Chapter 3
Digital Holography
Contents
3.1 Introduction . . . 19
3.2 Numerical reconstruction of digital holograms . . . 21
3.2.1 Operator formalism . . . . 22
3.2.2 Discretization of the Kirchho-Fresnel integral . . . . 24
3.3 Digital holographic microscopy . . . 24
3.3.1 Description of DHM . . . . 26
3.3.2 Extraction of the full interferometric information . . . . 28
3.1 Introduction
In digital holography (DH), the reference beam and the object beam are interfering on a electronic sensor, the CCD1 [9,10].The detected light distribution is digitized to give rise to a digital hologram.The reconstruction in depth of the wave eld is then performed digitally.
DH allows one to refocus the experimental volume, plane by plane, simulat- ing the depth adjustment of an imaging system.
The basis for numerical hologram reconstruction is given by the Fresnel-Kirchho diraction integral of Eq.(2.30):
u(ξ, η, d) = 1 jλ
+∞
−∞
+∞
−∞
h(x, y,0)ur(x, y,0)ejkρ ρ
cosθ+ cosθ
2 dx dy (3.1)
whereρ =
(x−ξ)2+ (y−η)2+d2, h(x, y,0) is the hologram function dened in the previous section and ur is the reference amplitude.
The coordinate system for numerical hologram reconstruction is shown in Fig.(3.1).If the angles θ and θ are small and lies close to the propagating axis, cosθ and cosθ are approximately equal to 1.This small-angle approximation is
1A Charged-Coupled Devices (CCD) is an electrical device used to create images of objects, store information or transfer electrical charge. The most popular application is image recording.
referred to the paraxial approximation. Eq.(3.1) computes the diraction patterns at a distance d behind the CCD plane, which means that it reconstructs the complex amplitude in the plane of the real image as illustrated in Fig.(3.2(b)). As explained in Section 2.6, the real image could be distorted by the spatially varying complex factor u2r of Eq.(2.31). To reconstruct an undistorted real image in digital holography, it is therefore necessary to insert u∗r instead of ur in Eq.(3.1):
u(ξ, η, d) = 1 jλ
+∞
−∞
+∞
−∞
h(x, y,0)u∗r(x, y,0)ejkρ
ρ dx dy (3.2)
with
ρ=
(x−ξ)2+ (y−η)2+d2 (3.3)
This reconstruction scheme is illustrated in Fig.(3.2(c)), where the three planes are perpendicular to the propagation axis (z). The real image is reconstructed at the position where the object was located during the recording.
To calculate Eq.(3.2) for the free-space propagation in the paraxial approxi- mation, some approximations have to be performed. Two major approximations can be done, referred to as Fresnel and Fraunhofer approximations. The rst approximation of the diraction formula is valid in the near eld of the aperture while the second one is valid in the far eld. As our application concerns digital holographic microscopy, the Fresnel diraction in the near eld will be derived in the next section.
Fig. 3.1: Coordinate system for numerical hologram reconstruction
3.2. Numerical reconstruction of digital holograms 21
(a) Recording
(b) Reconstruction with the reference waveur
(c) Reconstruction with the conjugate ref- erence waveu∗r
Fig. 3.2: Digital holography [11]
3.2 Numerical reconstruction of digital holograms
Forx, y, ξandηvalues small compared to the distance d, Eq.(3.3) can be expanded to a Taylor series:
ρ=d
1 +(ξ−x)2
d2 + (η−y)2 d2
=d+(ξ−x)2
2d +(η−y)2 2d −1
8
(ξ−x)2+ (η−y)2
d3 +. . . (3.4)
The fourth term can be neglected if it is small compared to the wavelength [1,12]:
1 8
(ξ−x)2+ (η−y)22
d3 λ ⇐⇒ d 3
1 8
[(ξ−x)2+ (η−y)2]2
λ (3.5)
The distance ρ becomes :
ρ≈d+ (ξ−x)2
2d +(η−y)2
2d (3.6)
which constitutes the Fresnel approximation. It can be shown that this approxima- tion is accurate only for small angles of diraction [3]. It is for this reason that the
Fresnel and the paraxial approximations are often considered as equivalent.
