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Subspace-based method for phase retrieval in interferometry

Abhijit Patil and Pramod Rastogi

Open Access Open Access

Abstract

A subspace-based method is applied to phase shifting interferometry for obtaining in real time values of phase shifts between data frames at each pixel point. A generalized phase extraction algorithm then allows for computing the phase distribution. The method is applicable to spherical beams and is capable of handling nonsinusoidal waveforms in an effective manner. Numerical simulations demonstrate phase measurement with high accuracy even in the presence of noise.

©2005 Optical Society of America

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Figures (4)
Fig. 1.
Fig. 1. Plot of the diagonal of S versus frequency for noiseless signal and signal with SNR=10 dB.

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Fig. 2.
Fig. 2. Plots of phase step values α (in degrees) obtained using forward-backward approach at an arbitrary pixel point with different values of N and m.

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Fig. 3.
Fig. 3. Plot shows typical wrapped phase φ (in radians) for phase step values obtained from Fig. 2(d) for 30dB noise.

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Fig. 4.
Fig. 4. Plot shows typical absolute error obtained in computation of phase φ (in radians) for phase step values determined from Fig. 2(d) for 30dB noise.

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Equations (21)

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I ( t ) = I d c + ∑ k = 1 κ a k e i k φ ︸ ℓ k u k t + ∑ k = 1 κ a k * e − i k φ ︸ ℓ k * ( u k * ) t + η ( t )
for  t = 0 , 1 , … m , … , N − 1
r ( p ) = E [ I ( t ) I * ( t − p ) ] = ∑ n = 0 2 κ A n 2 e i ω n p + σ 2 δ p , 0
R I = E [ I c ( t ) I ( t ) ] = [ r ( 0 ) r * ( 1 ) . . r * ( m − 1 ) r ( 1 ) . . . . . . . . . . . . . r * ( 1 ) r ( m − 1 ) . . . r ( 0 ) ]
R I = APA c ︸ R s + σ 2 I ︸ R ε
P = [ A 0 2 0 . 0 0 A 1 2 . . . . . . 0 . . A 2 κ 2 ]
R I G = G [ λ n + 1 0 . . 0 0 λ n + 2 . . . . . . . . . . . . 0 0 0 . . λ m ] = σ 2 G = APA c G + σ 2 G
a T ( z − 1 ) G ̂ G ̂ c a ( z ) = 0
φ ( x , y ) = 2 π λ ( x ′ − x ) 2 + ( x ′ − y ) 2
R ̂ I = 1 2 N ∑ t = m N [ [ I * ( t − 1 ) I * ( t − 2 ) . . I * ( t − m ) ] [ I ( t − 1 ) I ( t − 2 ) . . I ( t − m ) ] + [ I * ( t − m ) . . I * ( t − 2 ) I * ( t − 1 ) ] [ I ( t − m ) . . I ( t − 2 ) I ( t − 1 ) ] ]
[ e i κ α 0 e − i κ α 0 i ( κ − 1 ) α 0 1 e i κ α 1 e − i κ α 1 . 1 . . . . e i κ α ( N − 1 ) . . 1 ] [ ℓ κ ℓ κ * . I dc ] = [ I 0 I 1 . I N − 1 ]
I ( t ) = I dc + ∑ k = 1 κ a k e ik φ e i α kt + ∑ k = 1 κ a k e − ik φ e − i α kt + η ( t ) ;
r ( p ) = E { I ( t ) I * ( t − p ) }
I ( t ) = I dc + a 1 e i φ e i α t + a 1 e − i φ e − i α t + η ( t )
I * ( t − p ) = I dc + a 1 e i φ e i α ( t − p ) + a 1 e − i φ e − i α ( t − p ) + η * ( t − p )
r ( p ) = E { I ( t ) I * ( t − p ) } = E { I dc 2 + I dc a 1 e − i φ e − i α t + I dc a 1 e i φ e i α t + e i α p ( a 1 2 + I dc a 1 e − i φ e − i α t + a 1 2 e − 2 i φ e − 2 i α t ) + e − i α p ( a 1 2 + I dc a 1 e i φ e i α t + a 1 2 e 2 i φ e 2 i α t ) + η ( t ) η * ( t − p ) }
r ( p ) = E { I dc 2 + c 1 + e i α p ( a 1 2 + c 2 ) + e − i α p ( a 1 2 + c 3 ) + η ( t ) η * ( t − p ) }
r ( p ) = A 0 2 + A 1 2 e i α p + A 2 2 e − i α p + σ 2 δ p , 0
E { η ( k ) η * ( j ) } = σ 2 δ k , j E { η ( k ) η ( k ) } = 0 }
∫ 0 2 π e i ψ d ψ = 0
r ( p ) = E [ I ( t ) I * ( t − p ) ] = ∑ n = 0 2 κ A n 2 e i ω n p + σ 2 δ p , 0
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