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Hyperspectral ghost imaging camera based on a flat-field grating

Shengying Liu, Zhentao Liu, Jianrong Wu, Enrong Li, Chenyu Hu, Zhishen Tong, Xia Shen, and Shensheng Han

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Abstract

A spectral camera based on ghost imaging via sparsity constraints (GISC) acquires a three-dimensional (3D) spatial-spectral data cube of the target through a two-dimensional (2D) detector in a single snapshot. However, the spectral and spatial resolution are interrelated because both of them are modulated by the same spatial random phase modulator. In this paper, we theoretically and experimentally demonstrate a system by equipping the GISC spectral camera with a flat-field grating to disperse the light fields before the spatial random phase modulator, hence consequently decoupling the spatial and spectral resolution. By theoretical derivation of the imaging process we obtain the spectral resolution 1nm and spatial resolution 50μm about the new system which are verified by the experiment. The new system can not only modulate the spatial and spectral resolution separately, but also provide a possibility of optimizing the light field fluctuations of different wavelengths according to the imaging scene.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Figures (6)
Fig. 1
Fig. 1 Schematic of GISC hyperspectral camera based on a flat-field grating. (a) The object plane; (b) the first imaging plane; (c) the diffractive plane; (d) the speckles plane; (1) an objective lens; (2) a flat-field grating; (3) a spatial random phase modulator; (4) a microscope objective; (5) a Charge Coupled Device (CCD).

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Fig. 2
Fig. 2 Schematic diagram of coordinate conversion relations.

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Fig. 3
Fig. 3 The simple sketches of the calibration. (a) Point sources with different wavelengths are at the same pixel of field-of-view (FoV) and illuminate the flat-field grating, and then are spatially dispersed and imaged to the diffractive plane at different places with a certain adjacent distance determined by the dispersion coefficient. (b) Point sources with the same wavelength are at different pixels of FoV and illuminate the flat-field grating, and then are spatially dispersed and imaged to the diffraction plane at different places with a certain adjacent distance determined by the distance between two pixels.

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Fig. 4
Fig. 4 The experimental setup of GISC hyperspectral camera based on a flat-field grating. During the calibration, a calibration setup is put in front of the objective lens to acquire incoherent intensity impulse response functions of the system. After the calibration, an object is put before the objective lens to obtain the detected signal which is the overlay of the speckles from different pixels and wavelengths of the object.

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Fig. 5
Fig. 5 Experimentally determined resolution. (a) Two normalized second-order correlation functions of light fields at a pixel in FoV with two different wavelengths, g d r ( 2 ) (x0′, y0′, λ; x0′, y0′, λ′), and at two different pixels with the same wavelength, g d r ( 2 ) (x0, y0, λ′; x0′, y0′, λ′), whose half width respectively determine the spectral resolution (1nm) and spacial resolution (50μm). The experiment result (the blue line) is consist with the theoretical result (red dotted line). (b) Tow point light sources with two different wavelengths, 539 nm and 540 nm, whose distance is 40μm. (c) The analysis of spectral and spatial resolution (according to the reconstructed image of two points). The 3D schematic about the spectral and spatial distribution (upper). The red line and green line (lower left) are respectively the spectral distributions about two points with 539nm and 540nm center wavelength and the red line (lower right) is the spatial distributions about two points. The spectrum and position of the points are considered resolved if they are separated by a dip of at least 20%. The result verifies the resolution determined by the normalized second-order correlation function g d r ( 2 ) .

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Fig. 6
Fig. 6 Experimentally imaging result. (a) A little girl taken by conventional camera. (b,c) A little girl & institute logo passing through a 536–545 nm narrowed band pass filter detected by CCD1. (d) The reconstructed spectral images of institute logo & little girl of spectral 3D data-cube, displaying all the channels from 536 to 545 nm.

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Equations (22)