In the Fresnel approximation, Eq.(3.2) is rewritten:
u(ξ, η, d) = ejkd jλd
+∞
−∞
+∞
−∞
u(x, y,0) ejk2d[(ξ−x)2+(η−y)2] dx dy (3.7)
where h(x, y,0)u∗r(x, y,0) is replaced by u(x, y,0). This integral propagates the complex eld u(x, y,0) up to a distance d along the optical axis z. The complex function of the reconstructed wave eld u(ξ, η, d) permits to compute the intensity and the phase [13]:
I(ξ, η, d) =|u(ξ, η, d)|2
ϕ(ξ, η, d) = arctanIm [u(ξ, η, d)]
Re [u(ξ, η, d)] (3.8)
3.2.1 Operator formalism
Eq.(3.7) can be rewritten in a simpler way by adopting an operator formalism described in [14,15]. Dene three operators by the following relations:
• a quadratic phase multiplication operator2 Q[a]:
Q[a]f(x, y)≡ejka2 (x2+y2)f(x, y) (3.9)
• a scaling operatorV[a]:
V[a]f(x, y)≡f(ax, ay) (3.10)
• a Fourier transform (FT) operator (and an inverse FT):
F±f(x, y)≡ +∞
−∞
+∞
−∞
e∓2jπ(xνx+yνy) f(x, y)dx dy (3.11)
2Note that this operator acquires the argument of the function on which it operates, herexand y.
3.2. Numerical reconstruction of digital holograms 23 With these three operators, Eq.(3.7) can be rewritten after rearranging terms :
u(ξ, η, d) = ejkd jλd
+∞
−∞
+∞
−∞
ejk2d[(ξ−x)2+(η−y)2] u(x, y,0)dx dy
= ejkd
jλd ejk2d(ξ2+η2) +∞
−∞
+∞
−∞
e−jkd (xξ+yη) ejk2d(x2+y2) u(x, y,0)dx dy
= ejkd jλdQ
1 d
+∞
−∞
+∞
−∞
e−2jπλd (xξ+yη)Q 1
d u(x, y,0)dx dy
= ejkd jλdQ
1 d V
1 λd F Q
1
d u(x, y,0)
≡R[d]u(x, y,0) (3.12)
where the free-space propagation operator (FPO)R is dened in the last line3. An alternative expression of the FPOcan be derived by introducing the convolution operator∗ [3] in Eq.(3.7):
u(ξ, η, d) = ejkd jλdQ
1
d ∗u(x, y,0) (3.13)
By performing a Fourier transform on both sides of Eq.(3.13) and by using the convolution theorem that transforms a convolution into a product in the spatial frequency domain, the following equation is obtained :
F u(ξ, η, d) = ejkd jλd
F Q
1
d F u(x, y,0) (3.14)
In the previous expression, it can be easily shown that the FT transform of a quadratic phase factor is a quadratic phase factor (analogy with the Gaussian func- tion):
F Q 1
d =jλdQ
−λ2d
(3.15) By taking the inverse FT of Eq.(3.14), the Fresnel-Kirchho diraction integral can then be rewritten by another expression of the FPO:
u(x, y, d) = ejkd F−1Q
−λ2d
F+1u(x, y,0)
= ejkd Fx−1,yQ
−λ2d
Fν+1x,νyu(x, y,0) (3.16) where we changed the spatial coordinates(ξ, η)into(x, y)to simplify the notations.
The spatial frequencies are(νx, νy). This expression represents the Kirchho-Fresnel
3The Fraunhofer approximation is a more stringent approximation, and consists to add to the Fresnel approximation, the hypothesis that the quadratic phase function is approximately unity.
equation in the paraxial approximation that will be used in this thesis to propagate the complex eld on a distance d. This equation is read from right to left; rst a FT is performed on the complex amplitude of the initial plane (the recorded plane), then a multiplication by the quadratic phase factor and nally an inverse FT are performed to obtain the complex amplitude propagated to a distance d. To be implemented numerically, this expression has to be digitized. This is the subject of the next section.
3.2.2 Discretization of the Kirchho-Fresnel integral
For numerical evaluations, Eq.(3.16) has to be implemented in a discrete form. The hologram is sampled byN×N points (N andN are the number of pixels of the CCD camera) with Δ, the sampling distance in the spatial coordinates(x, y) and(x, y) andΔ the sampling distance in the spatial frequencies. The relation between these two sampling distance is given by the theory of Fourier transform [3,11]Δ = 1/NΔ.
