## Abstract

Transmitted-light coherence-controlled holographic microscope (CCHM) based on an off-axis achromatic interferometer allows us to use light sources of arbitrary degree of temporal and spatial coherence. Besides the conventional DHM modes such as quantitative phase contrast imaging and numerical 3D holographic reconstruction it provides high quality (speckle-free) imaging, improved lateral resolution and optical sectioning by coherence gating. Optical setup parameters and their limits for a technical realization are derived and described in detail. To demonstrate the optical sectioning property of the microscope a model sample uncovered and then covered with a diffuser was observed using a low-coherence light source.

©2010 Optical Society of America

## 1. Introduction

Image formation in all forms of light microscope relies on the interference of direct and diffracted light in the image plane. But a common light microscope is not capable to show or to measure the phase difference introduced by an observed object, e.g. a living cell. This door was opened by invention of an interference microscope with separate object and reference beams at the end of 19th century [1]. In the 1950s, Krug and Lau and then Horn had the boldness to take the lateral separation concept to its logical extreme and to design transmitted-light interference microscopes consisting essentially of two microscopes side by side with parallel axes. These microscopes allowed for optical path difference (OPD) measurement. But their precise optical elements, high cost and image processing possibilities of those days caused that these devices were not wide spread and that other techniques like Zernike phase contrast or differential interference contrast (DIC) were used, although they did not provide a quantitative phase image.

Potential of holography revealed after its invention in 1947, invention of laser in 1960 and a rapid development of computer technology and digital recording devices at the end of 20th century dust off the interference microscopy techniques and enabled utilization of its features like optical sectioning, quantitative phase contrast imaging, or *ex post* numerical refocusing.

Interference microscopes with separated object and reference beams can be classified into two groups with respect to an angle between the object and the reference beam in the output plane of the interferometer: in-line or off-axis systems.

In-line, interferometric microscopes. Object and reference beams interfere being parallel or nearly parallel. This allows use of low-coherence light sources, e.g. a halogen lamp that suppress coherence noise and give rise to optical sectioning effect by coherence gating. The use of substantially limited temporal and/or spatial coherence of light sources is described only for in-line setups [2–5]. But it is necessary to record more than one interferogram (usually 3-7) with different phase shifts (phase-stepping, phase-shifting techniques) to reconstruct the object wave, i.e. its amplitude and phase. This is a significant limitation, especially for observation of rapidly varying phenomena, but even if a stable object is observed, vibration and medium or ambience fluctuations can introduce significant noise.

Off-axis, holographic microscopes. Object and reference beams interfere being inclined to each other, forming an interferogram with the spatial frequency of interference fringes being sufficiently high for the object wave reconstruction from a single interference pattern recorded [6,7]. The advantage consists in a high frame rate, especially if Fourier reconstruction algorithms are used [8]. All of the published digital holographic microscopes (DHM) use in principle the same interferometers like the in-line systems and a non-zero angle between the object and the reference beam at the output plane is achieved by inclination of one beam to the other. This precludes a desirable reduction of coherence degree of the light source, because such an interferometer is not achromatic and lots of setups are neither space-invariant and would not form contrast interference fringes across the whole detector if a polychromatic and/or an extended light source was used. Hence the narrowband and spatially limited sources have to be used [5,9], e.g. a laser beam focused close to a rotating ground glass.

Typical application domain of the reflected-light DHM is surface profiling [6] including measurement of dynamic systems (MEMS, MOEMS) [10]. Subnanometer resolution can be achieved in *z-*axis [11]. Reflected-light short-coherence DHM, technically similar to optical coherence tomography (OCT) [9], proved the capacity to observe tissues. Anyway, the transmitted-light DHM is more frequently applied in biology [1,12], mostly for living cells observation in quantitative phase contrast imaging [13–15].

We achieved the possibility to reduce substantially both spatial and temporal coherence in the off-axis system by applying principles of incoherent-holography. Typically used conventional interferometer is replaced by so-called achromatic interferometer [16–18], which exhibits also the feature of spatial invariance [19]. Number of contrast interference fringes in the hologram is limited neither by temporal nor spatial incoherence, i.e. the off-axis holography is feasible with arbitrarily broad and broadband light sources [20]. Such an interferometer combines advantages of both the in-line and off-axis systems described above and moreover, it brings its own remarkable advantage, especially in the transmission setup, which is mentioned in the Discussion section of this paper.

