Abstract

We present a method to introduce a linear phase into an interference pattern. This phase is introduced when a grating is placed outside of the Fourier plane of a two-aperture common-path interferometer, which is built using a 4f optical imaging system. Based on an analysis of near-field diffraction, the introduction of the carrier fringes is mathematically justified. It is important to note that no tilt between the two beams is requested to produce this effect, and it turns out to be simpler, easier, and more versatile than other existing methods. The main attributes and advantages of the setup will be discussed and illustrated in detail with experimental fringe patterns.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. C. Meneses-Fabian, G. Rodriguez-Zurita, J. F. Vazquez-Castillo, C. Robledo-Sanchez, and V. Arrizon, Opt. Commun. 264, 13 (2006)
    [CrossRef]
  5. H. Schreiber and J. H. Bruning, in Optical Shop Testing, D.Malacara, ed. (Wiley, 2007), pp. 547–666.
    [CrossRef]
  6. J. Schwider, in Progress in Optics XXVIII, E.Wolf, ed. (Elsevier Science, 1990), pp. 271–359
    [CrossRef]
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    [CrossRef]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

2006 (2)

C. Meneses-Fabian, G. Rodriguez-Zurita, J. F. Vazquez-Castillo, C. Robledo-Sanchez, and V. Arrizon, Opt. Commun. 264, 13 (2006)
[CrossRef]

C. Meneses-Fabian, G. Rodriguez-Zurita, and V. Arrizon, J. Opt. Soc. Am. A 23, 298 (2006)
[CrossRef]

2004 (1)

1982 (1)

1964 (1)

Arrizon, V.

C. Meneses-Fabian, G. Rodriguez-Zurita, J. F. Vazquez-Castillo, C. Robledo-Sanchez, and V. Arrizon, Opt. Commun. 264, 13 (2006)
[CrossRef]

C. Meneses-Fabian, G. Rodriguez-Zurita, and V. Arrizon, J. Opt. Soc. Am. A 23, 298 (2006)
[CrossRef]

Arrizón, V.

Bruning, J. H.

H. Schreiber and J. H. Bruning, in Optical Shop Testing, D.Malacara, ed. (Wiley, 2007), pp. 547–666.
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

Ina, H.

Kobayashi, S.

Meneses-Fabian, C.

C. Meneses-Fabian, G. Rodriguez-Zurita, and V. Arrizon, J. Opt. Soc. Am. A 23, 298 (2006)
[CrossRef]

C. Meneses-Fabian, G. Rodriguez-Zurita, J. F. Vazquez-Castillo, C. Robledo-Sanchez, and V. Arrizon, Opt. Commun. 264, 13 (2006)
[CrossRef]

Robledo-Sanchez, C.

C. Meneses-Fabian, G. Rodriguez-Zurita, J. F. Vazquez-Castillo, C. Robledo-Sanchez, and V. Arrizon, Opt. Commun. 264, 13 (2006)
[CrossRef]

Rodriguez-Zurita, G.

C. Meneses-Fabian, G. Rodriguez-Zurita, J. F. Vazquez-Castillo, C. Robledo-Sanchez, and V. Arrizon, Opt. Commun. 264, 13 (2006)
[CrossRef]

C. Meneses-Fabian, G. Rodriguez-Zurita, and V. Arrizon, J. Opt. Soc. Am. A 23, 298 (2006)
[CrossRef]

Ronchi, V.

Sánchez-de-la-Llave, D.

Schreiber, H.

H. Schreiber and J. H. Bruning, in Optical Shop Testing, D.Malacara, ed. (Wiley, 2007), pp. 547–666.
[CrossRef]

Schwider, J.

J. Schwider, in Progress in Optics XXVIII, E.Wolf, ed. (Elsevier Science, 1990), pp. 271–359
[CrossRef]

Takeda, M.

Vazquez-Castillo, J. F.

C. Meneses-Fabian, G. Rodriguez-Zurita, J. F. Vazquez-Castillo, C. Robledo-Sanchez, and V. Arrizon, Opt. Commun. 264, 13 (2006)
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

C. Meneses-Fabian, G. Rodriguez-Zurita, J. F. Vazquez-Castillo, C. Robledo-Sanchez, and V. Arrizon, Opt. Commun. 264, 13 (2006)
[CrossRef]

Opt. Lett. (1)

Other (3)

H. Schreiber and J. H. Bruning, in Optical Shop Testing, D.Malacara, ed. (Wiley, 2007), pp. 547–666.
[CrossRef]

J. Schwider, in Progress in Optics XXVIII, E.Wolf, ed. (Elsevier Science, 1990), pp. 271–359
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

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Figures (3)

Fig. 1
Fig. 1

Experimental setup: the modified TACPI.

Fig. 2
Fig. 2

Basic optical system consisting of a lens showing two planes.

Fig. 3
Fig. 3

Experimental fringe patterns obtained with the TACPI: (a) acetate sheet and spherical lens with Δ f = 0 , (b) without phase objects Δ f = 35 mm and Δ f = 43.7 mm , and (c) with phase objects Δ f = 35 mm and Δ f = 43.7 mm .

Equations (11)

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t ( ξ , η ) = 1 i λ z 1 e i k 2 z 1 ( ξ 2 + η 2 ) d x d y · t A ( x , y ) · e i k 2 z 1 ( x 2 + y 2 ) e i k z 1 ( x ξ + y η )
t ˜ ( u , v ) = 1 i λ f e e i k 2 f f z 1 f e ( u 2 + v 2 ) d x d y · t A ( x , y ) · e i k 2 f f z 2 f e ( x 2 + y 2 ) e i k f e ( x u + y v ) ,
t ˜ ( u , v ) = 1 i λ f d x d y · t A ( x , y ) · e i k 2 f Δ f f ( x 2 + y 2 ) e i k f ( x u + y v ) .
t ( x , v ) = 1 i λ f e i k 2 Δ f f 2 ( x 2 + y 2 ) d u d v · t ˜ ( u , v ) R ˜ ( u , v ) · e i k f ( u x + v y ) ,
R ˜ ( u , v ) = 1 2 n sinc ( 1 2 n ) e i 2 π n u p u .
t ( x , y ) = 1 2 n sinc ( 1 2 n ) t A ( x + λ f u p n , y ) e i k 2 Δ f f 2 [ ( λ f u p n ) 2 2 λ f u p n x ] ,
t A ( x , y ) = w ( x + 1 2 x 0 , y ) + w ( x 1 2 x 0 , y ) e i ϕ ( x 1 2 x 0 , y ) ,
t ( x , y ) = n C n [ w ( x + ( n + 1 2 ) x 0 , y ) + w ( x + ( n 1 2 ) x 0 , y ) e i ϕ ( x + ( n 1 2 ) x 0 , y ) ] ,
C n = 1 2 sinc ( 1 2 n ) exp { i k 2 Δ f f 2 [ ( n x 0 ) 2 2 n x 0 x ] } .
t 1 , 0 ( x , y ) = C 1 + C 0 e i ϕ ( x , y ) .
I 1 , 0 ( x , y ) = | t 1 , 0 ( x , y ) | 2 = 1 π 2 + 1 4 + 1 π cos [ ϕ ( x , y ) + 2 π μ 0 x + π λ Δ f u p 2 ] ,

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