Abstract

An iterative algorithm to analyze three-flat test data for absolute planarity measurements is presented. Using the properties of Zernike polynomial representations, results are achieved in a fast and effective manner. Details and demonstrative examples are provided.

© 2008 Optical Society of America

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References

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  1. Lord Rayleigh, "Interference bands and their application," Nature (London) 48,212-214 (1893).
    [CrossRef]
  2. R. E. Parks, "Removal of test optics errors," Proc. SPIE 153, 56-63 (1978).
  3. B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 33,379-383 (1984).
  4. Q1. V. B. Gubin and V. N. Sharonov, "Absolute calibration of spherical surfaces," Sov. J. Opt. Technol. 57, 554-555 (1990).
  5. V. Greco, R. Tronconi, C. Del Vecchio, M. Trivi, and G. Molesini, "Absolute measurement of planarity with Fritz’s method: uncertainty evaluation," Appl. Opt. 38,2018-2027 (1999).
    [CrossRef]
  6. M. Vannoni and G. Molesini, "Iterative algorithm for three flat test," Opt. Express 15, 6809-6816 (2007).
    [CrossRef] [PubMed]
  7. J. Grzanna and G. Schulz, "Absolute testing of flatness standards at square-grid points," Opt. Commun. 77,107-112 (1990).
    [CrossRef]
  8. Q2. V. Greco and G. Molesini, "Micro-temperature effects on absolute flatness test plates," Pure Appl. Opt. 7,1341-1346 (1998).
    [CrossRef]
  9. S. Brandt, Statistical and Computational Methods in Data Analysis, 2nd ed. (North-Holland, Amsterdam, 1970), p. 216.
  10. International Bureau of Weights and Measures, International Electrotechnical Commission, International Federation of Clinical Chemistry, International Organization for Standardization, International Union of Pure and Applied Chemistry, International Union of Pure and Applied Physics, and International Organization of Legal Metrology, Guide to the Expression of Uncertainty in Measurements (International Organization for Standardization, Geneva, 1993).

2007 (1)

1999 (1)

1998 (1)

Q2. V. Greco and G. Molesini, "Micro-temperature effects on absolute flatness test plates," Pure Appl. Opt. 7,1341-1346 (1998).
[CrossRef]

1990 (2)

J. Grzanna and G. Schulz, "Absolute testing of flatness standards at square-grid points," Opt. Commun. 77,107-112 (1990).
[CrossRef]

Q1. V. B. Gubin and V. N. Sharonov, "Absolute calibration of spherical surfaces," Sov. J. Opt. Technol. 57, 554-555 (1990).

1984 (1)

B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 33,379-383 (1984).

1978 (1)

R. E. Parks, "Removal of test optics errors," Proc. SPIE 153, 56-63 (1978).

1893 (1)

Lord Rayleigh, "Interference bands and their application," Nature (London) 48,212-214 (1893).
[CrossRef]

Appl. Opt. (1)

Nature (London) (1)

Lord Rayleigh, "Interference bands and their application," Nature (London) 48,212-214 (1893).
[CrossRef]

Opt. Commun. (1)

J. Grzanna and G. Schulz, "Absolute testing of flatness standards at square-grid points," Opt. Commun. 77,107-112 (1990).
[CrossRef]

Opt. Eng. (1)

B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 33,379-383 (1984).

Opt. Express (1)

Proc. SPIE (1)

R. E. Parks, "Removal of test optics errors," Proc. SPIE 153, 56-63 (1978).

Pure Appl. Opt. (1)

Q2. V. Greco and G. Molesini, "Micro-temperature effects on absolute flatness test plates," Pure Appl. Opt. 7,1341-1346 (1998).
[CrossRef]

Sov. J. Opt. Technol. (1)

Q1. V. B. Gubin and V. N. Sharonov, "Absolute calibration of spherical surfaces," Sov. J. Opt. Technol. 57, 554-555 (1990).

