Abstract

A laser heterodyne apparatus for roughness measurements of polished surfaces has been developed. A new principle for generation of the local-oscillator light beam is introduced. Important features of the apparatus are: very high sensitivity (Ra values less than 0.1 nm can readily be measured), immunity against stray light of all kinds, noncritical aligning (insensitive to vibrations), simple configuration, microprocessor output unit for digital presentation of characteristic surface measures. A quantitative theory of the complete apparatus is presented together with experimental confirmation by comparisons with stylus and multiple interference measurements.

© 1978 Optical Society of America

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References

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  1. J. Westberg, Proc. Inst. Mech. Eng. 182, Pt. 3K, 260 (1967–68).
  2. I. J. Hodgkinson, J. Sci. Instrum. 5, 341 (1970).
  3. S. Hård, Y. Hamnerius, O. Nilsson, J. Appl. Phys. 47, 2433 (1976).
    [CrossRef]
  4. P. Beckmann, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).
  5. H. E. Bennett, J. O. Porteus, J. Opt. Soc. Am. 51, 123 (1961).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

1976 (1)

S. Hård, Y. Hamnerius, O. Nilsson, J. Appl. Phys. 47, 2433 (1976).
[CrossRef]

1970 (1)

I. J. Hodgkinson, J. Sci. Instrum. 5, 341 (1970).

1967 (1)

J. Westberg, Proc. Inst. Mech. Eng. 182, Pt. 3K, 260 (1967–68).

1961 (1)

Beckmann, P.

P. Beckmann, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).

Bennett, H. E.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

Hamnerius, Y.

S. Hård, Y. Hamnerius, O. Nilsson, J. Appl. Phys. 47, 2433 (1976).
[CrossRef]

Hård, S.

S. Hård, Y. Hamnerius, O. Nilsson, J. Appl. Phys. 47, 2433 (1976).
[CrossRef]

Hodgkinson, I. J.

I. J. Hodgkinson, J. Sci. Instrum. 5, 341 (1970).

Nilsson, O.

S. Hård, Y. Hamnerius, O. Nilsson, J. Appl. Phys. 47, 2433 (1976).
[CrossRef]

Porteus, J. O.

Westberg, J.

J. Westberg, Proc. Inst. Mech. Eng. 182, Pt. 3K, 260 (1967–68).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

J. Appl. Phys. (1)

S. Hård, Y. Hamnerius, O. Nilsson, J. Appl. Phys. 47, 2433 (1976).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Sci. Instrum. (1)

I. J. Hodgkinson, J. Sci. Instrum. 5, 341 (1970).

Proc. Inst. Mech. Eng. (1)

J. Westberg, Proc. Inst. Mech. Eng. 182, Pt. 3K, 260 (1967–68).

Other (2)

P. Beckmann, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, London, 1963).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

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Figures (15)

Fig. 1
Fig. 1

Laser light scattering due to small irregularities in a polished reflecting surface. Small arrows indicate scattered light.

Fig. 2
Fig. 2

Essential elements of the surface roughness measuring apparatus. A few additional lenses would be required if the curvature of the test surface S is significant.

Fig. 3
Fig. 3

Coordinate systems in the experiment.

Fig. 4
Fig. 4

Geometry of the rotating grating arrangement. The grating is positioned so that the radial rulings are parallel to the y1 axis when they are illuminated by the laser beam.

Fig. 5
Fig. 5

The detector unit with its six separate photodetectors.

Fig. 6
Fig. 6

The electronic analyzing system. The ac current In(t) from each of the channels 1–5 is first preamplified, bandpass filtered (center frequency n· 4 kHz, Q ≈ 10), and amplified to a convenient level (≈5 V) with electronically controlled amplification. Thereafter it is squared and low-pass filtered (1 Hz) to obtain I n 2 ¯ ( t ), which is then A/D converted. The dc signals Indc are decoupled after the preamplification, low-pass filtered (1 Hz), and A/D converted. The dc current Ir from channel 0 is A/D converted after preamplification.

Fig. 7
Fig. 7

Control and display unit of the surface roughness meter.

Fig. 8
Fig. 8

The primary measurement results for a polished carbon steel surface at eleven different positions (SIS 1650).

Fig. 9
Fig. 9

Multiple interference photography of the carbon steel surface: apparatus: Multimi 3000 C; light wavelength: 5471 Å; numerical aperture: 0.2.

Fig. 10
Fig. 10

Stylus profilogram of the carbon steel surface: apparatus: Talysurf 4; stylus tip width: 2.5 μm.

