Abstract

This Letter proposes a method to estimate phase derivatives of arbitrary order in digital holographic interferometry. Based on the desired order, the generalized complex-lag distribution is computed from the reconstructed interference field. Subsequently, the phase derivative is estimated by tracing the peak of the distribution. Simulation and experimental results are presented to validate the method’s potential.

© 2011 Optical Society of America

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References

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2010

2009

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef] [PubMed]

2007

C. Cornu, S. Stanković, C. Ioana, A. Quinquis, and L. Stanković, IEEE Trans. Signal Process. 55, 4831 (2007).
[CrossRef]

Q. Kemao, Opt. Lasers Eng. 45, 304 (2007).
[CrossRef]

2003

C. A. Sciammarella and T. Kim, Opt. Eng. 42, 3182 (2003).
[CrossRef]

1999

Barnes, T. H.

Cornu, C.

C. Cornu, S. Stanković, C. Ioana, A. Quinquis, and L. Stanković, IEEE Trans. Signal Process. 55, 4831 (2007).
[CrossRef]

Gorthi, S. S.

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, Opt. Express 18, 18041 (2010).
[CrossRef] [PubMed]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef] [PubMed]

Ioana, C.

C. Cornu, S. Stanković, C. Ioana, A. Quinquis, and L. Stanković, IEEE Trans. Signal Process. 55, 4831 (2007).
[CrossRef]

Kemao, Q.

Q. Kemao, Opt. Lasers Eng. 45, 304 (2007).
[CrossRef]

Kim, T.

C. A. Sciammarella and T. Kim, Opt. Eng. 42, 3182 (2003).
[CrossRef]

Quinquis, A.

C. Cornu, S. Stanković, C. Ioana, A. Quinquis, and L. Stanković, IEEE Trans. Signal Process. 55, 4831 (2007).
[CrossRef]

Rajshekhar, G.

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, Opt. Express 18, 18041 (2010).
[CrossRef] [PubMed]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef] [PubMed]

Rastogi, P.

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, Opt. Express 18, 18041 (2010).
[CrossRef] [PubMed]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef] [PubMed]

Sciammarella, C. A.

C. A. Sciammarella and T. Kim, Opt. Eng. 42, 3182 (2003).
[CrossRef]

Stankovic, L.

C. Cornu, S. Stanković, C. Ioana, A. Quinquis, and L. Stanković, IEEE Trans. Signal Process. 55, 4831 (2007).
[CrossRef]

Stankovic, S.

C. Cornu, S. Stanković, C. Ioana, A. Quinquis, and L. Stanković, IEEE Trans. Signal Process. 55, 4831 (2007).
[CrossRef]

Tan, S. M.

Watkins, L. R.

IEEE Trans. Signal Process.

C. Cornu, S. Stanković, C. Ioana, A. Quinquis, and L. Stanković, IEEE Trans. Signal Process. 55, 4831 (2007).
[CrossRef]

Opt. Eng.

C. A. Sciammarella and T. Kim, Opt. Eng. 42, 3182 (2003).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

Q. Kemao, Opt. Lasers Eng. 45, 304 (2007).
[CrossRef]

Opt. Lett.

Rev. Sci. Instrum.

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef] [PubMed]

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Figures (2)

Fig. 1
Fig. 1

(a) Original phase ϕ ( x , y ) in radians. (b) Fringe pattern. (c) Estimated first-order phase derivative ϕ ( 1 ) ( x , y ) in radians/pixel. (d) First-order phase derivative estimation error. (e) Estimated second-order phase derivative ϕ ( 2 ) ( x , y ) in radians / pixel 2 . (f) Second-order phase derivative estimation error.

Fig. 2
Fig. 2

(a) Experimental fringe pattern. (b) Estimated first-order phase derivative in radians/pixel. (c) Estimated second-order phase derivative in radians / pixel 2 .

Equations (20)

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Γ ( x , y ) = a ( x , y ) exp [ j ϕ ( x , y ) ] + η ( x , y ) ,
Γ ( x ) = a ( x ) exp [ j ϕ ( x ) ] + η ( x ) .
ϕ ( m ) ( x ) = m ϕ ( x ) x m .
GCM N m [ Γ ] ( x , τ ) = h ( τ ) k = 0 N 1 Γ ω N , k N m ( x + ω N , k m ! N τ m ) ,
GCM N m [ Γ ] ( x , τ ) = h ( τ ) exp [ j k = 0 N 1 ϕ ( x + ω N , k m ! N τ m ) ω N , k N m ] .
ϕ ( x + ω N , k m ! N τ m ) = p = 0 ϕ ( p ) ( x ) ω N , k p p ! ( m ! N τ m ) p ,
GCM N m [ Γ ] ( x , τ ) = h ( τ ) exp [ j k = 0 N 1 p = 0 ϕ ( p ) ( x ) ω N , k p + N m p ! ( m ! N τ m ) p ] .
k = 0 N 1 ω N , k p + N m = { N , if     p = 0   and   N = m N , if     p = N r + m 0 , otherwise ,
GCM N m [ Γ ] ( x , τ ) = h ( τ ) exp [ j N r = 0 ϕ ( N r + m ) ( x ) ( N r + m ) ! ( m ! N τ m ) N r + m ] = h ( τ ) exp [ j ϕ ( m ) ( x ) τ ] exp [ j U ( x , τ ) ] ,
U ( x , τ ) = N r = 1 ϕ ( N r + m ) ( x ) ( N r + m ) ! ( m ! N τ m ) N r + m .
GCM N m [ Γ ] ( x , τ ) = h ( τ ) exp [ j ϕ ( m ) ( x ) τ ] .
GCD N m [ Γ ] ( x , Ω ) = h ( τ ) exp [ j ϕ ( m ) ( x ) τ ] exp [ j Ω τ ] τ ,
GCD N m [ Γ ] ( x , Ω ) = H ^ ( Ω ϕ ( m ) ( x ) ) .
ϕ ( m ) ( x ) = arg max Ω | GCD N m [ Γ ] ( x , Ω ) | .
GCM N m [ Γ ] ( x , τ ) = h ( τ ) exp [ j ϕ ( m ) ( x ) τ ] exp [ j N ϕ ( x ) ] exp [ j U ( x , τ ) ] ,
GCM N m [ Γ ] ( x , τ ) = h ( τ ) exp [ j ϕ ( m ) ( x ) τ ] exp [ j N ϕ ( x ) ] ,
GCD N m [ Γ ] ( x , Ω ) = exp [ j N ϕ ( x ) ] H ^ ( Ω ϕ ( m ) ( x ) ) .
Γ ( x , y ) = exp [ j ϕ ( x , y ) ] ,
GCM 2 1 [ Γ ] ( x , τ ) = h ( τ ) Γ ( x + τ / 2 ) Γ 1 ( x τ / 2 ) .
GCM 2 2 [ Γ ] ( x , τ ) = h ( τ ) Γ ( x + τ ) Γ ( x τ ) .

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