The discretization of the dierent coordinates and frequencies are:
x−→sΔ y−→tΔ x −→sΔ y −→tΔ νx −→mΔ
νy −→nΔ (3.17)
wheres, t, s, t, mandnare integer numbers varying from0toN−1. The discrete form of Eq.(3.16) is written as:
u(sΔ, tΔ, d) = ejkd Fs−1,tQ
−λ2d
Fm,n+1 u(sΔ, tΔ,0)
= ejkd Fs−1,t e−jkλ22d(m2Δ2+n2Δ2) Fm,n+1 u(sΔ, tΔ,0)
= ejkd Fs−1,t e−jkλ2N2Δ22d(m2+n2) Fm,n+1 u(sΔ, tΔ,0) (3.18) where the FT operator is discretized in a form that allows the implementation of the FFT :
Fm,n± f(s, t) = 1 N
N−1
s,t=0
e∓2jπN (sm+tn) f(s, t) (3.19) Eq.(3.18) represents the discretization of the Kirchho-Fresnel diraction in the Fresnel approximation that is implemented for the propagation slice by slice in the experimental volume.
3.3 Digital holographic microscopy
Digital holography is applied in many eld, one of them is microscopy. Digital holographic microscopy (DHM) constitutes the eld of research of this thesis. As
3.3. Digital holographic microscopy 25 introduced in Section 1, this thesis is part of the HoloFlow project whose goal is the visualization and analysis of small particles, from1μmto a few hundred ofμm.
Optical microscopy is very limited by the small depths of eld due to the high numerical apertures of the microscope lenses and the high magnication ra- tios. Consequently, mechanical scanning along the optical axis has to be performed to investigate sample thicker than the depth of focus. As detailed in the previous section, digital holography is able to overcome this limitation.
DHM has been applied and demonstrated in many applications such as the observation of biological samples [1619], refractometry [20,21], analysis of living cells [2224], metrology [25] and velocimetry [26,27]. Because the complex amplitude signal is accurately determined with a digital holography setup, DHM is a very exible tool for implementing powerful processing of the holographic information. This technique allows for example to implement processing to improve the digital holographic reconstruction [28], to study concentration proles inside conned deformable bodies owing in microchannels [8], to perform 3D pattern recognition [29,30], to control the image size as a function of the distance and the wavelength [31], to achieve quantitative phase contrast imaging [3234], to process border artifacts [35] or to compensate aberrations [36].
Several optical methods have been implemented to extract, from the recorded interference pattern, the phase and the amplitude information [20,37]. For dynamic phenomena, a crucial point is the acquisition time. Indeed, phase stepping methods need the acquisition of several images, which restrict the applications to objects slowly varying in time. Conversely, the o-axis (Fourier) method [3840] is very suitable for fast phenomena because it requires only one recorded hologram to compute the complex amplitude. It is the reason why this method has been chosen to perform experiments in the HoloFlow project.
Usually the sources used in holography are lasers. However, highly coherent beams are very sensitive to any defect in the optical paths and the resulting recorded holograms are aected by coherent noise. To reduce this noise eect and avoid multiple reection interferences, several congurations were proposed with a partially coherent optical source [4143]. Among these, a DHM with a partial coherent illumination in a Mach-Zehnder conguration has been developed at MRC-ULB [44,45]. Because we are working with dynamic phenomena, it is mandatory to get the complete digital holographic information for every recorded image with a short exposure time. The illumination is performed by a laser diode, and the reduction of the spatial coherence is obtained by focusing the laser beam close to a rotating ground glass. It has been demonstrated that partial spatial illumination signicantly increases the quality of the holographic images [45]. In the next section, the optical setup of the DHM used during the thesis is described, and the technical specications of the microscope are provided.
3.3.1 Description of DHM
In this section, the DHM used to perform experiments is described. The optical setup, shown in Fig.(3.3) is a Mach-Zehnder interferometer in a microscope conguration working in transmission with a partial spatial coherent source. A coherent source (a monomode laser diode,λ= 635nm) is transformed into a partial spatial coherent source by focusing the incident beam, with lens L1, close to the plane of a rotating ground glass (RGG). The ground glass is rotating in its plane by a direct current motor. The light transmitted by the ground glass is scattered by the rough surface to create a speckle eld that varies with the rotation of the ground glass. When the ground glass is immobile, the sample is illuminated by a speckle eld with a characteristic grain size that depends on the focusing by lens L1. The scattering surface of the ground glass is placed in the front focal plane of lens L2 that is collimating the beam.