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I d ( x 2 ,   y 2 ) = ∭ I b ( x 0 , y 0 , λ ) h I ( x 2 ,   y 2 ; x 0 ,   y 0 , λ ) d x 0 d y 0 d λ ,
I d r ( x 2 ,   y 2 ; x 0 ′ , y 0 ′ , λ ′ ) = ∭ I b r ( x 0 ,   y 0 ,   λ ; x 0 ′ , y 0 ′ , λ ′ ) h I ( x 2 ,   y 2 , ; x 0 ,   y 0 , λ ) d x 0 d y 0 d λ = ∭ δ ( x 0 −   x 0 ′ , y 0 − y 0 ′ , λ − λ ′ ) h I ( x 2 ,   y 2 , ; x 0 ,   y 0 , λ ) d x 0 d y 0 d λ = h I ( x 2 ,   y 2 ; x 0 ′ ,   y 0 ′ , λ ′ ) .
I d t ( x 2 ,   y 2 ) = ∭ T 0 ( x 0 ,   y 0 ,   λ ) I d r ( x 2 ,   y 2 ; x 0 ,   y 0 , λ ) d x 0 d y 0 d λ ,
G ( 2 ) ( ( x 2 , y 2 ) d r , ( x 2 , y 2 ) d t ) = 〈 E d r * ( x 2 , y 2 ; x 0 ′ , y 0 ′ , λ ′ ) E d t * ( x 2 , y 2 ) E d t ( x 2 , y 2 ) E d r ( x 2 , y 2 ; x 0 ′ , y 0 ′ , λ ′ ) 〉 ,
G ( 2 ) ( ( x 2 , y 2 ) d r , ( x 2 , y 2 ) d r ) = 〈 I d r ( x 2 , y 2 ; x 0 ′ y 0 ′ , λ ′ ) ∭ T ( x 0 , y 0 , λ ) I d r ( x 2 , y 2 ; x 0 , y 0 , λ ) d x 0 d y 0 d λ 〉 = ∭ T ( x 0 , y 0 , λ ) G d r ( 2 ) ( x 0 ′ y 0 ′ , λ ′ ; x 0 , y 0 , λ ) d x 0 d y 0 d λ ,
G d r ( 2 ) ( x 0 ′ y 0 ′ , λ ′ ; x 0 , y 0 , λ ) = 〈 E d r * ( x 2 , y 2 ; x 0 ′ y 0 ′ , λ ′ ) E d r * ( x 2 , y 2 ; x 0 , y 0 , λ ) E d r ( x 2 , y 2 ; x 0 , y 0 , λ ) E d r ( x 2 , y 2 ; x 0 ′ y 0 ′ , λ ′ ) 〉 ,
G d r ( 2 ) ( x 0 ′ y 0 ′ , λ ′ ; x 0 , y 0 , λ ) = ( 〈 I d r ( x 2 , y 2 ; x 0 ′ y 0 ′ , λ ′ ) 〉 〈 I d r ( x 2 , y 2 ; x 0 , y 0 , λ ) 〉 ) × ( 1 + g d r ( 2 ) ( x 0 ′ y 0 ′ , λ ′ ; x 0 , y 0 , λ ) ) ,
g d r ( 2 ) ( x 0 ′ y 0 ′ , λ ′ ; x 0 , y 0 , λ ) = | J ( x 0 ′ y 0 ′ , λ ′ ; x 0 , y 0 , λ ) | 2 〈 I d r ( x 2 , y 2 ; x 0 ′ y 0 ′ , λ ′ ) 〉 〈 I d r ( x 2 , y 2 ; x 0 , y 0 , λ ) 〉 ,
J ( x 0 ′ y 0 ′ , λ ′ ; x 0 , y 0 , λ ) = 〈 E d r * ( x 2 , y 2 ; x 0 ′ y 0 ′ , λ ′ ) E d r ( x 2 , y 2 ; x 0 , y 0 , λ ) 〉 .
t g ( x , y ) = ( rect ( x a ) e j ϕ ) ⊗ 1 d eff comb ( x d eff ) ,
t p ( x m , y m , λ ) = exp [ j 2 π ( n − 1 ) h ( x m , y m ) λ ] ,
R h ( x m , y m ; x m ′ y m ′ ) = 〈 h ( x m , y m ) h ( x m ′ y m ′ ) 〉 = ω 2 exp [ − ( x m − x m ′ ) 2 + ( y m − y m ′ ) 2 ζ 2 ] ,
