Abstract

The complexity of laser material processing can be greatly reduced using computer-generated phase reflection holographic scanners. These scanners direct and focus the beam of a carbon dioxide laser into a spot on the workpiece and then translate this spot over some general 2-D pattern as the scanner undergoes a simple 1-D motion. Procedures for constructing these scanners are presented, and the first-order aberrations introduced by them are analyzed. The primary aberrations cause the diffracted beam to focus to an astigmatic spot on the work surface. The severity of the astigmatism is proportional to the scan rate, scan angle, and f/number. A technique is presented in which the design of the scanner is adjusted so that the astigmatic image is aligned with the scan direction. The resolution perpendicular to the scan direction is the same as that of a scanner without aberrations of the same f/number. Materials processed using these scanners are presented to show their capabilities for carbon dioxide laser material processing. Power densities on the order of 106/cm2 can be readily obtained using the proposed technique.

© 1978 Optical Society of America

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References

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  1. J. H. Wasko, D. Bennett, Electro.-Opt. Syst. Des. 9, 52 (Sept.1977).
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  10. Thick phase holograms can have a diffraction efficiency approaching 100%. However, there appears to be no method for realizing such holograms at wavelengths longer than a few microns.
  11. I. S. Sokolnikoff, R. M. Redheffer, Mathematics of Physics and Modern Engineering (McGraw-Hill, New York, 1958).
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
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1978 (1)

1977 (2)

1976 (2)

1975 (1)

1974 (2)

1971 (2)

1967 (1)

Angus, J. C.

Bennett, D.

J. H. Wasko, D. Bennett, Electro.-Opt. Syst. Des. 9, 52 (Sept.1977).

Bryngdahl, O.

Campbell, D. K.

Coffield, F. E.

Edwards, R. V.

Engel, A.

Gallagher, N. C.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Herziger, G.

Ih, C. H.

Lee, W. H.

Lohmann, A. W.

Mann, J. A.

Moran, J. M.

Paris, D. P.

Ransom, R. L.

Redheffer, R. M.

I. S. Sokolnikoff, R. M. Redheffer, Mathematics of Physics and Modern Engineering (McGraw-Hill, New York, 1958).

Rugh, R. W.

Shaffer, G.

Sokolnikoff, I. S.

I. S. Sokolnikoff, R. M. Redheffer, Mathematics of Physics and Modern Engineering (McGraw-Hill, New York, 1958).

Steffen, J.

Stevenson, W. H.

Sweeney, D. W.

Wasko, J. H.

J. H. Wasko, D. Bennett, Electro.-Opt. Syst. Des. 9, 52 (Sept.1977).

Appl. Opt. (10)

Electro.-Opt. Syst. Des. (1)

J. H. Wasko, D. Bennett, Electro.-Opt. Syst. Des. 9, 52 (Sept.1977).

Other (3)

Thick phase holograms can have a diffraction efficiency approaching 100%. However, there appears to be no method for realizing such holograms at wavelengths longer than a few microns.

I. S. Sokolnikoff, R. M. Redheffer, Mathematics of Physics and Modern Engineering (McGraw-Hill, New York, 1958).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (13)

Fig. 1
Fig. 1

A reflective holographic scanner used to scan a single spot over the workpiece.

Fig. 2
Fig. 2

The coordinate system for the holographic scanner.

Fig. 3
Fig. 3

Parametric representation of the scan pattern on the workpiece. The parameter x is the position of the scanner along the x axis defined in Fig. 2.

Fig. 4
Fig. 4

Representation of a straight line scan pattern at an angle α with respect to the xi axis.

Fig. 5
Fig. 5

Simplified versions of the computer plots used to generate the scanners: (a) strip-type scanner; (b) disk scanner.

Fig. 6
Fig. 6

A pair of reflection CGHS etched into 2.65 μm of aluminum deposited on a single silicon substrate.

Fig. 7
Fig. 7

Reconstruction geometry for the phase reflection CGHS. The x′,y′ axes are located at the center of the reconstruction beam, the x,y axes are attached to the scanner. Ur and U1 represent the reconstruction and first-order diffracted beam, respectively.

Fig. 8
Fig. 8

The coordinate system used in the scanner and image plane. The u,υ and ui,υi axes are aligned with the principal axis of the astigmatic image.

Fig. 9
Fig. 9

The intensity distribution for the astigmatic Gaussian wavefront. The ui,υi axes are defined in Fig. 8.

Fig. 10
Fig. 10

Experimental reconstruction geometry for the reflection CGHS. All reflective elements were used as close to on axis as possible.

Fig. 11
Fig. 11

Reconstructed images obtained with a transmission CGHS reconstructed with a He–Ne laser, (a) The reconstructed images through second order. In a properly designed phase reflection CGHS the zero-order and second-order reconstructed images have a diffraction efficiency approaching zero; the first orders each have a diffraction efficiency approaching 40.5%. (b) An enlarged view of the first-order image showing the astigmatic Gaussian. The beam scans along the ui axis.

Fig. 12
Fig. 12

Reconstructed spiral scanner: (a) reconstruction of a transmission CGHS with a He–Ne laser; (b) reconstruction of a reflection CGHS with an 8-W CO2 laser. The image is engraved in Plexiglass.

Fig. 13
Fig. 13

Reconstruction of a scanner to engrave the letters D C. Each letter is 1 cm high. The scanner contained thirteen segments, each segment scanned one straight line portion of the letters. In each region the astigmatic image was aligned with the scan direction: (a) reconstruction with a He–Ne laser; (b) reconstruction with an 11-W CO2 laser. The power density was about 103 W/cm2.

