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Multi-aperture Fourier transform imaging spectroscopy: theory and imaging properties

Samuel T. Thurman and James R. Fienup

Open Access Open Access

Abstract

Fourier transform imaging spectroscopy (FTIS) can be performed with a multi-aperture optical system by making a series of intensity measurements, while introducing optical path differences (OPD’s) between various subapertures, and recovering spectral data by the standard Fourier post-processing technique. The imaging properties for multi-aperture FTIS are investigated by examining the imaging transfer functions for the recovered spectral images. For systems with physically separated subapertures, the imaging transfer functions are shown to vanish necessarily at the DC spatial frequency. Also, it is shown that the spatial frequency coverage of particular systems may be improved substantially by simultaneously introducing multiple OPD’s during the measurements, at the expense of limiting spectral coverage and causing the spectral resolution to vary with spatial frequency.

©2005 Optical Society of America

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Supplementary Material (3)

Media 1: AVI (455 KB)     
Media 2: AVI (582 KB)     
Media 3: AVI (746 KB)     

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Figures (10)
Fig. 1.
Fig. 1. Illustration of multiple-telescope array with four subaperture telescopes.

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Fig. 2.
Fig. 2. Simplified refractive model for a multi-aperture optical system.

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Fig. 3.
Fig. 3. Pupil of optical system used in simulations.

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Fig. 4.
Fig. 4. Movie (455KB) showing the effect of the OPD’s on the optical system at ν=ν 0 as the time-delay variable is changed from τ=0 to τ=3/ν 0: (a) the magnitude of the relative phase delay of each subaperture, where white represents 0 and black represents ±π, (b) the PSF, and (c) the magnitude of the real part of the OTF.

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Fig. 5.
Fig. 5. Localization of FTIS signal in: (a) the raw intensity data cube, (b) the spectral image cube, and (c) the spectral-spatial transform cube. In each cube, the FTIS signal is localized to the darkly shaded regions.

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Fig. 6.
Fig. 6. Image intensity versus τ for the point object simulation: (a) at Point A, (b) at Point B, and contributions to the intensity at Point B due to the interference between subapertures: (c) 1 and 2, (d) 2 and 3, and (e) 1 and 3.

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Fig. 7.
Fig. 7. Spectral data from point object simulation at positive temporal frequencies in the ν′-domain: (a) at Point A (real-valued) and (b) at Point B (real and imaginary parts).

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Fig. 8.
Fig. 8. The extended object simulation: (a) movie (582KB) of the object data versus ν(the still frame shows the data at ν=1.03ν 0), (b) size of the pupil in spatial frequencies corresponding to ν=1.03ν 0, and (c) movie (746KB) of the image intensity versus τ(the still frame shows the image intensity at τ=0).

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Fig. 9.
Fig. 9. Spectral image data from the extended-object simulation. The top row shows the real part of spectral images at: (a) ν′=0.34ν 0, (c) ν′=0.68ν 0, and (e) ν′=1.03ν0. The bottom row shows the Fourier magnitude of each image. For the spectral images, note that dark grays represent negative values and light grays represent positive values.

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Fig. 10.
Fig. 10. Composite spectral image data from extended object simulation: (a) the real part of the spectral image at ν=1.03ν 0, (b) the corresponding Fourier magnitude, (c) the imaginary part of the same spectral image, and (d) the corresponding Fourier magnitude. For the spectral images, note that dark grays represent negative values, middle gray represents zero, and light grays represent positive values.

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Equations (44)

