Abstract

We have applied the technique of Fourier fringe analysis to microscopic interferograms of needle crystals that grow from a solution. We use a differential technique in which an empty field interferogram is compared with one that contains distortion and obscuration by the growing crystal, and we demonstrate both analytically and experimentally a phase shift sensitivity of 0.01 fringe with a spatial resolution of half of a fringe spacing (~1 μm). Following the analysis of the interferogram in two dimensions, we show that the three-dimensional refractive-index field around the crystal can be deduced, assuming that it is axially symmetric, by an iterative method.

© 1993 Optical Society of America

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References

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  1. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based tomography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  2. M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef] [PubMed]
  3. Y. Morimoto, Y. Seguchi, T. Higashi, “Application of moiré analysis of strain using Fourier transform,” Opt. Eng. 27, 650–656 (1988).
  4. W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
    [CrossRef] [PubMed]
  5. D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).
    [CrossRef] [PubMed]
  6. C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
    [CrossRef] [PubMed]
  7. E. Ben-Jacob, P. Garik, “The formation of patterns in non-equilibrium growth,” Nature (London) 343, 523–530 (1990).
    [CrossRef]
  8. E. Raz, S. G. Lipson, E. Polturak, “Dendritic growth of ammonium chloride crystals: measurement of the concentration field and a proposed nucleation model for growth,” Phys. Rev. A 40, 1088–1095 (1989).
    [CrossRef] [PubMed]
  9. G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980), Chap. 6.

1990 (1)

E. Ben-Jacob, P. Garik, “The formation of patterns in non-equilibrium growth,” Nature (London) 343, 523–530 (1990).
[CrossRef]

1989 (1)

E. Raz, S. G. Lipson, E. Polturak, “Dendritic growth of ammonium chloride crystals: measurement of the concentration field and a proposed nucleation model for growth,” Phys. Rev. A 40, 1088–1095 (1989).
[CrossRef] [PubMed]

1988 (1)

Y. Morimoto, Y. Seguchi, T. Higashi, “Application of moiré analysis of strain using Fourier transform,” Opt. Eng. 27, 650–656 (1988).

1987 (1)

1986 (1)

1983 (2)

1982 (1)

Bachor, H.-A.

Ben-Jacob, E.

E. Ben-Jacob, P. Garik, “The formation of patterns in non-equilibrium growth,” Nature (London) 343, 523–530 (1990).
[CrossRef]

Bone, D. J.

Garik, P.

E. Ben-Jacob, P. Garik, “The formation of patterns in non-equilibrium growth,” Nature (London) 343, 523–530 (1990).
[CrossRef]

Herman, G. T.

G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980), Chap. 6.

Higashi, T.

Y. Morimoto, Y. Seguchi, T. Higashi, “Application of moiré analysis of strain using Fourier transform,” Opt. Eng. 27, 650–656 (1988).

Ina, H.

Kobayashi, S.

Lipson, S. G.

E. Raz, S. G. Lipson, E. Polturak, “Dendritic growth of ammonium chloride crystals: measurement of the concentration field and a proposed nucleation model for growth,” Phys. Rev. A 40, 1088–1095 (1989).
[CrossRef] [PubMed]

Macy, W. W.

Morimoto, Y.

Y. Morimoto, Y. Seguchi, T. Higashi, “Application of moiré analysis of strain using Fourier transform,” Opt. Eng. 27, 650–656 (1988).

Mutoh, K.

Polturak, E.

E. Raz, S. G. Lipson, E. Polturak, “Dendritic growth of ammonium chloride crystals: measurement of the concentration field and a proposed nucleation model for growth,” Phys. Rev. A 40, 1088–1095 (1989).
[CrossRef] [PubMed]

Raz, E.

E. Raz, S. G. Lipson, E. Polturak, “Dendritic growth of ammonium chloride crystals: measurement of the concentration field and a proposed nucleation model for growth,” Phys. Rev. A 40, 1088–1095 (1989).
[CrossRef] [PubMed]

Roddier, C.

Roddier, F.

Sandeman, R. J.

Seguchi, Y.

Y. Morimoto, Y. Seguchi, T. Higashi, “Application of moiré analysis of strain using Fourier transform,” Opt. Eng. 27, 650–656 (1988).

Takeda, M.

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

Nature (London) (1)

E. Ben-Jacob, P. Garik, “The formation of patterns in non-equilibrium growth,” Nature (London) 343, 523–530 (1990).
[CrossRef]

Opt. Eng. (1)

Y. Morimoto, Y. Seguchi, T. Higashi, “Application of moiré analysis of strain using Fourier transform,” Opt. Eng. 27, 650–656 (1988).

Phys. Rev. A (1)

E. Raz, S. G. Lipson, E. Polturak, “Dendritic growth of ammonium chloride crystals: measurement of the concentration field and a proposed nucleation model for growth,” Phys. Rev. A 40, 1088–1095 (1989).
[CrossRef] [PubMed]

Other (1)

G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980), Chap. 6.

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the experimental system: 1, microscope stage; 2, tilting table; 3, water-cooled chamber; 4, thermoelectric (Peltier) coolers; 5, copper base; 6, thin mirror (sapphire); 7, wire spacers; 8, sample; 9, cover glass; 10, dry nitrogen flow (to prevent mist condensation); 11, Linnik interferometer.

