 ) 2 B + i π m x 0 2 λ f g i π m i k ( a x 2 + a y 2 ) 2 R ( z 1 ) i k z 1 + i k L ] ,
F I d m = 4 π 4 sin 2 ( m π 2 ) sinc 2 ( m 2 ) λ 2 B 2 [ 1 w 4 ( z 1 ) + ( k 2 R ( z 1 ) k A 2 B k m 2 f g ) 2 ] exp { [ a x k R ( z 1 ) a x k A B a x k m f g ] 2 + 2 [ 2 a x w 2 ( z 1 ) ] 2 + [ a y k R ( z 1 ) a y k A B a y k m f g ] 2 2 w 2 ( z 1 ) + 2 w 2 ( z 1 ) [ k 2 R ( z 1 ) k A 2 B k m 2 f g ] 2 2 ( a x 2 + b x 2 ) w 2 ( z 1 ) 2 ψ } ,
E t m ( x 2 , y 2 , z ) = F E t m exp [ ( k x 2 B + k c x + k m x 0 f g ) 2 4 w 2 ( z 1 ) + 2 k i R ( z 1 ) 2 k A i B 2 k m i f g ] exp [ ( k y 2 B + k c y ) 2 4 w 2 ( z 1 ) + 2 k i R ( z 1 ) 2 k A i B 2 k m i f g ] ,
I t m ( x 2 , y 2 , z ) = E t m ( x 2 , y 2 , z ) E t m * ( x 2 , y 2 , z ) = F I t m exp { ( x 2 + B c x + B m x 0 f ) 2 2 B 2 w 2 ( z 1 ) k 2 + 2 B 2 w 2 ( z 1 ) [ 1 2 R ( z 1 ) A 2 B m 2 f g ] 2 } exp { ( y 2 + B c y ) 2 2 B 2 w 2 ( z 1 ) k 2 + 2 B 2 w 2 ( z 1 ) [ 1 2 R ( z 1 ) A 2 B m 2 f g ] 2 } ,
F E t m = 2 π 2 exp ( ψ ) sin ( m π 2 ) sinc ( m 2 ) λ B [ 1 w 2 ( z 1 ) + i ( k 2 R ( z 1 ) k A 2 B k m 2 f g ) ] exp [ i π m x 0 2 λ f g + i k D ( x 2 2 + y 2 2 ) 2 B i k z 1 i π m + i k L ] ,
F I t m = 4 π 4 exp ( 2 ψ ) sin 2 ( m π 2 ) sinc 2 ( m 2 ) λ 2 B 2 [ 1 w 4 ( z 1 ) + ( k 2 R ( z 1 ) k A 2 B k m 2 f g ) 2 ] .
[ A B C D ] = [ 1 f 0 1 ] [ 1 0 1 f 1 ] = [ 0 f 1 f 1 ] ,
x d m = f m f g a x f m x 0 f g , y d m = f m f g a y ,
x t m = f c x f m x 0 f g , y t m = f c y ,
a x = f g 2 f d x + x o , a y = f g 2 f d y ,
c x = 1 f t x x o f g , c y = 1 f t y ,
θ = 2 φ ,
d 2 L φ ,
θ d / L .
a x = f g 2 z d x + x o , a y = f g 2 z d y .
c x = 1 z t x x 0 f g , c y = 1 z t y .

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