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T pup ( ξ , η , ν , τ ) = ∑ q = 1 Q T q ( ξ , η , ν ) exp ( i 2 π ν γ q τ ) ,
I ( x , y , τ ) = κ ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ 1 M 2 S o ( x ′ M , y ′ M , ν ) h ( x − x ′ , y − y ′ , ν , τ ) dx ′ dy ′ d ν ,
h ( x , y , ν , τ ) = ∑ q = 1 Q h q , q ( x , y , ν ) + ∑ p = 1 Q ∑ q = 1 q ≠ p Q h p , q ( x , y , ν ) exp [ i 2 π ν ( γ p − γ q ) τ ] .
h p , q ( x , y , ν ) = t p ( x , y , ν ) t q * ( x , y , ν ) ,
t q ( x , y , ν ) = 1 λ 2 f i 2 ∫ − ∞ ∞ ∫ − ∞ ∞ T q ( ξ , η , ν ) exp [ − i 2 π ( x λ f i ξ + y λ f i η ) ] d ξ d η .
H ( f x , f y , ν , τ ) = ∫ − ∞ ∞ ∫ − ∞ ∞ h ( x , y , ν , τ ) exp [ − i 2 π ( f x x + f y y ) ] d x d y ∫ − ∞ ∞ ∫ − ∞ ∞ h ( x , y , ν , τ ) d x d y
= T pup ( − λ f i f x , − λ f i f y , ν , τ ) ★ T pup ( − λ f i f x , − λ f i f y , ν , τ ) ∫ − ∞ ∞ ∫ − ∞ ∞ ∣ T pup ( ξ , η , ν , τ ) ∣ 2 d ξ d η
= ∑ q = 1 Q H q , q ( f x , f y , ν ) + ∑ p = 1 Q ∑ q = 1 q ≠ p Q H p , q ( f x , f y , ν ) exp [ i 2 π ν ( γ p − γ q ) τ ] ,
H p , q ( f x , f y , ν ) = ∫ − ∞ ∞ ∫ − ∞ ∞ h p , q ( x , y , ν ) exp [ − i 2 π ( f x x + f y y ) ] d x d y ∫ − ∞ ∞ ∫ − ∞ ∞ h ( x , y , ν , τ ) d x d y
= T p ( − λ f i f x , − λ f i f y , ν ) ★ T q ( − λ f i f x , − λ f i f y , ν ) ∫ − ∞ ∞ ∫ − ∞ ∞ ∣ T pup ( ξ , η , ν , τ ) ∣ 2 d ξ d η .
S i ( x , y , ν ′ ) = κ ∑ p = 1 Q ∑ q = 1 q ≠ p Q ∫ − ∞ ∞ ∫ − ∞ ∞ 1 ∣ γ p − γ q ∣ M 2 S o ( x ′ M , y ′ M , ν ′ γ p − γ q )
× h p , q ( x − x ′ , y − y ′ , ν ′ γ p − γ q ) d x ′ d y ′ .
G i ( f x , f y , ν ′ ) = κ ∑ p = 1 Q ∑ q = 1 q ≠ p Q 1 ∣ γ p − γ q ∣ G o ( M f x , M f y , ν ′ γ p − γ q ) H p , q ( f x , f y , ν ′ γ p − γ q ) ,
S comp ( x , y , ν ) = ∑ Δ γ > 0 Δ γ S i ( x , y , Δ γ ν ) for ν 1 ≤ ν ≤ ν 2 ,
S comp ( x , y , ν ) = κ ∑ p = 1 Q ∑ q = 1 Δ γ > 0 Q ∫ − ∞ ∞ ∫ − ∞ ∞ 1 M 2 S o ( x ′ M , y ′ M , ν )
× h p , q ( x − x ′ , y − y ′ , ν ) d x ′ d y ′ for ν 1 ≤ ν ≤ ν 2 .
S i ( x , y , ν ′ ) = S i * ( x , y , − ν ′ ) ,
G i ( f x , f y , ν ′ ) = G i * ( − f x , − f y , − ν ′ ) .
S i ( Re ) ( x , y , ν ′ ) = 1 2 [ S i ( x , y , ν ′ ) + S i * ( x , y , ν ′ ) ] ,
G i ( Re ) ( f x , f y , ν ′ ) = 1 2 [ G i ( f x , f y , ν ′ ) + G i * ( − f x , − f y , ν ′ ) ] ,
G i ( Re ) ( f x , f y , ν ′ ) = κ ∑ p = 1 Q ∑ q = 1 q ≠ p Q 1 ∣ γ p − γ q ∣ G o ( M f x , M f y , ν ′ γ p − γ q )
× 1 2 [ H p , q ( f x , f y , ν ′ γ p − γ q ) + H p , q * ( − f x , − f y , ν ′ γ p − γ q ) ] .
