Abstract

A new approach for designing diffractive optical corrective elements with zooming capability to convert nonlinear sinusoidal scanning into linear scanning is proposed. Such a device will be useful for linearizing the angular scan of a resonant mirror scanner. The design methodology is to create a graded index of a refraction device as the reference design with its index of refraction parameters based on beam retardation through propagation in an inhomogeneous medium. The diffractive element is designed by utilizing a binarizing algorithm of the accumulated phase from transmission through the refractive element. In contrast to a prior approach, which was introduced based on the beam propagation through inhomogeneous media, the new approach takes beam diameters into consideration. This makes both the refractive element and its associated diffractive element more robust against beam fanning.

© 2006 Optical Society of America

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References

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  1. S. Makki and J. Leger, "Solid-state laser resonators with diffractive optic thermal aberration correction," IEEE J. Quantum Electron. 35, 1075-1085 (1999).
    [CrossRef]
  2. J. Khoury, C. L. Woods, B. Haji-saeed, D. Pyburn, S. K. Sengupta, and J. Kierstead, "A mapping approach for image correction and processing for bidirectional resonant scanners," Opt. Eng. (to be published).
  3. GSI Lumonics CRS scanner manual, available for download at http://www.gsilumonics.com/process_download_open/01_optical_scanning/resources/7om025_CRS.pdf.
  4. J. Khoury, C. L. Woods, B. Haji-saeed, D. Pyburn, S. K. Sengupta, and J. Kierstead, "Diffractive element design for resonant scanner angular correction," Appl. Opt. 45, 6897-6902 (2006).
    [CrossRef] [PubMed]

2006 (1)

1999 (1)

S. Makki and J. Leger, "Solid-state laser resonators with diffractive optic thermal aberration correction," IEEE J. Quantum Electron. 35, 1075-1085 (1999).
[CrossRef]

Haji-saeed, B.

J. Khoury, C. L. Woods, B. Haji-saeed, D. Pyburn, S. K. Sengupta, and J. Kierstead, "Diffractive element design for resonant scanner angular correction," Appl. Opt. 45, 6897-6902 (2006).
[CrossRef] [PubMed]

J. Khoury, C. L. Woods, B. Haji-saeed, D. Pyburn, S. K. Sengupta, and J. Kierstead, "A mapping approach for image correction and processing for bidirectional resonant scanners," Opt. Eng. (to be published).

Khoury, J.

J. Khoury, C. L. Woods, B. Haji-saeed, D. Pyburn, S. K. Sengupta, and J. Kierstead, "Diffractive element design for resonant scanner angular correction," Appl. Opt. 45, 6897-6902 (2006).
[CrossRef] [PubMed]

J. Khoury, C. L. Woods, B. Haji-saeed, D. Pyburn, S. K. Sengupta, and J. Kierstead, "A mapping approach for image correction and processing for bidirectional resonant scanners," Opt. Eng. (to be published).

Kierstead, J.

J. Khoury, C. L. Woods, B. Haji-saeed, D. Pyburn, S. K. Sengupta, and J. Kierstead, "Diffractive element design for resonant scanner angular correction," Appl. Opt. 45, 6897-6902 (2006).
[CrossRef] [PubMed]

J. Khoury, C. L. Woods, B. Haji-saeed, D. Pyburn, S. K. Sengupta, and J. Kierstead, "A mapping approach for image correction and processing for bidirectional resonant scanners," Opt. Eng. (to be published).

Leger, J.

S. Makki and J. Leger, "Solid-state laser resonators with diffractive optic thermal aberration correction," IEEE J. Quantum Electron. 35, 1075-1085 (1999).
[CrossRef]

Makki, S.

S. Makki and J. Leger, "Solid-state laser resonators with diffractive optic thermal aberration correction," IEEE J. Quantum Electron. 35, 1075-1085 (1999).
[CrossRef]

Pyburn, D.

J. Khoury, C. L. Woods, B. Haji-saeed, D. Pyburn, S. K. Sengupta, and J. Kierstead, "Diffractive element design for resonant scanner angular correction," Appl. Opt. 45, 6897-6902 (2006).
[CrossRef] [PubMed]

J. Khoury, C. L. Woods, B. Haji-saeed, D. Pyburn, S. K. Sengupta, and J. Kierstead, "A mapping approach for image correction and processing for bidirectional resonant scanners," Opt. Eng. (to be published).

Sengupta, S. K.

J. Khoury, C. L. Woods, B. Haji-saeed, D. Pyburn, S. K. Sengupta, and J. Kierstead, "Diffractive element design for resonant scanner angular correction," Appl. Opt. 45, 6897-6902 (2006).
[CrossRef] [PubMed]

J. Khoury, C. L. Woods, B. Haji-saeed, D. Pyburn, S. K. Sengupta, and J. Kierstead, "A mapping approach for image correction and processing for bidirectional resonant scanners," Opt. Eng. (to be published).

Woods, C. L.

