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Super-resolution Fourier transform method in phase shifting interferometry

Rajesh Langoju, Abhijit Patil, and Pramod Rastogi

Open Access Open Access

Abstract

The paper proposes a super-resolution Fourier transform method for phase estimation in phase shifting interferometry. Incorporation of a super-resolution technique before the application of Fourier transform significantly enhances the resolution capability of the proposed method. The other salient features of the method lie in its ability to handle multiple harmonics, PZT miscalibration, and arbitrary phase steps in the optical configuration. The method does not need addition of any carrier fringes to separate the spectral contents in the intensity fringes. The proposed concept thus overcomes the limitations of other methods based on Fourier transform techniques. The robustness of the proposed method is studied in the presence of noise.

©2005 Optical Society of America

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Figures (6)
Fig. 1.
Fig. 1. Plots of power spectrum obtained for κ = 1 using a) Fourier transform b) Zero padded Fourier transform and c) super-resolution Fourier transform for determining the phase step values α.

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Fig. 2.
Fig. 2. Plots of power spectrum obtained for κ = 2 using a) Fourier transform b) Zero padded Fourier transform and c) super-resolution Fourier transform for determining the phase step values α.

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Fig. 3.
Fig. 3. Plots of phase step α (in degrees) versus SNR obtained using Eq. (15) at an arbitrary pixel location on a data frame for (a) - (d) κ = 1; N = 6,8,10, and 12, respectively, and (e) - (h) κ= 2; N = 10,12,14, and 16, respectively.

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Fig. 4.
Fig. 4. Plots of power spectrum for N = 12 data frames and κ= 1 for various orders of nonlinearity in the PZT response to the applied voltage.

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Fig. 5
Fig. 5 (a) Fringe pattern and (b) retrieved phase φ.

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Fig. 6.
Fig. 6. Typical phase error in computation of phase distributions φ (in radians), for the phase steps obtained from Fig 3(d) for SNR=30dB.

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Tables (1)

Tables Icon

Table 1. Phase step estimation for various values of NOF and N = 8 and 12. During simulation the phase step α is taken as 33°, κ=1, and SNR = 40 dB.

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Equations (35)