The beam is divided by beam splitter BS1. The transmitted part is the object
Fig. 3.3: Optical setup of the DHM [46]
beam, which is illuminating the sample S passing through the experimental cell (micro-channel MC) by transmission. The sample is inserted in the micro-channel
3.3. Digital holographic microscopy 27 by a pumping system PS (the uidic part is described in more details in Section 5.2 of the experiments description). The object beam, transmitted by microscope lens L3 , is reected by mirror M2 and by beam splitter BS2. The object beam is then transmitted by lens L5. Lenses L3-L5 image one plane of the sample onto the CCD camera sensor. The magnication G is then given by the ratio between the focal length L5 and the focal length L3. The reference beam, reected by beam splitter BS1 and by the mirror M3, is transmitted by microscope lens L4, by beam splitter BS2, and by lens L5. The beam is then incident on the CCD sensor, where it interferes with the object beam. The reference beam is tilted with respect to the object beam in such a way that a grating-like thin interference pattern is recorded on the sensor. This reference beam gives the direction from which the light comes from.
In an operating mode, the ground glass is spinning fast making the speckle eld change rapidly. The exposure time of the camera is adjusted to perform an averaging of the speckle eld in such a way that the illumination is almost uniform.
The partial spatial coherence is adjusted by changing the position of the focused spot on the RGG plane [39].
To quantify the spatial coherence, the speckle size in the plane of the sample is measured. It can be shown that the spatial coherence width of the beam illuminating the sample is equal to the average width of the speckle size [45]. For the experiments performed in the frame of the HoloFlow project (see Section 5.2), the coherence width has been adjusted to approximately 30 μm. The use of the spatial partial coherence reduces considerably the coherent noise and provides a temporal-coherence-like eect that also eliminates the coherent multiple reection eect [8].
The camera is a JAI CV-M4 camera with a CCD array of 1280 x 1024 pix- els, which is cropped in a 1024 x 1024 pixel window to match the 2D fast Fourier transform computation needed to perform the digital refocusing given by Eq.(3.18).
The angle between the reference and the object beam is adjusted in such a way that a gratinglike thin interference pattern is recorded on the sensor. It results in a high-fringe-density hologram which a fringe period corresponding to about 6 pixels.
Dierent objectives lenses can be mounted on the DHM depending on the species of interest. More details concerning the dierent adjustments of the CCD camera (exposure time, eld of view, etc.) are given in Section 5.2 of the experiments description.
Once one hologram is recorded, the interferometric information, composed of the intensity and the phase, is extracted. Through an example, the extraction process is explained in the next section.
3.3.2 Extraction of the full interferometric information 3.3.2.1 Intensity and phase
Once a digital hologram is recorded, the complex amplitude is rst computed using the Fourier method [38,40]. The intensity and the phase are then computed using Eq.(3.8). This process is illustrated in Fig.(3.4), where Fig.(3.4(a)) represents a digital hologram of a green alga, Pediastrum sp. The computed intensity and phase information is illustrated in Fig.(3.4(b)) and Fig.(3.4(c)), respectively. The phase map gives the optical path length integrated along the propagating axis of the illuminating wave.
The non-uniform background present on the phase map of Fig.(3.4(c)) is due to two sources of optical defects. Firstly, the experimental cell is not completely at and may have some defects. Secondly, there is always a small misalignment of the interferometer. Therefore it is necessary to implement a phase map correction that subtracts the background phase, which is described in the next section.
3.3.2.2 Phase map compensation
In the HoloFlow project, we are interested in the detection and the classication of alga which are considered as phase objects. Therefore, the non-uniform background of the phase has to be removed. Indeed, as we will see in the next chapters, the compensated phase is needed to perform the 3D detection as well as to extract the textural features based on the phase information.