J. Khoury, C. L. Woods, B. Haji-saeed, D. Pyburn, S. K. Sengupta, and J. Kierstead, "Diffractive element design for resonant scanner angular correction," Appl. Opt. 45, 6897-6902 (2006).
[CrossRef] [PubMed]

J. Khoury, C. L. Woods, B. Haji-saeed, D. Pyburn, S. K. Sengupta, and J. Kierstead, "A mapping approach for image correction and processing for bidirectional resonant scanners," Opt. Eng. (to be published).

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

S. Makki and J. Leger, "Solid-state laser resonators with diffractive optic thermal aberration correction," IEEE J. Quantum Electron. 35, 1075-1085 (1999).
[CrossRef]

Other (2)

J. Khoury, C. L. Woods, B. Haji-saeed, D. Pyburn, S. K. Sengupta, and J. Kierstead, "A mapping approach for image correction and processing for bidirectional resonant scanners," Opt. Eng. (to be published).

GSI Lumonics CRS scanner manual, available for download at http://www.gsilumonics.com/process_download_open/01_optical_scanning/resources/7om025_CRS.pdf.

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Figures (8)

Fig. 1
Fig. 1

Ray diagram of resonant scanner with refractive–diffractive corrective element.

Fig. 2
Fig. 2

Schematic illustrating wavefront bending due to phase lag between the two sides of the beam as the beam propagates in a graded index of refraction material.

Fig. 3
Fig. 3

Plots of sin - 1 ( x ) and its polynomial expansion to the seventh and the ninth orders. The solid curve is the plot for sin - 1 ( x ) , the dotted curve the polynomial expansion for the seventh order. The dashed curve is the polynomial expansion to the ninth order.

Fig. 4
Fig. 4

Linearized scanning geometries with FOV, greater than, equal to, or less than that for the sinusoidal scanner.

Fig. 5
Fig. 5

Plots of the graded index of refraction as a function of distance from the optical axis: A, for a constant magnification and constant beam diameter; B, for constant aperture and constant beam diameter; C, for constant aperture and constant magnification.

Fig. 6
Fig. 6

Plots of the graded index of refraction as a function of distance from the optical axis: A, for a constant magnification and constant beam diameter; B, for constant aperture and constant beam diameter; C, for constant aperture and constant magnification. These are the same as in Fig. 5 but for a beam with a larger diameter using Eq. (26).

Fig. 7
Fig. 7

Diffractive element generated from binarizing a refractive element based on Eq. (27). The rows are for constant apertures of y 0 = 1 and 2   cm , respectively. The columns are for constant beam diameters of D = 1 and 2   mm , respectively.

Fig. 8
Fig. 8

Illustration of beam fanning through diffraction from chirped grating: A, chirped grating taken from the diffractive element shown in Fig. 7D; B, 1D Fourier transform of A to show the diffraction orders; and C, plot of the 1D cross section of the Fourier transform taken from B.

Equations (53)