Equations (54)

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U object ( x , y ) = A ( x , y ) exp [ i ϕ ( x , y ) ] ,
U reference ( x , y ) = A 0 ,
θ x ( x ) = tan 1 [ f ( x ) F ] ,
θ y ( x ) = tan 1 [ g ( x ) F ] .
f x ( x ) = ( 1 / λ ) sin θ x ( x ) ,
f y ( x ) = ( 1 / λ ) sin θ y ( x ) ,
ϕ ( x , y ) x = 2 π f x ( x ) ,
ϕ ( x , y ) y = 2 π f y ( x ) .
ϕ ( x , y ) x = 2 π λ f ( x ) [ [ f ( x ) ] 2 + F 2 ] 1 / 2 ,
ϕ ( x , y ) y = 2 π λ g ( x ) [ [ g ( x ) ] 2 + F 2 ] 1 / 2 .
ϕ ( x , y ) = 2 π λ f ( x ) [ [ f ( x ) ] 2 + F 2 ] 1 / 2 d x + 2 π λ g ( x ) [ [ g ( x ) ] 2 + F 2 ] 1 / 2 y + G ( y ) .
G ( y = 0 ) = 0.
G ( y ) = [ ( 2 π ) / T ] y + [ ( π d ) / λ ] y 2 ,
ϕ ( x , y ) = 2 π T y + 2 π λ F f ( x ) d x + 2 π λ F y g ( x ) + π d λ y 2 .
q ( x , y ) = { sin 1 [ A ( x , y ) A max ] } / π ,
t a ( x , y ) = u s { cos [ ϕ ( x , y ) ] cos [ π q ( x , y ) ] } ,
u s ( x ) = { 1 when x 0 , 0 when x < 0.
f ( x ) = [ ( R cos α ) / L ] x ,
g ( x ) = [ ( R sin α ) / L ] x .
A ( x , y ) = 1 ,
ϕ ( x , y ) = ( 2 π T ) y + ( π R L λ F cos α ) x 2 + ( 2 π R L λ F sin α ) x y + ( π d λ ) y 2 ,
1 / T ( 1 / ( λ F ) ) ( height of scan pattern ) .
H ( x , y ) = exp [ i ( 4 π D λ ) t a ( x , y ) ] ,
H ( x , y ) = exp { i ( 4 π D λ ) u s [ cos ϕ ( x , y ) cos π q ( x , y ) ] } .
H ( x , y ) = 1 + ( { [ cos ( 4 π D λ ) 1 ] + i [ sin ( 4 π D λ ) ] } × m = sin [ m π q ( x , y ) ] m π exp [ i m ϕ ( x , y ) ] ) .
e m ( x , y ) = 4 sin 2 [ m π q ( x , y ) ] ( m π 2 ) sin 2 ( 2 π D λ ) .
U 1 ( x , y ) = H 1 ( x , y ) U r ( x , y ) ,
H 1 ( x , y ) = B A ( x , y ) exp [ i ϕ ( x , y ) ] .
ϕ ( x , y ) = [ ( 2 π ) / λ ] ( h + a x + b y + 1 2 c x 2 + e x y + 1 2 d y 2 ) ,
a = 1 F f ( X o ) ,
b = λ T + 1 F g ( X o ) ,
c = 1 F x [ f ( x + X o ) ] | x = 0 ,
e = 1 F x [ g ( x + X o ) ] | x = 0 ,
h = 1 F [ f ( x + X o ) d x ] | x = 0 .
U r ( x , y ) = A o r exp ( x 2 + y 2 w o 2 ) exp [ i π λ f s ( x 2 + y 2 ) ] exp [ i π λ f c ( 1 2 x 2 + x y + 1 2 y 2 ) ] .
U 1 ( x , y ) = A ˜ exp ( x 2 + y 2 w o 2 ) × exp [ i ( 2 π λ ) [ a x + b y + 1 2 c ˜ x 2 + 1 2 d ˜ y 2 + e ˜ x y ] ] ,
c ˜ = c 1 f s 1 2 f c ,
d ˜ = d 1 f s 1 2 f c ,
e ˜ = e 1 2 f c .
γ = 1 2 tan 1 [ 2 e ˜ / ( c ˜ d ˜ ) ] .
U 1 ( u , υ ) = A ˜ exp [ ( u 2 + υ 2 w o 2 ) ] exp [ i π λ ( u 2 R u o + υ 2 R υ o ) ] ,
1 R u o = 1 2 { c ˜ + d ˜ + [ 4 e ˜ 2 + ( c ˜ d ˜ ) 2 ] 1 / 2 } ,
1 R υ o = 1 2 { c ˜ + d ˜ [ 4 e ˜ 2 + ( c ˜ d ˜ ) 2 ] 1 / 2 } .
U 1 ( U i , υ i , z ) = A ˜ i exp [ ( u i 2 w u 2 + υ i 2 w υ 2 ) ] · exp [ i π λ ( u i 2 R u + υ i 2 R υ ) ] ,
w u ( z ) = w o [ 1 2 z R u o + ( z R u o ) 2 + ( λ z π w o 2 ) 2 ] 1 / 2 ,
R u ( z ) = 1 2 z R u o + ( z R u o ) 2 + ( λ z π w o 2 ) 2 1 R u o z R u o 2 z ( λ π w o 2 ) 2 .
I 1 ( u i , υ i ) = I o exp [ 2 ( u i 2 w u 2 + υ i 2 w υ 2 ) ] .
I o = ( 2 e 1 P ) / ( π w u w υ ) ,
a = R cos α L F X o ,
b = λ T + R sin α L F X o ,
c ˜ = R cos α L F 1 f s 1 2 f c ,
d ˜ = d 1 f s 1 2 f c ,
e ˜ = R sin α L F 1 2 f c .
d = [ R / ( L F ) ] sin α tan α ,

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