The first optical microscope being in principle off-axis DHM with achromatic interferometer was (to our knowledge) built and experimentally verified by Chmelik and Harna for reflected light [21]. Image plane hologram was digitally recorded and reconstructed using algorithms based on the fast Fourier transform. Confocal-like optical sectioning effect induced by an extended, spatially incoherent narrow-band light source was theoretically proved and experimentally demonstrated. This property was reflected in the name of the device: “parallel-mode confocal microscope”, which is a bit confusing from today’s point of view. Optical sectioning on a silicon sample surface [21] was demonstrated as well as within a biological sample [22]. Quantitative phase contrast combined with optical sectioning (intensity images) was applied to image and measure deep-surface objects in nanometer scale [23] by using spatially and spectrally broad source that introduced finer sectioning and approximately doubled the axial resolving power of the microscope [22]. Imaging process of this microscope was compared theoretically with that of a confocal scanning microscope, optical coherence tomography and optical coherence microscopy (OCM) in [24].

The concern of this paper is to describe the optical setup of the transmitted light coherence-controlled holographic microscope (CCHM) which we designed and built and which employs a diffraction grating achromatic interferometer and allows for wide range of both spatially and temporally partially coherent sources ranging from semi-coherent to nearly incoherent. The main optical setup parameters are proved theoretically and the first experimental results are demonstrated in the following sections.

## 2. Optical setup and principles of operation

#### 2.1 Basic setup parameters, origin of achromatic fringes

Diffraction grating (G) in the achromatic interferometer of CCHM (see Fig. 1 ) is imaged at the output plane (OP), as proposed by Leith [16], but each of the + 1 and the −1 diffracted order propagates through a separated optical path – the object and the reference arm of the interferometer.

Both arms are identical with a mirror symmetry along the interferometer axis (dash-dotted line in Fig. 1) and there are two identical microobjective lenses (O) and (C) of finite tube length in each arm. C acts as a condenser. Length of the arm is double the distance between the object plane and the image plane of the objective lens. Diffraction grating plane (G) and the interferometer output plane (OP) are (in both arms) optically conjugated with the object plane. The grating is thus imaged at the object plane, but without the fringe pattern, because here the image is always formed by one order only and the grating structure is finer than the resolving power of a condenser lens. The full image of the grating is formed in the output plane, where both the orders recombine.

Let the spatial frequency of the transmitted light phase diffraction grating (G) be *f*
_{G} and trace the axial ray coming out from the source (S) and passing through the relay lens (L). This ray traverses the grating and is split into two beams, each deflected by an angle *ϑ* from the interferometer axis:

*λ*is a wavelength of light. Interferometer arms are deflected by an angle

*ϑ*

_{0}corresponding to the central wavelength

*λ*

_{0}= 650 nm (see sec. 2.3).

An extended, spatially and temporally incoherent source (S) (a halogen lamp and IR cut-off filter, possibly combined with an interference filter) is imaged by L approximately to the front focal planes of both the condenser lenses (C) (the primary image). This induces the Köhler illumination of an object. Image of the source is spectrally dispersed with respect to the dispersive power of G so that the central point of the source lies on the optical axis of C for *λ* = *λ*
_{0} while for *λ* ≠ *λ*
_{0} it is laterally shifted. According to Eq. (1) the longer is *λ* the bigger is *ϑ*, so the primary images of the light source are shifted for longer *λ* further from the interferometer axis and from each other.

Secondary images of the light source are formed in the rear focal planes of both the objective lenses. These images are laterally shifted by the same distance but in an opposite direction compared to the case of condensers. Here, the longer *λ* the closer are the secondary images to the interferometer axis and to each other. So, if the beams emitted from the secondary images of the source were propagated directly to the output plane (OP), it would give rise to interference fringes of a different spatial frequency for a different wavelength *λ*. Longer wavelength light beams would recombine being deflected by a smaller angle, thus producing interference fringes of lower spatial frequency and vice versa. For this reason the reference and the object interferometer arms has to be crossed, e.g. using mirrors (M) in the sense that the light beam coming from the upper objective lens (see Fig. 1) enters the output plane from bellow and vice versa. The arms can be crossed on the objectives side or on the condenser side of the interferometer, but not both.