Other (2)

S. Brandt, Statistical and Computational Methods in Data Analysis, 2nd ed. (North-Holland, Amsterdam, 1970), p. 216.

International Bureau of Weights and Measures, International Electrotechnical Commission, International Federation of Clinical Chemistry, International Organization for Standardization, International Union of Pure and Applied Chemistry, International Union of Pure and Applied Physics, and International Organization of Legal Metrology, Guide to the Expression of Uncertainty in Measurements (International Organization for Standardization, Geneva, 1993).

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Figures (5)

Fig. 1.
Fig. 1.

Flow chart of data processing by the iterative approach with Zernike polynomials.

Fig. 2.
Fig. 2.

Normalized error function versus number of iteration cycles.

Fig. 3.
Fig. 3.

Map of uncorrelated residuals [interferogram Δ(KFM)] after processing a real data set with the iterative algorithm (a) and the constrained fitting approach (b).

Fig. 4.
Fig. 4.

Difference map of decorrelation residuals computed with the iterative algorithm and the constrained fitting procedure for the interferogram Δ(KFM).

Fig. 5.
Fig. 5.

Profile of surface L along the y-axis by (dotted curve) the classic three flat method and (solid curve) the iterative method.

Tables (1)

Tables Icon

Table 1. Peak-to-valley (P-V) and root mean square (rms) departures from absolute planarity of the surfaces K, L, M.

Equations (169)

