## Abstract

We propose a novel method to obtain non-lens holographic images of micro-objects in white light with diffraction limit quality, based on fourier-spectroscopy principles. We developed a simple method for numerical acquisition of digital holograms of micro-objects at any spectral component from the set of two-dimensional interferograms, registered by fourier-spectrometer. In our experiments we used spectrally-spatial holographic fourier-spectrometer (SSHFS), equipped with supercontinuum light source and CCD camera for registration. Holographic images of several test objects acquired experimentally at different spectral components are presented. Visualization of local spatially-spectral inhomogeneities of micro-objects is discussed through the example of silver berry scaly hair sample.

© 2013 Optical Society of America

## 1. Introduction

There are several recent comprehensive reviews of digital holography [1–3]. Phase shifting (PS) technique, proposed for micro-object non-lens image acquisition by means of digital holography microscopy (DHM) in coherent light with zero order diffraction suppression is one of the most advanced methods for holographic micro-object imaging [4–7]. However, PS faces certain difficulties due to coherent light implementation, such as speckle effect and high demands on laser spectrum stability. In order to eliminate those difficulties we have suggested new approach and shown that micro-object’s holographic images can be obtained in white light [8]. Experimental verification of the proposed method, based on modified Fourier spectrometer setup was presented [9]. In current work we develop principles proposed earlier and consider a transparent object, placed into the arm of interferometer with fixed mirror.

## 2. Principle

Without loss of generality, lets consider a transparent object with complex transmission function *a*(*x*) = |*a*(*x*)|*exp*[*iφ*(*x*)]. /In case of reflecting object - complex function of reflection/. Let *U* is the amplitude of the wave, illuminating the object,
$U=\frac{1}{\mathrm{\Delta}\sigma}{\int}_{\mathrm{\Delta}\sigma}E(\sigma )\mathit{exp}(2\pi i\sigma z)d\sigma $, *σ* = 1/*λ* - spectral component, *λ* - wavelength, *E*(*σ*) - amplitude spectral density function, which is considered to be known, Δ*σ* - spectral width, where *E*(*σ*) is non-zero, *z* - distance from the object to CCD.

Consider one spectral component of the plane wave *E*(*σ*)*exp*(2*πiσz*). This component would be modulated by transmittance function of the object: *a*(*x*)*E*(*σ*)*exp*(2*πiσz*). If the size of the object *D* and distance *z* fulfill Fraunhofer diffraction condition, then the spectral component of the filed in CCD-plane will be: *E*(*σ*)*exp*(2*πiσz*) ∫* _{D}a*(

*x*)

*exp*(−2

*πiσθx*)

*dx*,

*θ*=

*ξ/z*, where

*ξ*is a coordinate in the CCD-plane.

Interference occurs only between spectral components with the same wavenumber and thus:

*I*(

*σ*,

*θ*,

*δ*) is the intensity of the field at

*θ*-point, formed by the interference of the object field and reference field

*E*(

*σ*)

*exp*[2

*πiσ*(

*z*+

*δ*)], reflected by moving mirror, which is shifted on

*δ*from the zero path-length difference position. Expression (1) coincides with corresponding formulae of fourier-spectromery for asymmetric interferometer, where sample complex coefficient of reflection is substituted instead of ∫

*(*

_{D}a*x*)

*exp*[−2

*πiσθx*]

*dx*[10]. Transforming (1) we get:

*A*(

*σθ*) = ∫

*(*

_{D}a*x*)

*exp*[−2

*πiσθx*]

*dx*- complex amplitude of the Fourier transform of the object,

*S*(

*σ*) = |

*E*(

*σ*)|

^{2}- light source power density at a given

*σ*. At

*θ*-point CCD registers total intensity, which is an integral over all frequencies from Δ

*σ*spectral range:

*G*(

*θ*,

*δ*) can be considered as a spatially-spectral interferogram of the object, since it is simultaneously determined by spatial frequencies of the object and spectral components of the light source. With

*δ*→ ∞,

*G*(∞) = ∫

_{Δσ}

*S*(

*σ*)

*dσ*represents background, uniform illumination, when second summand in (3) goes to zero due to rapid oscillations of exponential factor under the integral. For simplicity further we omit

*G*(∞).

*exp*[2

*πiσ′δ*] and integrating over

*δ*yields: ∫

*G*(

*θ*,

*δ*)

*exp*[2

*πiσ′δ*]

*dδ*= ∫

_{Δσ}

*S*(

*σ*){

*A*(

*σθ*)

*exp*[−2

*πi*(

*σ*−

*σ′*)

*δ*]+

*A*(

^{*}*σθ*)

*exp*[2

*πi*(

*σ*−

*σ′*)

*δ*]}

*dσdδ*, which finally gives

*S*(

*σ*) and interferograms

*G*(

*θ*,

*δ*) in every point

*θ*of CCD, performing inverse Fourier transform, one can obtain the image of the object for any spectral frequency

*σ*.