Geometry and deflection of beams after traversing the diffraction grating then corresponds symmetrically to the geometry and deflection of these beams upon arriving at the output plane, where they recombine giving rise to interference fringes parallel with lines of the diffraction grating and of a spatial frequency:

which is constant for all the wavelengths*λ*, because the interferometer is achromatic. If an object is observed, an image plane off-axis hologram with the spatial carrier

*f*

_{OP}is formed in the output plane. Formation of the achromatic interference fringes is analysed in subsection 2.5.

#### 2.2 Image processing

Reconstruction of the image amplitude and phase follows the method proposed by Kreis [8] which is based on the carrier removal in the Fourier plane. First, the image of the holographic pattern is Fourier transformed using the two-dimensional fast Fourier transform algorithm (FFT). The image spectrum in the sideband is then extracted by a windowing operation. The circular window is centered at the carrier frequency *f*
_{OP}; size of the window corresponds to the maximum image frequency *f*
_{OM}. The frequency origin is then translated to the centre of the window and the windowed image spectrum is multiplied by the Hann window function in the form 0.5(1 – cos π*ρ*), where *ρ* is the normalized distance from the centre of the window with *ρ* = 1 on the edge of the window. Finally, the image complex amplitude is computed using the two-dimensional inverse FFT and the image amplitude and phase are computed from the complex amplitude.

#### 2.3 Spatial frequency of the diffraction grating

Diffraction grating spatial frequency has to meet two conditions. The first one is to be sufficiently high to separate the zero order primary image of the light source out from the pupil of condensers. Although the grating is designed to have zero intensity in all the even orders including the zero order, it is true only for the ideal one. A real grating is not ideal and the intensity of even orders is thus non-zero, but weak and it does not have to be considered. The zero order image for *λ* ≠ *λ*
_{0} has generally non-zero intensity that cannot be neglected and has to be separated out from the pupils of condensers. It might decrease the contrast of interference fringes in the worst case if arrived at the output plane. Let us suppose the pupil of condenser to be identical with its mechanical aperture of a diameter *d*, and to be placed in a distance *l* from the grating, measured along the optical axis of the condenser, and the primary image of the light source to be imaged in the aperture plane and to be also *d* in diameter. The distance from the centre of the zero order primary image to the centre of the aperture can be expressed approximately according to Eq. (1) as *l*sin*ϑ*
_{0} = *lf*
_{G}
*λ*
_{0}. To ensure the zero order image be formed out of the condenser aperture, the distance *lf*
_{G}
*λ*
_{0} has to be greater or equal to *d*, and this gives the first condition for the spatial frequency:

The second condition relates to the hologram reconstruction process. To be possible to reconstruct both the amplitude and the phase of the object wave from one interferogram, it has to be a hologram, i.e. the carrier frequency *f*
_{OP} has to meet the holographic condition [24]:

*f*

_{OM}is the maximum spatial frequency of the image complex amplitude limited by the objective lens with a numerical aperture NA and a magnification

*m*:

Equation (5) in terms of Eq. (2) and Eq. (6) gives the second condition for the spatial frequency of the diffraction grating:

There were other methods of the image amplitude reconstruction proposed that do not require fulfilment of the Eq. (5), but each of them is more or less approximative. An interesting one is described in [25].

The upper limit for *f*
_{G} is given by Eq. (1) and by the condition |sin*ϑ*| ≤ 1 which gives *f*
_{G} ≤ 1/*λ*. The greater is *f*
_{G}, the finer is the interference structure in the output plane and the larger magnification is needed to resolve the fringes by a detector and thus the smaller is the field of view. Therefore it is convenient to keep *f*
_{G} as low as possible.