Fig. 11
Fig. 11

Calibration curve for CEJ glossmeter 8510-1. The dot represents the measured value for the carbon steel surface (13 ≲ Ra ≲ 25 nm).

Fig. 12
Fig. 12

Profilogram of the gauge block surface perpendicular to the lapping scratches: apparatus: Talysurf 4; stylus tip width: 2.5 μm.

Fig. 13
Fig. 13

Profilogram of the gauge block surface parallel to the lapping scratches: apparatus: Talysurf 4; stylus tip width: 2.5 μm.

Fig. 14
Fig. 14

Interferogram of the step surface: Apparatus: Multimi 3000 C; light wavelength: 5471 Å; numerical aperture: 0.2.

Fig. 15
Fig. 15

Results for a step surface. Circles represent measured values, and the broken line is obtained from Eq. (19) with hs = 16 nm.

Equations (42)

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E k ¯ ( t , u , υ ) u > 0 a k ¯ 2 cos θ π 3 / 2 | n ˆ 2 1 | D ( Z 0 · P i ) 1 / 2 R λ 2 | [ ( n ˆ 2 sin 2 θ ) 1 / 2 + cos θ ] 2 | × cos [ Ω t + ϕ k ¯ π λ R ( u 2 + υ 2 ) ] · exp { [ ( u u x ) 2 ( π D 2 λ R ) 2 + ( υ υ x ) 2 ( π D 2 λ R ) 2 ] } .
E n ( t , u , υ ) u > 0
E n ( t , u , υ ) u > 0 D e λ R ( π Z 0 P n ) 1 / 2 × cos [ ( Ω + ω n ) t π λ ( R l ) ( u 2 + υ 2 ) + ϕ n ] · exp { [ ( u u n ) 2 + υ 2 ] ( π D e 2 λ R ) 2 } ,
D e = D [ 1 + ( D 2 N n 8 r 2 cos θ 0 ) 2 ] 1 / 2 if D 2 N n 8 r 2 cos θ 0 1 ( see Appendix A ) ;
I = I n d c + I n ( t ) = η λ e h c 1 Z 0 S [ E n ( t , u , υ ) + k ¯ E k ¯ ( t , u , υ ) ] 2 d u d υ ,
I n dc η λ e h c 1 Z 0 S E n 2 ( t , u , υ ) d u d υ = P n λ h c e η ;
I n ( t ) η λ e h c 1 Z 0 S 2 E n ( t , u , υ ) k ¯ E k ¯ ( t , u , υ ) d u d υ .
I n ( t ) η e h c · 8 π cos θ | n ˆ 2 1 | ( P i · P n ) 1 / 2 | [ ( n ˆ 2 sin 2 θ ) 1 / 2 + cos θ ] 2 | · D D e ( D 2 + D e 2 ) exp [ 1 2 ( l l 0 ) 2 ] × k ¯ a k ¯ cos ( ω n t + ϕ k ¯ + ϕ l + ϕ n ) × exp { 1 16 ( D 2 D e 2 D 2 + D e 2 ) [ ( k x k n cos θ ) 2 + k y 2 ] } ,
ϕ l λ l 4 π cos 2 θ ( D 2 k x + D e 2 k n D 2 + D e 2 ) 2 ,
l 0 ( D 2 + D e 2 2 ) 1 / 2 π r n N λ cos θ 0 ,
k n = cos θ cos θ 0 n N r .
Σ b n cos ( ω n t + φ n ) .
I n 2 ( t ) ¯ I n dc = η e λ h c 64 π 2 cos 2 θ P r D 2 D e 2 ( D 2 + D e 2 ) 2 exp [ ( l l 0 ) 2 ] · 1 2 b n 2 K b n 2 ,
| ( n ˆ 2 1 ) [ ( n ˆ 2 sin 2 θ ) 1 / 2 + cos θ ] 2 | 2 .
I r [ ( P r λ ) / ( h c ) ] e η ,