E d r ( x 2 , y 2 ; x 0 ′ y 0 ′ , λ ′ ) = e j ϕ a λ z 1 − λ ′ 2 z 2 z 3 exp [ j k ( z 0 + z 1 ) ] j λ ′ z 0 z 1 exp ( j 2 π λ ′ x 0 2 + y 0 2 2 z 0 ) × exp ( j 2 π λ ′ x 1 2 + y 1 2 2 z 1 ) sinc ( a λ ′ z 1 f x ) 1 d eff ∑ n = − ∞ ∞ δ ( f x − λ ′ z 1 d eff n ) × exp { j 2 π λ ′ [ ( z 2 + z 3 ) + ( X 2 − X 1 ) 2 + ( y 2 − y 1 ) 2 2 ( z 2 + z 3 ) ] } ∬ t p ( X m , y m , λ ′ ) × exp { j π λ ′ z 2 + z 3 z 2 z 3 [ ( X m − z 3 X 1 + z 2 X 2 z 2 + z 3 ) 2 + ( y m − z 3 y 1 + z 2 y 2 z 2 + z 3 ) 2 ] } d X m d y m ,
g d r ( 2 ) ( x 0 , y 0 ; x 0 ′ y 0 ′ , λ ′ ) = ( z 2 + z 3 ) 4 λ 4 z 2 4 z 3 4 × | ∭ 〈 t p ( X m , y m , λ ) t * ( X m ′ y m ′ , λ ′ ) 〉 × exp { j π ( z 2 + z 3 ) z 2 z 3 [ 1 λ ( α 2 + α ′ 2 ) − 1 λ ′ ( β 2 + β ′ 2 ) ] } d X m d y m d X m ′ d y m ′ | 2 ,
〈 t p ( X m , y m , λ ) t p * ( X m ′ , y m ′ , λ ′ ) 〉 = 〈 exp { j 2 π ( n − 1 ) [ h ( X m , y m ) λ − h ( X m ′ , y m ′ , ) λ ′ ] } 〉 = M ˜ H ( X m , y m ) H ( X m ′ , y m ′ ) ( 2 π ( n − 1 ) λ , − 2 π ( n − 1 ) λ ′ ) ,
M ˜ H ( X m , y m ) H ( X m ′ , y m ′ ) ( 2 π ( n − 1 ) λ − 2 π ( n − 1 ) λ ′ ) = exp { − 1 2 [ 2 π ( n − 1 ) ] 2 [ ( 1 λ 2 + 1 λ ′ 2 ) ω 2 − 2 R h ( X m , y m , X m ′ , y m ′ ) λ λ ′ ] } .
g d r ( 2 ) ( x 0 , y 0 , λ ; x 0 ′ , y 0 ′ , λ ′ ) ≈ exp { − [ 2 π ω ( n − 1 ) ] 2 [ ( 1 λ − 1 λ ′ ) 2 + 2 λ λ ′ ] } × exp ( 2 [ 2 π ω ( n − 1 ) ] 2 λ λ ′ exp { − ( z 3 z 1 sec β H ) 2 [ ( λ − λ ′ ) d eff − ( x 0 − x 0 ′ ) z 0 ] 2 − z 3 2 ( y 0 − y 0 ′ ) 2 ( z 2 + z 3 ) 2 ζ 2 } ) .
Δ G ( 2 ) ( ( x 2 , y 2 ) d r , ( x 2 , y 2 ) d t ) = G ( 2 ) ( ( x 2 , y 2 ) d r , ( x 2 , y 2 ) d t ) − 〈 I d r ( x 2 , y 2 ; x 0 ′ , y 0 ′ , λ ′ ) 〉 〈 I d t ( x 2 , y 2 ) 〉 ≈ k ′ 2 a 4 ( z 2 + z 3 ) 4 z 0 4 d eff 4 [ sinc ( a d eff ) ] 4 T ( x 0 ′ , y 0 ′ , k ′ ) ⊗ exp { 2 [ 2 π ω ( n − 1 ) ] 2 × exp [ − z 3 2 z 1 2 ( sec β H ) 2 ( z 2 + z 3 ) 2 ζ 2 z 0 2 x 0 ′ 2 − z 3 2 y 0 ′ 2 ( z 2 + z 3 ) 2 ζ 2 ] × [ k ′ 2 − 2 z 3 2 z 1 2 ( sec β H ) 2 ( z 2 + z 3 ) 2 ζ 2 d eff z 0 k ′ x 0 ′ ] − 3 [ 2 π ω ( n − 1 ) ] 2 k ′ 2 } .
Y = AX ,
A = ( A 1 , 1 λ 1 ⋯ A 1 , N λ 1 A 1 , 1 λ 2 ⋯ A 1 , N λ 2 ⋯ ⋯ A 1 , 1 λ L ⋯ A 1 , N λ L A 2 , 1 λ 1 ⋯ A 2 , N λ 1 A 2 , 1 λ 2 ⋯ A 2 , N λ 2 ⋯ ⋯ A 2 , 1 λ L ⋯ A 2 , N λ L ⋮ ⋯ ⋮ ⋮ ⋯ ⋮ ⋯ ⋯ ⋮ ⋯ ⋮ A M , 1 λ 1 ⋯ A M , N λ 1 A M , 1 λ 2 ⋯ A M , N λ 2 ⋯ ⋯ A M , 1 λ L ⋯ A M , N λ L )
y p = ∑ λ = 1 L ∑ q = 1 N A p , q λ x q λ ,
X = arg   min X ≥ 0 ‖ Y − A X ‖ 2 2 + μ 1 ‖ Φ X ‖ 1 + μ 2 ‖ X ‖ * ,
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