Spatial frequency of the diffraction grating used in our microscope is *f*
_{G} = 71 mm^{−1} which meets both the conditions given by Eq. (3) and Eq. (7) for most of wavelengths from a spectral transmittance interval of the microscope. At the same time it allows for a reasonable field of view width. The central wavelength was set to be *λ*
_{0} = 650 nm with respect to application of the microscope in biology. Equation (1) then gives the deflection of the interferometer arms *ϑ*
_{0} ≈2.7°. The microscope employs four identical pieces of planachromatic microobjective lenses of 160 mm tube length and *l* ≈150 mm. Aperture diameters for some of the lenses used are specified in Table 1 together with the corresponding overlap Δ*d* that indicates fulfilment of the first condition represented by Eq. (3) and the lower limit of *λ* meeting the second condition given by Eq. (7). The last two columns of Table 1 specify the spectral transmittance intervals of the microscope for each of the microobjectives.

The first condition, Eq. (3) is apparently valid for objective lenses 20 × and 40 × ; in case of 10 × lens the zero order overruns by 0.6 mm the objective’s aperture. It was proved experimentally that this overlap does not noticeably affect the imaging process. The holographic condition, Eq. (7) is also not quite valid for the 10 × lens. This could cause a little deformation of the image complex amplitude in the highest spatial frequencies of the structures parallel with diffraction grating lines. Such a deformation was not observed and it is even not expected because of use of the Hann window function during the hologram reconstruction.

#### 2.4 Output lens and the field of view

Magnification *m*
_{OL} of the output lens (OL) is dependent on the highest spatial frequency present in hologram at the output plane that has to be resolved and recorded digitally. Maximum spatial frequency equals to *f*
_{OP} + *f*
_{OM}, sum of the carrier frequency and the maximum image frequency [24]. The sampling rate should be at least 2.3 times higher [26]. This gives a condition for the spatial frequency *f*
_{D} (pixel density) of a detector:

Pixels of the detector used are square-shaped, 6.45 μm, thus the sampling rate *f*
_{D} = 155 mm^{−1} and Eq. (9) for *f*
_{G} = 71 mm^{−1} gives *m*
_{OL} ≥ 2.8. OL used in the microscope is a standard 4 × /0.12 microobjective lens, 160/–, i.e. *m*
_{OL} = 4 and the condition by Eq. (9) is thus fulfilled. Field of view with this lens is 2.2 mm × 1.7 mm (2.8 mm diagonal), measured in OP.

Numerical aperture NA_{OL} of OL is the second parameter that should be found. All the light passing through OP has to pass also through OL in order to form a hologram in the image plane of OL where a detector is placed. Maximum angle of a beam entering OL is given by sum of *ϑ*
_{0} and α’, where α’ = arcsin(NA/*m*) ≈1.5° is the aperture angle of the objective lenses in the image space (see Fig. 3b
for illustration). The condition can be expressed as follows: NA_{OL} ≥ sin(*ϑ*
_{0} + α’) ≈sin*ϑ*
_{0} + sinα’. After substitution from Eq. (1) we get:

The limit value for the microobjective lenses listed in Table 1 is for the lens 10 × /0.25: NA_{OL} ≥ 0.071. Numerical aperture of the output lens used in our microscope is NA_{OL} = 0.12 which is in agreement with (10).

Let us finally compare CCHM and a conventional (incoherent) microscope with respect to magnification of an output lens or a field of view size. If we consider the limit case of Eq. (5), i.e. *f*
_{OP} = 3*f*
_{OM}, then the highest spatial frequency present at the output plane is *f*
_{OP} + *f*
_{OM} = 4*f*
_{OM}. Maximum output spatial frequency in case of conventional incoherent imaging process is 2*f*
_{OM}. Therefore CCHM requires the magnification *m*
_{OL} of an output lens two times (at least) higher than the conventional microscope and the field of view size is thus two times (at least) smaller.

#### 2.5 Spectral transmissivity of the microscope

As the primary image of a spectrally broad source is extended by dispersive power of the diffraction grating and the mechanical aperture (pupil) of condensers is circular with diameter *d*, a limitation in the spectral transmittance of the microscope has to be considered.