I n 2 ( t ) ¯ I n dc · I r = 2 · ( 2 π b n cos θ λ ) 2 · 4 D 2 D e 2 ( D 2 + D e 2 ) 2 exp [ ( l / l 0 ) 2 ] .
R = R 1 + 1 R 1 ( π w 2 4 λ ) 2 ,
σ e 2 π 2 8 D 2 ( Δ u ) 2 R 2 λ 2 cos θ n = 1 5 n b n 2 ,
α e 2 π 4 2 D 2 ( Δ u ) 4 cos 3 θ R 4 λ 4 n = 1 5 n 3 b n 2 .
τ e 2 = 2 · k 1 2 · n = 1 5 n b n 2 n = 1 5 n 3 b n 2 ,
σ e 21 nm τ e 28 μ m ( τ e max = 72.0 μ m , τ e w = 18.6 μ m ) .
0.5 h 2.5 h σ e = 1.10 nm , σ e = 0.47 nm , τ e = 24 μ m , τ e = 31 μ m .
I n 2 ( t ) ¯ I r I n dc 128 π h s 2 ( R cos θ u n ) 2 1 ( D 2 + D e 2 ) · exp [ l 2 ( R D e 2 2 u n ) 2 ] ,
E ( x 1 , y 1 ) = 4 D · ( Z 0 P i π ) 1 / 2 exp [ ( 4 cos 2 θ 0 x 1 2 D 2 ) ( 4 y 1 2 D 2 ) ] .
E ( t , x 1 , y 1 ) = E ( x 1 , y 1 ) · cos [ Ω t 2 π λ ( Δ r 1 r 1 ) ] ,
E ( x 1 , y 1 ) · T ( x 1 , y 1 ) ,
T ( x 1 , y 1 ) = T 0 + n T n · cos [ ω n t + n N arctan ( x 1 r + y 1 ) + ϕ n 1 ]
E ( u , υ , t ) cos θ 0 λ R 1 E ( x 1 , y 1 ) × cos ( Ω t 2 π λ Δ r 1 π 2 ) · { T 0 + n T n cos [ ω n t + n N arctan ( x 1 r + y 1 ) + ϕ n 1 ] } d x 1 d y 1 .
Δ r 1 1 2 R 1 ( 2 u x 1 cos θ 0 + u 2 + 2 υ y 1 + υ 2 )
arctan ( x 1 r + y 1 ) x 1 r x 1 y 1 r 2 ,
E n ( u , υ , t ) Re { 1 2 · cos θ 0 λ R 1 · 4 D ( Z 0 P i π ) 1 / 2 · T n · exp { j [ ( Ω + ω n ) t π λ R 1 ( u 2 + υ 2 ) π 2 + ϕ n 1 ] } × exp [ ( 4 cos 2 θ 0 x 1 2 D 2 ) ( 4 y 1 2 D 2 ) ] · exp [ j ( 2 π u cos θ 0 λ R 1 x 1 + 2 π υ λ R 1 y 1 n N x 1 r + n N x 1 y 1 r 2 ) ] d x 1 d y 1 } .
E n ( u , υ , t ) 1 2 cos θ 0 λ R 1 4 D ( Z 0 P i π ) 1 / 2 T n · π D 2 · cos [ ( Ω + ω n ) t π λ R 1 ( u 2 + υ 2 ) π 2 + ϕ n 1 × υ ( u u n ) cos θ 0 ( π D D e 4 λ R 1 r ) 2 ] · π [ ( 2 cos θ 0 D ) 2 + ( D n N 4 r 2 ) 2 ] 1 / 2 · exp [ ( u u n ) 2 + υ 2 ( 2 R 1 λ π D e ) 2 ] ,
u n R 1 λ n N 2 π r cos θ 0 ,
1 D e 2 1 D 2 + ( D n N 8 r 2 cos θ 0 ) 2 .
ξ ( x , y ) = k ¯ a k ¯ · cos ( k ¯ · r ¯ + ϕ k ¯ ) ,
k ¯ · r ¯ = k x · x + k y · y , k x > 0 , < k y < .
σ 2 1 S S ξ 2 ( x , y ) d x d y = 1 S S k ¯ 1 k ¯ 2 a k ¯ 1 a k ¯ 2 × cos ( k ¯ 1 · r ¯ + ϕ k ¯ 1 ) · cos ( k ¯ 2 · r ¯ + ϕ k ¯ 2 ) d x d y .
σ 2 = π 2 0 a 2 ( k ) k · d k ,
b n 2 a 2 ( k n ) · 8 π cos θ ( D 2 + D e 2 D 2 D e 2 ) .
σ 2 = 1 2 · D 2 16 cos θ 0 b 2 ( k ) k d k .
k = n · k 1 ( n = 1 , , 5 ) , where k 1 = ( 2 π Δ u cos θ ) / ( λ R ) .
σ 2 D 2 32 cos θ · k 1 2 · n = 1 5 n · b n 2 σ e 2 .

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