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K = j = 1 N k j Z j , L = j = 1 N l j Z j , M = j = 1 N m j Z j
M R = j = 1 N m R j Z j .
K F = j = 1 N k F j Z j .
K F M = j = 1 N ( k F m ) j Z j ,
L F M = j = 1 N ( l F m ) j Z j ,
L F M R = j = 1 N ( l F m R ) j Z j ,
L F K = j = 1 N ( l F k ) j Z j .
K F M ¯ = j = 1 N ( k F m ) j ¯ Z j = j = 1 N ( k F j ¯ + m j ¯ ) Z j ,
L F M ¯ = j = 1 N ( l F m ) j ¯ Z j = j = 1 N ( l F j ¯ + m j ¯ ) Z j ,
L F M R ¯ = j = 1 N ( l F m R ) ¯ j Z j = j = 1 N ( l F j ¯ + m R j ¯ ) Z j ,
L F K ¯ = j = 1 N ( l F k ) j ¯ Z j = j = 1 N ( l F j ¯ + k j ¯ ) Z j .
Δ ( K F M ) = j = 1 N Δ ( k F m ) j Z j = K F M ¯ K F M = j = 1 N [ ( k F m ) j ¯ ( k F m ) j ] Z j
Δ ( K F M ) 2 = j = 1 N [ Δ ( k F m ) j ] 2 .
EF = Δ ( K F M ) 2 + Δ ( L F M ) 2 + Δ ( L F M R ) 2 + Δ ( L F K ) 2 .
K ¯ new = K ¯ + α [ Δ ( K F M ) ] F + α [ Δ ( L F K ) ] ,
L ¯ new = L ¯ + β [ Δ ( L F K ) ] F + β [ Δ ( L F M ) ] F + β [ Δ ( L F M R ) ] F ,
M ¯ new = M ¯ + β [ Δ ( K F M ) ] + β [ Δ ( L F M ) ] + β [ Δ ( L F M R ) ] R ,
( k F m ) j = k j + m j
( l F m ) j = l j + m j
( l F m R ) j = l j + m j
( l F k ) j = l j + k j
( k F m ) j , h ¯ = k j , h ¯ + m j , h ¯
( l F m ) j , h ¯ = l j , h ¯ + m j , h ¯
( l F m R ) j , h ¯ = l j , h ¯ + m j , h ¯
( l F k ) j , h ¯ = l j , h ¯ + k j , h ¯
k j , h + 1 ¯ = 9 10 k j , h ¯ + 1 10 k j + 1 20 ( l j l j , h ¯ ) + 1 20 ( m j m j , h ¯ )
l j , h + 1 ¯ = 9 10 l j , h ¯ + 1 10 l j + 1 30 ( k j k j , h ¯ ) + 1 15 ( m j m j , h ¯ )
m j , h + 1 ¯ = 9 10 m j , h ¯ + 1 10 m j + 1 30 ( k j k j , h ¯ ) + 1 15 ( l j l j , h ¯ )
n odd, sine term k F j = k j n odd, cosine term k F j = k j n even, sine term k F j = k j n even, cosine term k F j = k j
a = ( k F m ) j = k j + m j
b = ( l F m ) j = l j + m j
c = ( l F m R ) j = l j + m j = b
d = ( l F k ) j = l j + k j
K = k h , j ¯ , L = l h , j ¯ , M = m h , j ¯ .
A = ( k F m ) j , h ¯ = k j , h ¯ + m j , h ¯ = K + M
B = ( l F m ) j , h ¯ = l j , h ¯ + m j , h ¯ = L + M
C = ( l F m R ) j , h ¯ = l j , h ¯ + m j , h ¯ = L + M = B
D = ( l F k ) j , h ¯ = l j , h ¯ + k j , h ¯ = L + K
EF = ( a A ) 2 + 2 ( b B ) 2 + ( d D ) 2
K = k j , h + 1 ¯ = K + α ( a A ) + α ( d D )
L = l j , h + 1 ¯ = L + β ( d D ) + 2 β ( b B )
M = m j , h + 1 ¯ = M + β ( a A ) + 2 β ( b B )
A = K + M = K + M + ( α + β ) ( a A ) + 2 β ( b B ) + α ( d D )
B = L + M = L + M + β ( a A ) + 4 β ( b B ) + β ( d D )
D = L + K = L + K + α ( a A ) + 2 β ( b B ) + ( α + β ) ( d D )