Let the central point of the light source in the first order image for *λ* = *λ*
_{0} coincide with the pupil centre. Then its displacement *p* for *λ* ≠ *λ*
_{0} due to the dispersive power of the diffraction grating can be derived by differentiation of Eq. (1) and after approximations: cos*ϑ* ≈1 and Δ*ϑ* ≈*p/l* for small angle *ϑ*, it gives:

For *λ* = *λ*
_{0}, both the pupils are fully filled by the same section of the broad-source (see Fig. 2a
). For *λ* ≠ *λ*
_{0}, both the broad-source images are displaced from the pupil centre by distance *p*, but in opposite directions (see Fig. 2b) and thus slightly different sections of the source appear in each pupil.

Their conjunction shown in Fig. 2c is the only section of the source (effective area) that contributes (for a particular *λ*) to the imaging process (see the out-of-area points Z, Z’ in Fig. 2b, c). This is a consequence of spatial incoherence of the broad-source. The effective area is of a lenticular shape, *d –* 2*p* width. The interference structure is formed only if *|p*| ≤ *d/*2. Spectral transmittance interval of the microscope is given by this condition substituted to Eq. (11): *|λ – λ*
_{0}| ≤ *d*/2*lf*
_{G}. Spectral function of the microscope is a function of the effective area dependent on *λ*. This function is represented by autocorrelation of a constant function, non-zero on a circular area and it corresponds (quite formally) to the optical transfer function for a purely incoherent system (for the course of the function see e.g [27].). It is an even function, taking its maximum for zero argument and a half of the maximum is taken approximately for 2/5 of the limit of argument. Hence the spectral function of the microscope takes a half of its maximum for |*p*| ≈(2/5)(*d*/2) = *d*/5. If substituted into Eq. (11) it gives Δ*λ* and double of this value is the full width in a half maximum (FWHM), see Table 1:

#### 2.6 Broad-source diameter and interference fringes contrast

The negative consequence of an extended source employment is a decreasing contrast of the interference pattern towards edges of the field of view [28].

Field of view in the output plane is primarily limited by an effective area of the diffraction grating and it is a square of about 3 mm by 3 mm. Although there is the detector which limits the field of view (diagonal of the chip that we use, measured in OP, is 2.8 mm, see subsection 2.4), we will consider the full area of diffraction grating in this subsection, because the detector is usually rotated to have the fringes approximately perpendicular to the chip diagonal.

To analyse the cause of decreasing contrast we consider a monochromatic spatially incoherent broad source of *λ* = *λ*
_{0}, fully filling the condenser pupils (see Fig. 2a). The axial point of the source is imaged in both arms to the centre of condenser’s pupil and further to the centre of objective’s pupil (points A, A’ in Fig. 3a), thus creating a pair of mutually coherent quasi-point secondary sources forming a 3D interference structure with maxima represented by circular hyperboloids of foci in points A and A’.

Interferogram in OP is a planar cross-section of this structure and its maxima forms a set of hyperbolas. There is about 400 interference fringes across the field of view (*f*
_{OP} = 142 mm^{−1}, field of view width is about 3 mm), the central fringe is linear and a curvature of the marginal fringes is up to 1% of the fringe width, difference of the fringe width between the marginal and the central fringe is 0.01%, so this interference structure in the output plane can be approximated (with respect to the system arrangement) by linear equidistant interference fringes of spatial frequency *f*
_{OP}.

The other points of the source form their own structures that do not interfere with the other structures, but add up in intensity, because of the spatial incoherence of the light source. To understand how the fringe pattern formed by a general pair of points differs from the pattern formed by points A, A’, we will analyse the following two special cases: points B, B’ and C, C’ (see Fig. 3a). First we state that all the pairs of points lie on a sphere with the central point Q and the radius |AQ| = *l* ≈150 mm (see Fig. 3b).

Interference structure formed by points B, B’ can be seen as the structure of points A, A’ rotated around the central point Q of the output plane by an angle *α*’ (see Fig. 3b, c) that corresponds to the aperture angle (in the image space) of the objective lenses. Rotation introduces a difference in spatial frequency *f*
_{OP} of the interference pattern being a cross-section of the rotated structure with the output plane. The spatial frequency is lower (wider fringes) compared to the structure from points A, A’. Figure 3c shows the difference *δ* of the fringe patterns formed by points A, A’ and points B, B’ at a distance *x* from the centre Q of the output plane. The relation is: *δ* = *x*(1/cos *α*’ – 1) and its maximum occurs at the margin of the field of view (*x* ≈1.5 mm). Maximum aperture angle for the objective lenses in use is *α*’ ≈1.5° and we get *δ*
_{max} ≈500 nm, i.e. about 7% of the fringe width.

