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Super-resolution in digital holography by a two-dimensional dynamic phase grating

M. Paturzo, F. Merola, S. Grilli, S. De Nicola, A. Finizio, and P. Ferraro

Open Access Open Access

Abstract

An approach that uses an electro-optically tunable two dimensional phase grating to enhance the resolution in digital holographic microscopy is proposed. We show that, by means of a flexible hexagonal phase grating, it is possible to increase the numerical aperture of the imaging system, thus improving the spatial resolution of the images in two dimensions. The augment of the numerical aperture of the optical system is obtained by recording spatially multiplexed digital holograms. The grating tuneability allows one to adjust the intensity among the spatially multiplexed holograms maximizing the grating diffraction efficiency. Furthermore we demonstrate that the flexibility of the numerical reconstruction allows one to use selectively the diffraction orders carrying useful information for increasing the spatial resolution. The proposed approach can improve the capabilities of digital holography in three-dimensional imaging and microscopy.

©2008 Optical Society of America

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Figures (6)
Fig. 1.
Fig. 1. (a). Scheme of the DH recording setup: Fourier configuration in off-axis mode; the ray diagrams of the object waves: (b) without the grating in the setup and (c) with the grating in the setup.

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Fig. 2.
Fig. 2. (a). Amplitude reconstruction of the digital hologram when no voltage is applied to the electro-optic grating; (b) magnified view showing the reticules with the shortest pitches (31.6 µm, 25.1 µm, 20.0 µm, 15.8 µm).

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Fig. 3.
Fig. 3. Amplitude reconstruction of the multiplexed digital hologram when the phase grating is switched-on (applied voltage of 2.5 kV). This numerical reconstruction has been obtained without introducing the transmission function of the phase diffraction grating in the reconstruction algorithm (i.e. T(x,y)=1). The labels of the reconstructed images indicates the corresponding diffraction orders.

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Fig. 4.
Fig. 4. (a). The coloured ellipses encircle the reconstructed images along the three typical directions of the hexagonal grating. Magnified view of the image obtained by superimposing only the -1a, 0th, +1a diffraction orders (blue ellipse in (a)); (c) the -1b, 0th, +1b orders (red ellipse); (d) the -1c, 0th, +1c orders (yellow ellipse). The reticule with a pitch of 25.1 µm, completely blurred in (b), is resolved in (c) and (d) thanks to an improvement of the optical resolution. Plot of intensity profile along the white lines in the images (b), (c) and (d) are shown in (e), (f) and (g), respectively. Axes of the plots have a.u. for intensities on ordinates while pixel number on abscissa.

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Fig. 5.
Fig. 5. (a). Amplitude reconstruction of the target obtained by superimposing all the first diffraction orders, ±1a, ±1b and ±1c, on the zero order 0th; (c), (e) amplitude reconstruction obtained by ignoring the ±1a orders and by using different or same weight for the orders ±1b, ±1c, respectively; (b), (d), (f) corresponding profiles calculated along the ruling with 25.1 µm pitch.

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Fig. 6.
Fig. 6. (a, b). Super-resolved images optimized as to the geometrical issues, (c,d) their magnified view concerning the reticules with the shortest pitches and (e,f) the profile of the 25.1 µm pitch grating (along the white line) for Δφ=3π/5 and Δφ=π, respectively. By Fig. 6(e), (f) it is clear that the increase of the optical resolution is higher when the phase step is Δφ=π ; (g) the profile of the 20.0 µm pitch grating, clearly resolved for Δφ=π.

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Equations (3)

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b ( x 1 , y 1 ) = 1 i λ d 2 e i π λ d 2 ( x 1 2 + y 1 2 ) ∬ r ( x 2 , y 2 ) h ( x 2 , y 2 ) e i π d 2 λ [ x 2 2 + y 2 2 ] e − 2 i π λ d 2 [ x 2 x 1 + y 2 y 1 ] d x 2 d y 2
T ( x 1 , y 1 ) = 1 + a cos ⁡ ( 2 π x 1 p ) + b cos ⁡ ( ( x 1 + 3 y 1 ) π p ) + c cos ⁡ ( ( x 1 + 3 y 1 ) π p )
b ( x 0 , y 0 ) = 1 i λ d 1 e i π λ d 1 ( x 0 2 + y 0 2 ) ∬ r ( x 1 , y 1 ) h ( x 1 , y 1 ) e i π d 1 λ [ x 1 2 + y 1 2 ] e − 2 i π λ d 1 [ x 1 x 0 + y 1 y 0 ] d x 1 d y 1
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