EF = ( a A ) 2 + 2 ( b B ) 2 + ( d D ) 2
EF = [ a A ( α + β ) ( a A ) 2 β ( b B ) α ( d D ) ] 2
+ 2 [ b B β ( a A ) 4 β ( b B ) β ( d D ) ] 2
+ [ d D α ( a A ) 2 β ( b B ) ( α + β ) ( d D ) ] 2
EF = ( a A ) 2 ( 1 2 α 2 β + 2 α 2 + 3 β 2 + 2 α β )
  + 2 ( b B ) 2 ( 1 8 β + 20 β 2 )
    + ( d D ) 2 ( 1 2 α 2 β + 2 α 2 + 3 β 2 + 2 α β )
+ ( a A ) ( b B ) ( 8 β + 20 β 2 + 8 α β )
+ ( a A ) ( d D ) ( 4 α + 4 α 2 + 4 α β + 4 β 2 )
+ ( b B ) ( d D ) ( 8 β + 20 β 2 + 8 α β )
8 β + 20 β 2 + 8 α β = 0
4 α + 4 α 2 + 4 α β + 4 β 2 = 0
EF = 169 361 [ ( a A ) 2 + 2 ( b B ) 2 + ( d D ) 2 ] 0.468 EF
w 11 α + w 12 β + w 13 = 0
w 21 α + w 22 β + w 23 = 0
w 11 = 4 ( a A ) 2 + 4 ( d D ) 2 + 8 ( a A ) ( d D )
w 12 = 2 ( a A ) 2 + 2 ( d D ) 2 + 8 ( a A ) ( b B ) + 4 ( a A ) ( d D )
+ 8 ( b B ) ( d D )
w 13 = 2 ( a A ) 2 2 ( d D ) 2 4 ( a A ) ( d D )
w 21 = 2 ( a A ) 2 + 2 ( d D ) 2 + 8 ( a A ) ( b B ) + 4 ( a A ) ( d D )
+ 8 ( b B ) ( d D )
w 22 = 6 ( a A ) 2 + 80 ( b B ) 2 + 6 ( d D ) 2 + 40 ( a A ) ( b B )
+ 8 ( a A ) ( d D ) + 40 ( b B ) ( d D )
w 23 = 2 ( a A ) 2 16 ( b B ) 2 2 ( d D ) 2 8 ( a A ) ( b B ) 8 ( b B ) ( d D )
α sd = w 13 w 22 + w 12 w 23 w 11 w 22 w 12 w 21 , β sd = w 11 w 23 + w 21 w 13 w 11 w 22 w 12 w 21
m R , p = cos φ m p sin φ m q
m R , q = cos φ   m q + sin φ m p
a p = k F p + m p a q = k F q + m q
b p = l F p + m p b q = l F q + m q
c p = l F p + m Rp c q = l Fq + m Rq
d p = l Fp + k p d q = l Fq + k q
A p = K F p + M p = K p + M p A q = K Fq + M q = K q + M q
B p = L Fp + M p = L p + M p B q = L Fq + M q = L q + M q
C p = C Fp + M Rp = L p + ρ M p σ M q C q = L Fq + M Rq = L q + ρM q + σ M p
D p = L Fp + K p = L p + K p D q = L Fq + K q = L q + K q
EF = ( a p A p ) 2 + ( b p B p ) 2 + ( c p C p ) + ( d p D p ) 2
+ ( a q A q ) 2 + ( b q B q ) 2 + ( c q C q ) + ( d q D q ) 2
K p = K p + α ( a p A p ) + α ( d p D p )
L p = L p + β ( b p B p ) + β ( c p C p ) + β ( d p D p )
M p = M p + β ( a p A p ) + β ( b p B p ) + β ρ ( c p C p ) + σ ( c q C q )
K q = K q α ( a q A q ) + α ( d q D q )
L q = L q β ( b q B q ) β ( c q C q ) β ( d q D q )
M q = M q + β ( a q A q ) + β ( b q B q ) + β ρ ( c q C q ) σ ( c p C p )
EF = a p A p ( α + β ) ( a p A p ) β ( b p B p ) β ρ ( c p C p ) α ( d p D p )
β σ ( c q C q ) ] 2
+ b p B p β ( a p A p ) 2 β ( b p B p ) ( β + ρ β ) ( c p C p ) β ( d p D p )
β σ ( c q C q ) ] 2
+ c p C p β ρ ( a p A p ) ( β + ρ β ) ( b p B p ) 2 β ( c p C p ) β ( d p D p )
+ β σ ( a q A q ) + β σ ( b q B q ) ] 2