The structure from points C, C’ can also be identified with the structure formed by points A, A’, but in this case by translation by the distance |AC|. Maximum difference occurs again at the edges (higher differences are in direction perpendicular to the fringes) and especially in the corners of the field of view, where it reaches 10% of the fringe width.

The final interference pattern is an intensity summation of all the patterns formed by all the points within the effective area of the light source. The highest contrast of fringes will be in a vicinity of the point Q and it will decrease towards the edges of the field of view. Numerical calculation performed in [28] shows that contrast of the final interference pattern in OP will not decrease below 0.75 across the field of view. Experimental data obtained with an extended spatially incoherent light source fully filling condensers’ pupils, equipped with an interference filter 650 nm, of FWHM 10 nm, prove the decreasing tendency of the fringe pattern’s contrast towards edges with values of contrast ranging from 0.73 to 0.25 in one example, in another example from 0.57 to 0.37.

## 3. Experiment

An experiment has been carried out using the setup described in the previous chapter and shown in Fig. 1. Plan-achromatic objective lenses 20 × /0.4, 160/0.17, long working distance (LWD) were employed. The light source was a halogen lamp combined with an interference filter λ = 650 nm, FWHM = 70 nm. Diameter of the broad-source in pupils of condenser lenses was 6.5 mm. The detector used was a digital camera Softhard Technology Astropix 1.4 (the Peltier cooling, BW, 12 bit, 1376 pixels × 1038 pixels, cell size 6.45 μm × 6.45 μm, 12 fps).

#### 3.1 Optical sections

Two specimens: an amplitude object (AO) and a phase object (PO) were observed in order to prove optical sectioning effect of CCHM induced by extra low-coherent illumination. Figure 4 compares the images from CCHM (the second and the third row) to the conventional bright field images (the first row) acquired with the shutter closed in the reference arm of CCHM.

Both the samples were imaged alone (the first and the third column) and then covered with a diffuser (frosted glass – both sides ground cover slip 150 μm thick) placed 3 mm far from the specimen towards the objective lens (the second and the fourth column). Figure 4 shows that the conventional bright field image is destroyed by the light dispersion within the diffuser, while the CCHM intensity and phase images still clearly reveal features of the imaged structure. The phase in CCHM phase images is wrapped.

#### 3.2 Quantitative phase contrast

Eppendorf Cellocate coverslip with etched letter “M” (see Fig. 4, PO) was used as a transparent specimen. Depth of the relief derived from an atomic force microscope (AFM) measurement ranges in interval (359 ± 6) nm (95.4% probability).

The wrapped phase profiles *φ* acquired from the CCHM phase images as cross-sections along the white lines shown in Fig. 4 (from the upper left to the lower right; the same place of the specimen in both the phase images) were unwrapped first and then both the profiles were transformed by subtraction of a linear function *y* = *ax* + *b*. Coefficients *a* and *b* were different for both the profiles and were chosen to compensate for a tilt and to get the best alignment between the two profiles. Thereafter, the height profile *h* was calculated from the transformed phase profiles *φ*: *h* = *φ*/*k*(*n* – *n*
_{0}), where *k* = 2*π*/*λ*
_{0}, *λ*
_{0} = 650 nm, *n* ≈1.520 is the refractive index of Cellocate (*n* = 1.525 for *λ* = 546 nm) and *n*
_{0} ≈1 is the value for air. The result is shown in Fig. 5
.

The continuous line shows the specimen only and the dotted line shows the specimen “hidden” behind the diffuser. The depth 367 nm estimated from CCHM profiles roughly corresponds to the AFM measurement.

The noise apparent in the profile with the diffuser applied is a consequence of a low signal forming the image. Contrast of interference fringes in this case (PO) was between 0.5 and 0.3 for the specimen without diffuser and it was lower than 0.03 when the diffuser was applied.