Fig. 2
Fig. 2

Illustration of the Fourier plane operations: (a) contour map of the imposed phase variation and obstruction (contours at intervals of 0.02π), (b) the fringe field corresponding to (a), (c) Fourier transform of (b) (amplitude contours) with the origin at the center of the field, (d) transform shifted so that the first order is at the origin, (e) transform after removal of zero and minus first orders, (f) reconstructed contour map of phases with the same scale as (a).

Fig. 3
Fig. 3

Fourier transform of the model field: (a) in its initial form, (b) after the apodized window operation, (c) the two vectors that contribute to the phase field reconstruction, (d) comparison of the amplitudes of the two contributions to the reconstruction for σ = ω0/3.

Fig. 4
Fig. 4

Geometry of the tomographic analysis.

Fig. 5
Fig. 5

Results obtained for a needle crystal growing from supersaturated solution: (a) the interferogram, (b) the optical path l(x, y) (contour map at intervals of 0.1π), (c) the concentration field c(x, y, 0) in the plane that contains the axis of the crystal [contour map at intervals of 0.1% from Eq. (2)]. Note the impressed axial symmetry and the restricted region of the field for which calculations were possible.

Equations (23)

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l ( x , y ) = z 0 z 0 n ( x , y , z ) 1 d z .
n ( c ) = 1 . 3325 + 0 . 0019 c ,
g 0 ( x , y ) = a ( x , y ) + 2 b ( x , y ) cos [ ω 0 x + ϕ 0 ( x , y ) ] ,
g ( x , y ) = d ( x , y ) { a ( x , y ) + 2 b ( x , y ) cos [ ω 0 x + ϕ ( x , y ) + ϕ 0 ( x , y ) ] .
G 0 ( u , υ ) = A ( u , υ ) + B ( u , υ ) * [ δ ( u ω 0 ) δ ( υ ) * P 0 ( u , υ ) + δ ( u + ω 0 ) δ ( υ ) * P 0 * ( u , υ ) ] ,
G 0 ( u , υ ) = 2 B ( u , υ ) * P 0 ( u , υ ) ,
G ( u , υ ) = 2 B ( u , υ ) * D ( u , υ ) * P 0 ( u , υ ) * P ( u , υ ) .
r ( x , y ) = g ( x , y ) g 0 ( x , y ) = d ( x , y ) exp [ i ϕ ( x , y ) ] .
d ( x , y ) = | r ( x , y ) | ,
ϕ ( x , y ) = phase [ r ( x , y ) ] .
g 0 ( x , y ) = 1 + cos ω 0 x ,
g ( x , y ) = [ 1 rect ( x / a 0 ) rect ( y / b 0 ) ] [ 1 + cos ( ω 0 x + ϕ ) ] .
G ( u , υ ) = [ δ ( u ) δ ( υ ) 4 a 0 b 0 sinc ( a 0 u ) sinc ( b 0 υ ) ] * [ δ ( u ) δ ( υ ) + 1 2 exp ( i ϕ ) δ ( u ω 0 ) δ ( υ ) + 1 2 exp ( i ϕ ) δ ( u + ω 0 ) δ ( υ ) ] .
G 1 ( u , υ ) = exp ( i ϕ ) [ 1 2 δ ( u ) δ ( υ ) 2 a 0 b 0 sinc ( a 0 u ) × sinc ( b 0 υ ) ] exp [ ( u 2 + υ 2 ) / 2 σ 2 ] ,
G 2 ( u , υ ) = 4 a 0 b 0 sinc [ a 0 ( u ω 0 ) ] × sinc ( b 0 υ ) exp [ ( u 2 + υ 2 ) / 2 σ 2 ] + exp [ ( ω 0 2 / 2 σ 2 ) ] δ ( u ω 0 ) δ ( υ ) .
g 1 ( x , y ) = 1 2 exp ( i ϕ ) [ 1 rect ( x / a 0 ) rect ( y / b 0 ) ] * σ 2 π exp [ ( x 2 + y 2 ) σ 2 / 2 ] ,
g 2 ( x , y ) = exp ( i x ω 0 ) exp ( ω 0 2 / 2 σ 2 ) [ exp ( i x ω 0 ) rect ( x / a 0 ) rect ( y / b 0 ) ] * σ 2 π exp [ ( x 2 + y 2 ) σ 2 / 2 ] .
l ( x , y ) = 2 A E n ( x , y , z ) d z = 2 A F n ( x , y , 0 ) ( d z d x ) d x ,
l ( x , y ) = 2 A F n ( x , y , 0 ) x ( x 2 x 2 ) 1 / 2 d x .
l 1 ( x , y ) = 2 α 0 A F l ( x , y ) x ( x 2 x 2 ) 1 / 2 d x .
α 1 ( x , y ) = α 0 l ( x , y ) / l 1 ( x , y ) .
l 2 ( x , y ) = 2 A F α 1 ( x , y ) l ( x , y ) x ( x 2 x 2 ) 1 / 2 d x .
χ n 2 = [ l n ( x , y ) l ( x , y ) ] 2 l 2 ( x , y ) d x d y

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