+ [ d p D p α ( a p A p ) β ( b p B p ) β ( c p C p ) ( α + β ) ( d p D p ) ] 2
+ a q A q ( α + β ) ( a q A q ) β ( b q B q ) β ρ ( c q C q ) + α ( d q D q )
+ β σ ( c p C p ) ] 2
  + b q B q β ( a q A q ) 2 β ( b q B q ) ( β + ρ β ) ( c q C q ) β ( d q D q )
+ β σ ( c p C p ) ] 2
+ c q C q βρ ( a q A q ) ( β + ρ β ) ( b q B q ) 2 β ( c q C q ) β ( d q D q )
β σ ( a p A p ) β σ ( b p B p ) ] 2
+ [ d q D q + α ( a q A q ) β ( b q B q ) β ( c q C q ) ( α + β ) ( d q D q ) ] 2
EF ʹ = ( a p A p ) 2 ( 1 + 2 α 2 + 2 α β + 3 β 2 2 α 2 β )
+ ( b p B p ) 2 [ 1 + ( 8 + 2 ρ ) β 2 4 β ]
+ ( c p C p ) 2 [ 1 + ( 8 + 2 ρ ) β 2 4 β ]
+ ( d p D p ) 2 ( 1 + 2 α 2 + 2 α β + 3 β 2 2 α 2 β )
+ ( a q A q ) 2 ( 1 + 2 α 2 + 2 α β + 3 β 2 2 α 2 β )
+ ( b q B q ) 2 [ 1 + ( 8 + 2 ρ ) β 2 4 β ]
+ ( b q B q ) 2 [ 1 + ( 8 + 2 ρ ) β 2 4 β ]
+ ( d q D q ) 2 ( 1 + 2 α 2 + 2 α β + 3 β 2 2 α 2 β )
+ ( a p A p ) ( b p B p ) ( 8 + 2 ρ ) β 2 + 4 α β 4 β
+ ( a p A p ) ( c p C p ) ( 2 + 8 ρ ) β 2 + ( 2 + 2 ρ ) α β 4 ρ β
+ ( a p A p ) ( d p D p ) [ 4 α 2 + 4 α β + ( 2 + 2 ρ ) β 2 4 α ]
+ ( b p B p ) ( c p C p ) [ ( 10 + 10 ρ ) β 2 ( 4 + 4 ρ ) β ]
+ ( b p B p ) ( d p D p ) [ ( 8 + 2 ρ ) β 2 + 4 α β 4 β ]
+ ( c p C p ) ( d p D p ) ( 8 + 2 ρ ) β 2 + ( 2 + 2 ρ ) α β 4 β
+ ( a q A q ) ( b q B q ) ( 8 + 2 ρ ) β 2 4 β
+ ( a q A q ) ( c q C q ) ( 2 + 8 ρ ) β 2 ( 2 2 ρ ) α β 4 ρ β
+ ( a q A q ) ( d q D q ) [ 4 α 2 4 α β + ( 2 + 2 ρ ) β 2 + 4 α ]
+ ( b q B q ) ( c q C q ) ( 10 + 10 ρ ) β 2 ( 4 + 4 ρ ) β
+ ( b q B q ) ( d q D q ) ( 8 + 2 ρ ) β 2 4 β
+ ( c q C q ) ( d q D q ) ( 8 + 2 ρ ) β 2 + ( 2 2 ρ ) α β 4 β
+ ( a p A p ) ( b q B q ) ( 2 σ β 2 )
+ ( a p A p ) ( c q C q ) ( 8 σ β 2 + 2 σ α β 4 σ β )
+ ( a p A p ) ( d q D q ) ( 2 σ β 2 )
+ ( b p B p ) ( a q A q ) ( 2 σ β 2 )
+ ( b p B p ) ( c q C q ) ( 10 σ β 2 4 σ β )
+ ( b p B p ) ( d q D q ) ( 2 σ β 2 )
+ ( c p C p ) ( a q A q ) ( 8 σ β 2 2 σ α β + 4 σ β )
+ ( c p C p ) ( b q B q ) ( 10 σ β 2 + 4 σ β )
+ ( c p C p ) ( d q D q ) ( 2 σ β 2 + 2 σ α β )
+ ( d p D p ) ( a q A q ) ( 2 σ β 2 )
+ ( d p D p ) ( b q B q ) ( 2 σ β 2 )
+ ( d p D p ) ( c q C q ) ( 2 σ β 2 + 2 σ α β )
w 11 = 4 ( a p A p ) 2 + 4 ( d p D p ) 2 + 4 ( a q A q ) 2 + 4 ( d q D q ) 2
      + 8 ( a p A p ) ( d p D p ) 8 ( a q A q ) ( d q D q )
w 12 = 2 ( a p A p ) 2 + 2 ( d p D p ) 2 + 2 ( a q A q ) 2 + 2 ( d q D q ) 2
+ 4 ( a p A p ) ( b p B p ) + ( 2 + 2 ρ ) ( a p A p ) ( c p C p ) + 4 ( a p A p ) ( d p D p )