Thanks to the symmetry between the object and the reference beams of the interferometer, we implemented a fast method based on the phase map derivative, described in [8]. The computed phaseϕ(x, y) of Fig.3.4(c) is the sum of the phase induced by the object and the background phase that corresponds to the phase obtained when no object is present in the optical system:
ϕ(x, y) =β(x, y) +φ(x, y) (3.20)
where β(x, y) is the background phase and φ(x, y) is the object phase we want to measure. This phase addition results in global phase dierences (over the eld of view) that can be larger than 2π, introducing jumps in the phase map. To solve this problem, one solution is to unwrap the phase. Several methods have been proposed to solve the complex problem of phase unwrapping [47,48]. However, phase unwrapping does not compensate for aberations and is therefore not a good way to measure the object phase. For the studied organisms4 in the HoloFlow project, the optical thickness dierence between these species and the surrounding medium (water) is suciently small to avoid phase jumps bigger than2π when the phase map is adequately corrected. Therefore, we can use a compensation method
4See Section5.2.3of Chapter5for a description of the studied organisms
3.3. Digital holographic microscopy 29
(a)
(b) (c)
Fig. 3.4: Extraction of the intensity and the phase information. (a) Recorded hologram of a Pedistrum sp. and the extracted intensity image (b) and phase image (c).
to remove the background phase5.
Because of the symmetry of the interferometer, we assume that the background phaseβ(x, y) is modeled by a quadratic phase expressed by:
β(x, y) =σx(x−x0)2+σy(y−y0)2 (3.21) where(x0, y0) are the coordinates of the center of the parabola and σx and σy are the curvatures along the x and y axes. Note that a similar modeling of the phase
5For other species with phase jumps, a solution consists to mask those objects before the compensation of the background.
map by a polynomial function has been successfully proposed and implemented in [49] for phase compensation aberration. As the phase change induced by the objects is relatively small, one can thus assume that the measured phase ϕ(x, y) is relatively close to the background phase β(x, y) and can be determined by tting ϕ(x, y)by Eq.(3.21). However, this tting cannot be computed directly fromϕ(x, y) because of phase jumps. To remove these discontinuities, the tting is performed on the derivative of the phase map [50]. The parameters x0, y0, σx and σy of the simulated quadratic phase result from a least mean squares method, by minimizing the following expression:
Ψ(σx, σy, x0, y0) = N x=1
N x=1
∂ϕ(x, y)
∂x −∂β(x, y)
∂x
2+
∂ϕ(x, y)
∂y −∂β(x, y)
∂y
2 (3.22) which gives the values ofx0, y0, σx andσy, and determines the simulated quadratic background phase β(x, y). The compensated phase φ(x, y) is then obtained by subtracting this computed background phaseβ(x, y)to the measured phaseϕ(x, y).
Fig.(3.5(a)) illustrates the computed background phase of Fig.(3.4(c)). The obtained compensated phase is shown on Fig.(3.5(b)). Thanks to this aberrations compensation of the optical phase, the permanent defects of the optical cells and some possible misalignment of the microscope are eliminated. From this compen- sated phase, the Pediastrum alga can be detected and quantitative information can be measured as illustrated in Fig.(3.6) where the cropped alga is shown with its pseudo-3D representation (with inverted contrast).
In addition to the quantitative phase information, digital holography oers
(a) (b)
Fig. 3.5: Compensation of the phase. Computed background phase (a) and compensated phase image (b)
3.3. Digital holographic microscopy 31
(a) (b)
Fig. 3.6: Cropped compensation phase of the Pediastrum sp. (a) Pseudo-3D representa- tion with inverted contrast (b) (the z-axis corresponds to the optical tickness).
the possibility to perform digital refocusing in a post-processing by using Eq.(3.18).
It counters the key limitation of 2Dimaging instruments, which is the narrow depth of focus. Thanks to DHM refocusing, particles that are initially recorded out of focus, can be refocused. This refocusing step is indispensable to extract the 3Dposition of the recorded object and to perform further processing like feature extraction for a classication procedure. However, digital holography does not provide any criterion to indicate when the best focus plane of an object is reached.
In this thesis, we use a criterion proposed in [51] based on the invariances of both energy and amplitude. In the next chapter, this criterion is rst reviewed.
Then, its dependency is derived through analytical expressions, which are conrmed by simulations and experiments. The robustness of the proposed criterion is then investigated. A robust criterion is crucial for the automatization of the objects detection (see Section 5). Indeed, since a large volume of sample has to be monitored, an automation of the detection process is needed. As the refocusing step is the main processing step, we have to ensure that the used criterion is reliable.
This is the goal of the detailed study of this criterion presented in the next chapter.