## 4. Discussion

It is obvious from the first experimental results that the CCHM is capable to show structures covered by a dispersive object. This implies and proves its properties like noise reduction or optical sectioning. These and other features of CCHM are discussed in the next paragraphs.

#### 4.1 Numerical refocusing

If both the intensity (amplitude) and phase of the object wave are known in any plane of observation (usually the image plane), they can be calculated in any other plane within a coherence volume. This is fully true for CCHM and it allows for *ex-post* numerical refocusing up and down within the specimen [2]. This is advantageous in case of objects moving in 3D space. Reconstruction of 3D trajectories of living cancerous cells in matrix gels [3] or a 3D particle-flow analysis [4] demonstrate this capability. Limited coherence of illumination reduces the axial range of numerical refocusing [2].

#### 4.2 Low-coherence illumination consequences – noise reduction

Coherent-light sources (usually used in holography) introduce both the coherence-noise (speckles) and unintended interference in optical systems, thus impairing microscopic images quality. Reduction of spatial coherence of the light source limits substantially the coherence noise occurrence and hence it improves the image quality [2–5]. This effect can be described mathematically as a reduction of the high spatial frequencies coming from out-of-focus planes, otherwise degrading in-focus plane image [29]. Coherence-noise originating from scatterers lying outside the plane being imaged (out-of-plane scatterers) can be substantially reduced in the achromatic systems by combining both the low temporal and the low spatial coherence [18]. Achromatic system behaves like an incoherently illuminated imaging system in its noise suppression [30].

Figure 4 shows no coherence noise originating by an extremely strong scatterer. Unintended interference of the light scattered on surfaces of optical elements is also suppressed.

#### 4.3 Low-coherence illumination consequences – optical sectioning and related phenomena

Noise reduction caused by limited coherence of the light source goes hand in hand with optical sectioning effect, i.e. a limited contribution of light scattered in out-of-focus planes of the specimen to the resulting image and significantly improved in-focus image contrast, especially for objects embedded in a scattering media. It is a typical attribute of confocal microscopy achieved by tandem scanning of the object plane by point-apertures, or of other techniques like OCT or OCM in reflection mode [31], where the coherence gating effect is employed. Optical sectioning in DHM with partially coherent illumination was proved only for the phase-stepping DHM (in-line setup) [3,5]. Low spatial coherence suppresses influence of a scattering media (e.g. collagenous gels) surrounding an observed object in such a way that it limits interference of low-coherence non-ballistic photons.

However, an achromatic interferometer in off-axis CCHM allows to reach the optical sectioning effect by coherence gating too. Image-plane spatially incoherent holography allows to limit substantially both the spatial and the temporal coherence of illumination. It has been identified with confocal imaging both theoretically [22,24] and experimentally [23,32]. Optical sectioning effect of both the techniques is equivalent if a monochromatic but extended i.e. spatially incoherent light source is used. Optical sectioning in reflected-light DHM has been proved [21]. If the temporal coherence is reduced, the optical sectioning effect in case of incoherent holography techniques is stronger [3,5,22,23].

Applicability of incoherent holography for imaging through a scattering media was proved experimentally: in a classical holography setup [33], in a setup combining imaging systems and grating interferometers [34,35], or in a microscopical phase-shifting setup [36–38].

The effect of a ballistic light separation and optical sectioning for the transmitted light CCHM is proved in Fig. 4 in both the CCHM intensity and phase images of both the amplitude and phase objects hidden behind a diffuser. In the conventional bright field image, there is no structure of an object visible, because the diffuser spreads the image of the object in many directions. In the holographic mode, the reference arm acts as a filter that selects always a single direction (one image) from the object arm.

#### 4.4 Low-coherence illumination consequences – signal to noise ratio

Moreover, the coherence gating techniques can detect a low ballistic signal from a scattering media better than the confocal techniques due to the optical amplification of the signal over a detector basic noise [39]. This is applicable for CCHM as well.

#### 4.5 Remarkable features of incoherent off-axis holography

It comes out that it is convenient to use achromatic interferometers for construction of interference microscopes, because these allow for optical sectioning by coherence gating effect also in case of the off-axis holography that has the advantage of one shot full field reconstruction of image intensity and phase.