+ 4 ( b p B p ) ( d p D p ) + ( 2 + 2 ρ ) ( c p C p ) ( d p D p )
( 2 2 ρ ) ( a q A q ) ( c q C q ) 4 ( a q A q ) ( d q D q )
+ ( 2 2 ρ ) ( c q C q ) ( d q D q ) + 2 σ ( a p A p ) ( c q C q )
2 σ ( c p C p ) ( a q A q ) + 2 σ ( c p C p ) ( d q D q ) + 2 σ ( d p D p ) ( c q C q )
w 13 = 2 ( a p A p ) 2 2 ( d p D p ) 2 2 ( a q A q ) 2 2 ( d q D q ) 2
4 ( a p A p ) ( d p D p ) + 4 ( a q A q ) ( d q D q )
w 21 = 2 ( a p A p ) 2 + 2 ( d p D p ) 2 + 2 ( a q A q ) 2 + 2 ( d q D q ) 2
+ 4 ( a p A p ) ( b p B p ) + ( 2 + 2 ρ ) ( a p A p ) ( c p C p )
+ 4 ( a p A p ) ( d p D p ) + 4 ( b p B p ) ( d p D p ) + ( 2 + 2 ρ ) ( c p C p ) ( d p D p )
( 2 2 ρ ) ( a q A q ) ( c q C q ) 4 ( a q A q ) ( d q D q )
+ ( 2 2 ρ ) ( c q C q ) ( d q D q ) + 2 σ ( a p A p ) ( c q C q )
2 σ ( c p C p ) ( a q A q ) + 2 σ ( c p C p ) ( d q D q ) + 2 σ ( d p D p ) ( c q C q )
w 22 = 6 ( a p A p ) 2 + 2 ( 8 + 2 ρ ) ( b p B p ) 2 + 2 ( 8 + 2 ρ ) ( c p C p ) 2 + 6 ( d p D p ) 2
+ 6 ( a q A q ) 2 + 2 ( 8 + 2 ρ ) ( b q B q ) 2 + 2 ( 8 + 2 ρ ) ( c q C q ) 2 + 6 ( d q D q ) 2
+ 2 ( 8 + 2 ρ ) ( a p A p ) ( b p B p ) + 2 ( 8 + 2 ρ ) ( a p A p ) ( c p C p )
+ 2 ( 8 + 2 ρ ) ( a p A p ) ( d p D p ) + 2 ( 10 + 10 ρ ) ( b p B p ) ( c p C p )
+ 2 ( 8 + 2 ρ ) ( b p B p ) ( d p D p ) + 2 ( 8 + 2 ρ ) ( c p C p ) ( d p D p )
+ 2 ( 8 + 2 ρ ) ( a q A q ) ( b q B q ) + 2 ( 8 + 2 ρ ) ( a q A q ) ( c q C q )
+ 2 ( 8 + 2 ρ ) ( a q A q ) ( d q D q ) + 2 ( 10 + 10 ρ ) ( b q B q ) ( c q C q )
+ 2 ( 8 + 2 ρ ) ( b q B q ) ( d q D q ) + 2 ( 8 + 2 ρ ) ( c q C q ) ( d q D q )
+ 4 σ ( a p A p ) ( b q B q ) + 16 σ ( a p A p ) ( c q C q ) + 4 σ ( a p A p ) ( d q D q )
4 σ ( b p B p ) ( a q A q ) + 20 σ ( b p B p ) ( c q C q ) + 4 σ ( b p B p ) ( d q D q )
    16 σ ( c p C p ) ( a q A q ) 20 σ ( c p C p ) ( b q B q ) 4 σ ( c p C p ) ( d q D q )
    4 σ ( d p D p ) ( a q A q ) 4 σ ( d p D p ) ( b q B q ) + 4 σ ( d p D p ) ( c q C q )
w 23 = 2 ( a p A p ) 2 4 ( b p B p ) 2 4 ( c p C p ) 2 2 ( d p D p ) 2 2 ( a q A q ) 2
4 ( b q B q ) 2 4 ( c q C q ) 2 2 ( d q D q ) 2 4 ( a p A p ) ( b p B p )
4 ρ ( a p A p ) ( c p C p ) ( 4 + 4 ρ ) ( b p B p ) ( c p C p ) 4 ( b p B p ) ( d p D p )
    4 ( c p C p ) ( d p D p ) 4 ( a q A q ) ( b q B q ) 4 ρ ( a q A q ) ( c q C q )
( 4 + 4 ρ ) ( b q B q ) ( c q C q ) 4 ( b q B q ) ( d q D q ) 4 ( c q C q ) ( d q D q )
4 σ ( a p A p ) ( c q C q ) 4 σ ( b p B p ) ( c q C q ) + 4 σ ( c p C p ) ( a q A q )
+ 4 σ ( c p C p ) ( b q B q )

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