Abstract

For the purpose of measuring aspheric surfaces, an improved type of computer generated hologram, named the inclined bar type hologram, is proposed. This hologram is characterized by inclined bars. The inclination angle for each bar is obtained from the calculated wavefront aberration of the test lens. A few problems with these computer generated holograms are analyzed, and the advantage of the inclined bar type hologram over the conventional one is demonstrated by experiment.

© 1976 Optical Society of America

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References

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  1. A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [CrossRef] [PubMed]
  2. A. J. MacGavern, J. C. Wyant, Appl. Opt. 10, 619 (1971).
    [CrossRef]
  3. J. C. Wyant, V. P. Bennett, Appl. Opt. 11, 2833 (1972).
    [CrossRef] [PubMed]
  4. T. Takahashi, K. Konno, M. Kawai, Jpn. J. Appl. Phys. 14, 247 (1975), Suppl. 14–1.
  5. W. H. Lee, Appl. Opt. 13, 1677 (1974).
    [CrossRef] [PubMed]

1975 (1)

T. Takahashi, K. Konno, M. Kawai, Jpn. J. Appl. Phys. 14, 247 (1975), Suppl. 14–1.

1974 (1)

1972 (1)

1971 (1)

1967 (1)

Bennett, V. P.

Kawai, M.

T. Takahashi, K. Konno, M. Kawai, Jpn. J. Appl. Phys. 14, 247 (1975), Suppl. 14–1.

Konno, K.

T. Takahashi, K. Konno, M. Kawai, Jpn. J. Appl. Phys. 14, 247 (1975), Suppl. 14–1.

Lee, W. H.

Lohmann, A. W.

MacGavern, A. J.

Paris, D. P.

Takahashi, T.

T. Takahashi, K. Konno, M. Kawai, Jpn. J. Appl. Phys. 14, 247 (1975), Suppl. 14–1.

Wyant, J. C.

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of an intensity distribution on the hologram plane along a cross section CC′. Shadowed squares indicate the apertures of the hologram.

Fig. 2
Fig. 2

Ha(tanζ) vs tanζ. Solid lines show the values of nm that make am the nearest value to 0.1 mm.

Fig. 3
Fig. 3

Relation between the reference and object wavefront at the hologram plane.

Fig. 4
Fig. 4

Phase error vs Δam/am.

Fig. 5
Fig. 5

Interferometer: 1, laser; 2, objective lens; 3, collimator lens; 4, beam splitter; 5, mirror; 6, test lens; 7, CGH; 8, projection lens; 9, spatial filter; 10, projection screen.

Fig. 6
Fig. 6

Comparison of interference patterns: left, using the Lohmann-type hologram; right, using the inclined bar type hologram.

Equations (30)

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F b ( y ) = m = Rect [ y y m Δ y ] ,
Rect = { 1 1 2 ξ 1 2 , 0 elsewhere .
a m = y m + 1 y m .
F m ( y ) = n = C n exp ( 2 π i n y a m ) ,
C n = 1 a m Δ y 2 + y m Δ y 2 + y m exp ( 2 π i n y a m ) d y ,
F m ( y ) = n = Δ y a m sinc [ n π Δ y a m ] exp [ 2 π n i a m ( y y m ) ] ,
sinc ( ξ ) = sin ξ / ξ .
F b ( y ) = m = Rect [ y y m a m 1 2 ] Δ y a m n = sinc [ n π Δ y a m ] exp [ 2 π n i a m ( y y m ) ] .
E 0 ( y ) = exp [ i k Φ ( y ) ] , k = 2 π / λ ( λ = wavelength ) , and
H a ( y ) = F b ( y ) E 0 ( y ) .
E 0 ( y ) E ( y ) = m = Rect ( y y m a m 1 2 ) exp { i k [ Φ ( y m ) + Φ ( y y m ) tan ζ m ] } ,
tan ζ m = Φ ( y m ) Φ ( y m + 1 ) a m
H a ( y ) = F b ( y ) E ( y ) = m = Rect ( y y m a m 1 2 ) Δ y a m n = sinc ( π n Δ y a m ) exp { 2 n π i a m ( y y m ) + i k [ ( y y m ) Φ ( y m ) Φ ( y m + 1 ) a m + Φ ( y m ) ] } .
Φ ( y m ) Φ ( y m + 1 ) = n m λ , ( n m , integer )
a m = n m λ / tan ζ m
H a ( y ) = m = Rect ( y y m a m 1 2 ) 1 n m π × sin ( π Δ y tan ζ m λ ) exp [ i k Φ ( y 0 ) ] .
E 0 ( y ) = exp { i k [ Φ ( y ) + Δ Φ ( y ) ] } ,
H a ( y ) = m = Rect ( y y m a m 1 2 ) n = sinc ( π n Δ y a m ) × exp { 2 n π i a m ( y y m ) + i k [ ( y y m ) Φ ( y m ) Φ ( y m + 1 ) a m + Δ Φ ( y m ) ] }
H a ( y ) = m = Rect ( y y m a m 1 2 ) 1 n m π sin ( tan ζ m Δ y π λ ) × exp { i k [ Δ Φ ( y ) + Φ ( y 0 ) ] } .
x = x u + l 2 r , y = y u + m 2 r , z = z u + n 2 r . }
l 1 x + m 1 y + n 1 z = 0 .
r = l 1 x u + m 1 y u + n 1 z u l 1 l 2 + m 1 m 2 + n 1 n 2 .
Q + r = l λ + S ,
cos θ = [ ( n 1 2 + m 1 2 ) l 2 ( n 1 n 2 + m 1 m 2 ) l 1 ] l 2 + [ ( l 1 2 + n 1 2 ) m 2 ( l 1 l 2 + n 1 n 2 ) m 1 ] m 2 + [ ( m 1 2 + l 1 2 ) n 2 ( m 1 m 2 + l 1 l 2 ) n 1 ] n 2
y p = y u + S cos θ { [ ( l 1 2 + n 1 2 ) m 2 ( l 1 l 2 + n 1 n 2 ) m 1 ] m 2 l 2 [ ( n 1 2 + m 1 2 ) l 2 ( n 1 n 2 + m 1 m 2 ) l 1 ] }
z p = z u + S cos θ { [ ( m 1 2 + l 1 2 ) n 2 ( m 1 m 2 + l 1 l 2 ) n 1 ] m 2 l 2 [ ( n 1 2 + m 1 2 ) l 2 ( n 1 n 2 + m 1 m 2 ) l 1 ] } .
L 2 = 0 , M 2 = n 2 ( m 1 m 2 + l 1 l 2 ) n 1 ( m 2 2 + l 2 2 ) , N 2 = [ m 2 ( l 1 l 2 + n 1 n 2 ) m 1 ( l 2 2 + n 2 2 ) ] .
a m = 1 2 f ( ξ ) a m ( Lohmannt type )
a m = 1 8 f ( ξ ) a m 2 ( inclined bar type )
M = Δ a m n m a m + Δ a m

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