Moreover, a generalized analysis of optical sectioning by holographic coherence imaging [20] gives remarkable conclusions and it shows that achromatic interferometers for off-axis holography have unique properties especially in transmission mode. The main conclusions from [20] follow:

In the case in which the light is reflected from the object:

In a transmission mode, there are two special cases: a basic transmission situation (a confocal tandem scanning or a low-coherence image plane holography with a conventional interferometer, i.e. without a diffraction grating, e.g. in-line):

- • Low temporal coherence is second order effect for the optical sectioning, while the spatial incoherence is the prime cause; and a diffraction grating achromatic interferometer based low-coherence image plane holography (off-axis):
- • Diffraction grating transforms the temporal incoherence into the spatial incoherence. Spectrally broad point source is converted by the dispersive power of the grating into an linearly extended spatially incoherent source. Broad-spectrum source thus causes a similar effect of optical sectioning as a spatially incoherent monochromatic source.
- • In the case when the source is both spectrally and spatially broad (extended) the sectioning is finer than would be expected from the summation of the two broadening processes – those of broad source and those of broad spectrum and than it is in case of non-grating achromatic interferometers. This statement is valid also for the reflected-light systems.

The above analysis shows the key point and the principal advantage of the optical setup with the grating achromatic interferometer used for construction of the off-axis holographic microscopes. Spatial broadening and temporal broadening of the light source (i.e. extended and polychromatic source) is limited only by apertures of the optical components and by the dispersive power of the diffraction grating. Coherence degree of a light source can thus be adapted according to the object and to the required image properties. Higher degree of coherence allows for wider range of numerical refocusing, while the lower degree of coherence makes the optical sectioning stronger, i.e. finer optical section. As both the broadening processes enhance each other’s optical sectioning effect [20] in the grating achromatic interferometer, one can expect extremely fine optical sections.

On top of that, the lateral resolution limit of such a microscope corresponds to incoherent imaging process [24] which means that it is half of the value for coherent illumination, the mode used in current DHMs.

A single point of the light source does not contribute to the resulting image in intensity as it is in case of a conventional microscope, but its contribution is added as amplitude and phase, so the resulting image (image complex amplitude) is coherent similarly to a standard confocal microscope [24].

#### 4.6 Applications

Most of the applications of the transmitted-light CCHM deal with living cells. The quantitative phase contrast of CCHM is a marker-free (non-toxic) possibility for real-time dynamic observations of living cells motility, e.g. areal density of dry mass distribution within cells [1,40]. Applications of the reflected light CCHM are focused mostly on surface profiling [23,41].

## 5. Conclusion

- • We have built a transmitted-light holographic microscope that employs an achromatic and space-invariant interferometer for off-axis image plane holography which allows to use extended light sources with substantially reduced spatial and temporal coherence (CCHM) and we described its technical parameters and imaging properties in this paper.
- • CCHM allows for quantitative phase-contrast imaging.
- • CCHM is capable to image structures hidden behind a dispersive layer. This property arises from low-coherence of illumination. It is obvious from the experiment that the light multiply scattered outside the plane being imaged does not affect the imaging process.
- • Coherence degree of the light source in CCHM can be adapted according to the object and to the required image properties. Closer to coherent illumination allows for wider range of numerical refocusing, while closer to incoherent light makes the optical sectioning stronger, i.e. finer optical section [22–24] and it allows for ballistic light separation.
- • Optical sectioning of the holographic imaging process is primarily a consequence of spatial incoherence. Low temporal coherence added reduces significantly thickness of the optical section in reflected light. It can also achieve optical sectioning in itself in the reflected light and in the special case of transmitted-light setup – that with the diffraction grating achromatic interferometer [20], i.e. in our CCHM.
- • Coherent noise originating from out-of-plane scatterers is substantially reduced by combining both the low temporal and low spatial coherence [18].
- • Lateral resolution limit corresponds to incoherent imaging process [24] and it is half of the value for coherent illumination.

## Acknowledgments

The work is supported by the Grant Agency of the Czech Republic (grant 202/08/0590), and Ministry of Education of CR (grant MSM0021630508).

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