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I ( t ) = I dc + ∑ k = 1 κ a k exp ⁡ ( jkφ ) ︸ ℓ k u k t + ∑ k = 1 κ a k exp ⁡ ( − jkφ ) ︸ ℓ k * ( u k * ) t
+ η ( t ) ; for t = 0,1 , … , N − 1
r ( p ) = E [ I ( t ) I * ( t − p ) ] = ∑ k = 0 2 κ A k 2 exp ⁡ ( j ω k p ) + σ 2 δ p , 0 .
R I = E [ I ( t ) I * ( t ) ] = [ r ( 0 ) r * ( 1 ) . r * ( m − 1 ) r ( 1 ) r ( 0 ) . r * ( m − 2 ) . . . . r ( m − 1 ) . . r ( 0 ) ]
R I = SPS H ︸ R s + σ 2 I d ︸ R ε
P = [ A 0 2 0 . 0 0 A 1 2 . . . . . . 0 . . A 2 κ 2 ]
R I = [ υ ν ] [ λ 0 … 0 ⋮ ⋱ ⋮ 0 … λ m − 1 ] [ υ ν ] H
R I υ = SPS H υ + σ 2 υ
= υ [ λ 0 . . . 0 . λ 1 . . . . . . . . . . . . . 0 . . . λ 2 κ ]
= υγ + σ 2 υ
γ = [ λ 0 − σ 2 . . . 0 . λ 1 − σ 2 . . . . . . . . . . . . . 0 . . . λ 2 κ − σ 2 ]
υ = S ( PS H υ γ − 1 ) ≅ SC
S H ν = [ s 0 H . . s 2 κ H ] ν = 0
= s k H ν i = 0 ∀ i = ( 0,1,2 , …. , m − 2 κ − 2 ) and k = ( 0,1,2 , …. , 2 κ )
〈 s k , ν i 〉 = ∑ n = 0 m − 1 s k H ( n ) ν i ( n )
〈 s k , ν i 〉 = 0 ∀ i = ( 0,1,2 , … , m − 2 κ − 2 ) , k = ( 0,1 , ….. , 2 κ )
〈 s k , ν i 〉 = ∑ n = 0 m − 1 exp ⁡ ( − jn ω k ) ν i ( n )
= 〈 s , ν i 〉 | ω = ω k
= 0
P i [ exp ⁡ ( j ω f ) ] = 1 ∣ 〈 s f , ν i 〉 ∣ 2 ; ω f = 0,2 π / NOF , ……….. , 2 π ( NOF − 1 ) / NOF
P [ exp ⁡ ( j ω f ) ] = 1 ∑ i = 0 m − 2 κ − 2 1 λ 2 κ + 1 + i ∣ 〈 s f , ν i 〉 ∣ 2 ; ω f = 0,2 π / NOF , ……….. , 2 π ( NOF − 1 ) / NOF
I ̂ ( k ) = ∑ t = 0 N − 1 I ( t ) exp ⁡ ( − j 2 πkt N ) ; k = 0,1,2 , …. , N − 1
I ̂ ZP ( k ) = ∑ t = 0 ZP − 1 I ( t ) exp ⁡ ( − j 2 πkt N ) ; k = 0,1,2 , . . N − 1 , … , ZP − 1
[ 1 1 . . 1 exp ⁡ ( jκα ) exp ⁡ ( − jκα ) . . 1 exp ⁡ ( 2 jκα ) exp ⁡ ( − 2 jκα ) . . 1 . . . . . . . . . . exp ⁡ [ ( N − 1 ) jκα ] . . . 1 ] [ ℓ κ ℓ κ * ℓ κ − 1 . . I dc ] = [ I 0 I 1 . . . I N − 1 ]
I ( t ) = I dc + ∑ k = 1 κ a k exp ⁡ ( jkφ ) exp ⁡ ( jαkt ) + ∑ k = 1 κ a k exp ⁡ ( − jkφ ) exp ⁡ ( − jαkt ) + η ( t ) ;
for t = 0,1,2 , … , m … , N − 1
r ( p ) = E [ I ( t ) I * ( t − p ) ]
I ( t ) = I dc + a 1 exp ⁡ ( jφ ) exp ⁡ ( jαt ) + a 1 exp ⁡ ( − jφ ) exp ⁡ ( − jαt ) + η ( t )
I * ( t − p ) = I dc + a 1 exp ⁡ ( jφ ) exp ⁡ [ jα ( t − p ) ] + a 1 exp ⁡ ( − jφ ) exp ⁡ [ − jα ( t − p ) ] + η * ( t − p )
r ( p ) = E [ I ( t ) I * ( t − p ) ] = E { I dc 2 + I dc a 1 exp ⁡ ( − jϕ ) exp ⁡ ( − jαt ) + I dc a 1 exp ⁡ ( jφ ) exp ⁡ ( jαt ) + exp ⁡ ( jαp ) [ a 1 2 + I dc a 1 exp ⁡ ( − jφ ) exp ⁡ ( − jαt ) + a 1 2 exp ⁡ ( − 2 jφ ) exp ⁡ ( − 2 jαt ) ] + exp ⁡ ( − jαp ) [ a 1 2 + I dc a 1 exp ⁡ ( jφ ) exp ⁡ ( jαt ) + a 1 2 exp ⁡ ( 2 jφ ) exp ⁡ ( 2 jαt ) ] + η ( t ) η * ( t − p ) }
r ( p ) = E [ I dc 2 + c 1 + exp ⁡ ( jαp ) ( a 1 2 + c 2 ) + exp ⁡ ( − jαp ) ( a 1 2 + c 3 ) + η ( t ) η * ( t − p ) ]
r ( p ) = A 0 2 + A 1 2 exp ⁡ ( jαp ) + A 2 2 exp ⁡ ( − jαp ) + σ 2 δ p , 0
E [ η ( g ) η * ( h ) ] = σ 2 δ g , h
∫ 0 2 π exp ⁡ ( jψ ) dψ = 0
r ( p ) = E [ I ( t ) I * ( t − p ) ] = ∑ n = 0 2 κ A n 2 exp ⁡ ( j ω n p ) + σ 2 δ p , 0
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