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tan   θ 1 = h Z 1 ,
h = Z 1   tan   θ 1 ,
tan   θ 2 = h 2 Z 2 ,
h 2 = Z 2   tan   θ 2 .
tan   θ 3 = h h 2 Z 1 Z 2 = Z 1   tan   θ 1 Z 2   tan   θ 2 Z 1 Z 2 ,
θ 3 = h h 2 Z 1 Z 2 = Z 1 θ 1 Z 2 θ 2 Z 1 Z 2 .
θ 2 = θ 20   sin   ω t = y Z 2 ,
t = 1 ω sin - 1 y Z 2 θ 20 .
θ 1 = g t ,
θ 1 = g ω sin - 1 ( y Z 2 θ 20 ) .
tan   θ 3 θ 3 = Z 1 g ω sin - 1 ( y Z 2 θ 20 ) Z 2 y Z 2 ( Z 1 Z 2 ) = g Z 1 sin - 1 ( y Z 2 θ 20 ) y ω ( Z 1 Z 2 ) ω ,
θ 4 = 2 π ( n 0 + Δ n ) l λ ,
θ 5 = 2 π ( n 0 ) l λ .
Δ θ 45 = θ 4 θ 5 = 2 π ( Δ n ) l λ .
Δ θ 45 2 π λ n y ( Δ D ) l .
Δ θ 45 = Δ θ 32 = θ 3 θ 2 .
2 π λ n y ( Δ D ) l = θ 3 θ 2 = Z 1 θ 1 Z 2 θ 2 Z 1 Z 2 θ 2 .
2 π λ n y ( Δ D ) l = A g ω sin - 1 ( y z 2 θ 20 ) A y z 2 ,
n ( y ) = λ A 2 π ( Δ D ) l [ g ω y sin - 1 ( y z 2 θ 20 ) + g ω z 2     2 θ 20         2 y 2 y 2 2 z 2 ] + C ,
A = Z 1 Z 1 - Z 2 ,
n ( y ) = λ 2 π ( Δ D ) l Z 1 Z 1 - Z 2 [ g ω y sin - 1 ( y z 2 θ 20 ) + g ω z 2 2 θ 20 2 - y 2 y 2 2 Z 2 ] + C .
ln ( n ) = 1 l ( Z 1 Z 1 - Z 2 ) { g ω [ y sin - 1 ( y y 0 ) + y 0 2 y 2 ] y 2 2 Z 2 } .
Δ n = n y ( Δ D ) + 1 2 ! ( 2 n y 2 ) ( Δ D ) 2 + 1 3 ! ( 3 n y 3 ) ( Δ D ) 3 .
Δ θ = 2 π λ Δ n l = 2 π λ [ n y ( Δ D ) + 1 2 ! ( 2 n y 2 ) ( Δ D ) 2 + 1 3 ! ( 3 n y 3 ) ( Δ D ) 3 ] l .
Δ θ 32 = θ 3 θ 2 .
2 π λ [ n y Δ D + 1 2 ! ( 2 n y 2 ) ( Δ D ) 2 + 1 3 ! ( 3 n 2 y 3 ) ( Δ D ) 3 ] l = θ 3 θ 2 = Z 1 θ 1 Z 2 θ 2 Z 1 Z 2 θ 2 .
[ n y Δ D + 1 2 ! ( 2 n y 2 ) ( Δ D ) 2 + 1 3 ! ( 3 n 2 y 3 ) ( Δ D ) 3 ] = A g λ 2 π ω l sin - 1 ( y y 0 ) A λ y 2 π l Z 2 .
a 3 3 n y 3 + a 2 2 n y 2 + a 1 n y = b 1 sin - 1 ( y y 0 ) b 2 y ,
a 3 = 1 3 ! ( Δ D ) 3 ,
a 2 = 1 2 ! ( Δ D ) 2 ,
a 1 = 1 1 ! ( Δ D ) ,
b 1 = A g λ 2 π ω l ,
b 2 = A λ 2 π l Z 2 .
a 3 3 n y 3 + a 2 2 n y 2 + a 1 n y = b 1 A P 9 ( y y 0 ) b 2 y .
n [ y ] = ( 1 / 72 ) b 2 ( Δ D ) 2 [ ( Δ D ) 2 6 Δ D y + 6 y 2 ] + 1 5806080 y 0 9 { b 1 ( Δ D ) 2 [ u 0 + u 1 ( y ) + u 2 ( y ) + u 3 ( y ) + u 4 ( y ) + u 5 ( y ) + u 6 ( y ) + u 7 ( y ) + u 8 ( y ) + u 9 ( y ) + u 10 ( y ) ] } + C 1 ,
u 0 = 986,125 ( Δ D ) 10 ,
u 1 ( y ) = 1,286,250 ( Δ D ) 9 y ,
u 2 ( y ) = 99,750 ( Δ D ) 8 ( 2 y 0 2 + 49 y 2 ) ,
u 3 ( y ) = 73,500 ( Δ D ) 7 ( 6 y 0 2 y + 49 y 3 ) ,
u 4 ( y ) = 210 ( Δ D ) 6 ( 24 y 0 4 + 300 y 0 2 y 2 + 1225 y 4 ) ,
u 5 ( y ) = 3780 ( Δ D ) 5 ( 24 y 0 4 y + 100 y 0 2 y 3 + 245 y 5 ) ,
u 7 ( y ) = 2520 ( Δ D ) 3 ( 16 y 0 6 y + 24 y 0 4 y 3 + 30 y 0 2 y 5 + 35 y 7 ) ,
u 8 ( y ) = 630 ( Δ D ) 2 ( 128 y 0 8 + 64 y 0 6 y 2 + 48 y 0 4 y 4 + 40 y 0 2 y 6 + 35 y 8 ) ,
u 9 ( y ) = 12 ( Δ D ) ( 40,320 y 0 8 y + 6720 y 0 6 y 3 + 3024 y 0 4 y 5 + 1800 y 0 2 y 7 + 1225 y 9 ) ,
u 10 ( y ) = 12 y 2 ( 40,320 y 0 8 + 3360 y 0 6 y 2 + 1008 y 0 4 y 4 + 450 y 0 2 y 6 + 245 y 8 ) .
T = { 2 ( m 1 ) π   <   2 π λ nl   <   2 m π = 0 ( 2 m 1 ) π   <   2 π λ n l   <   ( 2 m + 1 ) π = 1 ,
T = { 2 ( m 1 ) π   <   2 π λ nl   <   2 m π = 1 ( 2 m 1 ) π   <   2 π λ n l   <   ( 2 m + 1 ) π = 1 .
2 π n ( 0 ) l λ = ( 2 m + 1 ) π ,
( 2 m + 1 ) λ 2 l 1 ,
m 1 2 ( 2 l λ 1 ) .
m = round [ 1 2 ( 2 l λ 1 ) ] ,
C = { 2round [ 1 2 ( 2 l λ 1 ) ] + 1 } λ 2 l ,
C 1 = ( 2round { 1 2 [ 1 72 A π Z 2 ( Δ D ) 4 986,125 5,806,080 y 0     9 A g π ω ( Δ D ) 12 ] } + 1 ) λ 2 l .

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