## 3. Experiment and results

To verify new principle of spectral holograms recording experimentally, we have built two-beam interferometer Fig. 1. Diffracted by the object *4* wave is combined by BS *5* with the reference wave. Hologram registration with megapixel camera *8* was synchronized with sequential shift of dove-prism *6* mounted on *PZT 7*. *PZT* full travel range was 100*μ*, with 0.1*μ* pitch. Once hologram set is registered, image construction computation procedure falls into two simple steps Fig. 2. Firstly, perform *1D-FFT’s* over interferograms along *δ*–dimension for each pixel and truncate the result to the half-length due to the symmetry of the *FFT*. Each plane of the new data set, according to (5), represents complex amplitude of the spatial fourier-spectrum of the object *A*(*θ*) at a given spectral component *σ*. Thus, performing *2D-FFT* over selected plane yields a complex amplitude of the object at selected component of the spectrum.

We used standard line target #1 as a test amplitude object Fig. 3. One can see natural contrast fading of the resolved lines of the target while their spatial frequency increases. It happens due to the limited angular aperture, which in our case was about 1/5 radians. Diffraction limit of resolution, corresponding to that aperture is 6.1*μ* for *λ* = 1*μ* and 2.75*μ* for *λ* = 0.45*μ*. Stroke thickness from 10th block is 5.9*μ* and 3*μ* for strokes from 22nd block. Both results closely correlate with diffraction limit estimation which shows that proposed method of recording and reconstruction of spectral holograms doesn’t bring any artifacts into optical wavefield, which could affect image resolution. Reconstruction of pure phase object profile, a sin-shaped phase gitter, has also shown good results.

Each pixel of the CCD can be considered as a single-point receiver, which registers individual interferogram, as it does in classical FT-spectrometer. On Fig. 4 two pixel-averaged spectrum intervals of the supercontinuum source are presented, which we have registered with our setup with the set of filters: SZS4 and SZS5 meant for visible range and IKS7 meant for NIR range. Been multiplied on CCD spectral sensitivity function this curve closely correlates with the supercontinuum spectrum given in the laser datasheet.

In our experiments we have chosen the travel length and the pitch of the PZT-shifter in such a way that minimum wavelength of the source would be resolved and acceptable spectral resolution would be obtained. Spectrum range is sampled at 500 *σ*-points for any of which spatial distribution of intensity may be obtained.

We used silver berry scaly hair sample to synthesize a pseudo-colored image reflecting its spectrum characteristics Fig. 5. Spatial distribution of intensities, obtained at *λ _{r,g,b}* = 650

*nm*, 550

*nm*, 450

*nm*, were spectrally normalized and scaled. They were combined as RGB-layers of the image correspondingly. The sample was masked with nontransparent screen with round aperture so that the region of interest of the object would be illuminated; an artifact seen on the upper part of the image is a shadow projection of the part of this mask. Correlation of the visual perception with spectral information is more the task of coloristics, so we didn’t set a mission of spectral interpretation in the frames of this work.

## 4. Discussion

(5) was derived with implicit assumption that the spectral interval width is large. Lets discuss what large means. Δ*ξ* is the spatial spectrum width for that part of argument domain *ξ* = *σθ* of function *A*(*ξ*), where it significantly differs from zero. *γ* is the angular dimension of CCD - typical angle at which matrix boundaries are seen from object plane. If Δ*σγ* > Δ*ξ*, then integration limits might be considered infinite. And thus, our restriction of ’infinite limit’ reduces to a simple condition:

*σ*= 1/

*l*, where

_{c}*l*- coherence length and that the spatially-spectral width of the object is Δ

_{c}*ξ*= 1/

*D*, (6) can be represented as: If

*L*≈ 5

*mm*- typical CCD size,

*z*≈ 1

*cm*and

*d*≈ 1

*μ*is the minimum size of the resolving element of the object, then above restriction fulfils when

*l*= 0.5

_{c}*μ*, which coresponds to white light.

## 5. Conclusion

In this paper we presented a novel method of spatially-spectral interferometry of micro-objects based on a spatial analogue of FT-spectrometer, which we have built and called spectrally-spatial holographic fourier-spectrometer. Varying temporal delay between interfering fields coverts into corresponding spatial distribution of intensity, registered by CCD-camera. Since the typical size of diffraction patterns of micro-objects is comparable with modern CCD matrix dimensions, proposed method is particularly effective in application to holographic registration of spatially-spectral characteristics of micro-objects and their subsequent digital image formation. Suggested method unites two modern and highly advanced branches of computer optics: Fourier spectroscopy and digital non-lens microscopy. Such combination opens vast opportunities for development of new type devices: spectrally-spatial holographic fourier-spectrometers, which could be used to examine micro-objects of various origins. We have shown that spectrally-spatial fourier-holography not only enables to reconstruct spatial characteristics of micro-object, basically to acquire its holographic image, but also to determine its internal optical characteristics variations. Indeed, if a micro-object is optically heterogeneous, i.e. has locally varying dispersion and absorption due to its physical origin, then its spectrally-spatial holograms, registered at different spectral components, would vary. You might say as well that its holographic images would vary depending on the spectral component they were reconstructed at. Thus, micro-object holographic images analyses at different spectral components will allow to investigate its inner structure more accurately and get better understanding of its physical nature. Lastly, we’ll note that when talked of holographic recording, it’s commonly meant that object wavefield information is being recorded and reconstructed totally: spatial distribution of both - amplitude and phase. Proposed method adds spectral information to those and, thus, makes it total indeed.

## Acknowledgments

This research was supported by the Russian Foundation for Basic Research (Grant Nos 11-07-00755 and 09-07-00502-a).

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