G i ( Im ) ( f x , f y , ν ′ ) = κ ∑ p = 1 Q ∑ q = 1 q ≠ p Q 1 ∣ γ p − γ q ∣ G o ( M f x , M f y , ν ′ γ p − γ q )
× 1 2 i [ H p , q ( f x , f y , ν ′ γ p − γ q ) − H p , q * ( − f x , − f y , ν ′ γ p − γ q ) ] .
S i ( x , y , ν ′ ) = κ ∑ p = 1 Q ∑ q = 1 q ≠ p Q ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ 1 M 2 S o ( x ′ M , y ′ M , ν ) h p , q ( x − x ′ , y − y ′ , ν )
× 2 τ max sin⁡ c [ 2 τ max ( γ p − γ q ) ( ν ′ γ p − γ q − ν ) ] d x ′ d y ′ d ν ,
S comp ( x , y , ν ) ≈ κ ∑ p = 1 Q ∑ q = 1 Δ γ > 0 Q ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ 1 M 2 S o ( x ′ M , y ′ M , ν ′ ) h p , q ( x − x ′ , y − y ′ , ν ′ )
× 2 τ max ( γ p − γ q ) sin⁡ c [ 2 τ max ( γ p − γ q ) ( ν − ν ′ ) ] d x ′ d y ′ d ν ′ for ν 1 ≤ ν ≤ ν 2 .
S o ( x ′ , y ′ , ν ) = E rect [ ( ν − ν 0 ) ⁄ ( ν 2 − ν 1 ) ] δ ( x ′ , y ′ ) ,
I ( x , y , τ ) = κ E ∫ ν 1 ν 2 h ( x , y , ν , τ ) d ν .
t q ( x , y , ν ) = π R 2 λ 2 f i 2 jinc ( 2 R λ f i x 2 + y 2 ) exp [ − i 2 π λ f i ( x ξ q + y η q ) ] ,
τ p , q = x B ( ξ p − ξ q ) + y B ( η p − η q ) c f i ( γ p − γ q ) .
W ( z ) ( x 1 , y 1 , x 2 , y 2 , ν ) δ ( ν − ν ' ) = 〈 V ( x 1 , y 1 , ν ) V * ( x 2 , y 2 , ν ′ ) 〉 ,
W ( d ) ( x 1 , y 1 , x 2 , y 2 , ν ) = 1 λ 2 d 2 ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ d x 1 ′ d y 1 ′ d x 1 ′ d y 2 ′ W ( 0 ) ( x 1 ′ , y 1 ′ , x 2 ′ , y 2 ′ , ν )
× exp { i π λ d [ ( x 1 − x 1 ′ ) 2 + ( y 1 − y 1 ′ ) 2 − ( x 2 − x 2 ′ ) 2 − ( y 2 − y 2 ′ ) 2 ] } ,
V trans ( x , y , ν ) = T ( x , y , ν ) V inc ( x , y , ν ) ,
W trans ( 0 ) ( x 1 , y 1 , x 2 , y 2 , ν ) = T ( x 1 , y 1 , ν ) T * ( x 2 , y 2 , ν ) W inc ( 0 ) ( x 1 , y 1 , x 2 , y 2 , ν ) ,
W ( i ) ( x 1 , y 1 , x 2 , y 2 , ν , τ ) = ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ d x 1 ′ d y 1 ′ d x 2 ′ d y 2 ′ 1 M 2 W ( o ) ( x 1 ′ M , y 1 ′ M , x 2 ′ M , y 2 ′ M , ν )
× exp [ i π λ f o ( 1 − d 1 f o ) ( x ′ 1 2 M 2 + y ′ 1 2 M 2 − x ′ 2 2 M 2 − y ′ 2 2 M 2 ) ]
× exp [ i π λ f i ( 1 − d 2 f i ) ( x 1 2 + y 1 2 − x 2 2 − y 2 2 ) ]
× { ∑ q = 1 Q ∑ p = 1 Q t q ( x 1 − x 1 ′ , y 1 − y 1 ′ , ν ) t p * ( x 2 − x 2 ′ , y 2 − y 2 ′ , ν )
× exp [ i 2 π ν ( γ q − γ p ) τ ] ,
I ( x , y , τ ) = ∫ − ∞ ∞ W ( i ) ( x , y , x , y , ν , τ ) d ν .
W ( o ) ( x 1 ′ , y 1 ′ , x 2 ′ , y 2 ′ , ν ) = κ S o ( x ′ 1 , y 1 ′ , ν ) δ ( x 1 ′ − x 2 ′ , y 1 ′ − y 2 ′ )
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