Numerical Optics in Digital Holography

Abstract

As digital holography achieves numerical reconstruction of the complex wavefront diffracted by specimen, a numerical optics formalism has been developed: numerical filters replace pinholes in Fourier planes, numerical lenses allow aberrations compensation and image magnification.

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Numerical optics in digital holography
Tristan Colomba, Florian Charrièreb, Jonas Kühnb, Frédéric
Montfortc, Christian Depeursingeb
aCentre de Neurosciences Psychiatriques, Département de psychiatrie DP-CHUV, Site de Cery,
1008 Prilly- Lausanne, Switzerland
bEcole polytechnique fédérale de Lausanne, Institute of imaging and applied optics, CH-1015
Lausanne, Switzerland
cLyncée Tec SA, PSE-A, CH-1015 Lausanne, Switzerland
Abstract: Asdigitalholographyachievesafullynumericalreconstructionofthecomplexwavefront
diffracted by specimen, a numerical optics formalism has been developed: numerical filters replace
pinholes in Fourier planes, numerical lenses allow aberrations compensation and image magnifica-
tion.
c ? 2007 Optical Society of America
OCIS codes: (090.1760) Computer holography; (090.1000) Aberration compensation; (110.0180) Microscopy
1.Introduction
In classical microscopy, and in optics in general, optical elements are used and optimized in order to achieve the best
results. For example, lenses arrangement could be optimized for the best lateral resolution with minimum aberrations
and/or distortion; pinholes are sometimes used to filter out different diffraction orders after a grating element.
On the other hand, a first application of holography is to use the holograms as optical elements to compensate for
aberrations. These holograms are sometimes recorded on photographic plates [1,2] or are generated by computer and
written on spatial light modulators [3,4].
A new branch of holography called digital holography or digital holographic microscopy showed that it is possi-
ble to recover at video frequency the complex wavefront reflected by or transmitted through a specimen [5] by using
a digital camera instead of photographic plate. Most of the time, the digital hologram of a physical diffracted wave-
front is recorded on a CCD camera [6] and then the digitally reconstructed wavefront is propagated digitally from the
hologram plane to the image plane (see for example Refs. [7,8]).
It was already demonstrated in several papers, that numerical procedure can be used to perform frequency filter-
ing [9], high order aberrations and anamorphism compensation [8,10–19] in digital holography. The purpose of this
paper consists to demonstrate that numerical optics can replace physical optics. These numerical optics are also free
from obstruction problems and any kind of shape for numerical lenses (NLs) are possible, allowing aberration and
distortion compensation, magnification and frequency filtering.
To demonstrate the effectiveness of the numerical optics, the microscope objective of the digital holographic
microscope (DHM) is replaced by a cylindrical lens to produce aberrations, distortion and different magnifications
along horizontal and vertical directions. We show that numerical optics allows to reconstruct amplitude and phase
images without aberration or image distortion and with the same magnification.
2.Hologram acquisition and wavefront reconstruction
The setup used is presented on Fig. 1(a): O is the object wave and R the reference wave, CL is the cylindrical lens
used as microscope objective. The object is a USAF test target. The object and reference waves interfere in off-axis
geometry on the CCD camera to produce the digital hologram IH = R2+ O2+ R∗O + RO∗[Fig. 1(b)]. The
wavefront is reconstructed in the Fresnel approximation by using the convolution formalism [7,8]:
Ψ(m,n) = NLI(m,n) · A · FFT−1?FFT?NLH(k,l)IF
where, FFT is the Fast Fourier Transform; m, n, k, l are integers (−N/2 < m,n,k,l ≤ N/2); d, the reconstruction
distance; A = exp(i2πd/λ)/(iλd); λ, the wavelength; νk= k/(N∆x), νl= l/(N∆y) are the spatial frequencies
coordinates, NLPare defined in the hologram (P = H) or image (P = I) plane. Instead of propagating the digital
hologram IH, we propagate a filtered apodized hologram
H(k,l)?· exp?−iπλd(ν2
k+ ν2
l)??,
(1)
IF
H= FFT−1[FFT(IH· AP)FM],
(2)

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where AP is a numerical amplitude filter used to apodized the hologram and therefore to suppress numerical diffrac-
tion created by the finite window of the hologram (cf. Ref. [20] for details); FM is a numerical mask filtering the
hologram to propagate only the real (RO∗) or the virtual (R∗O) image. FM can be seen as a numerical pinhole with
any shape definition [9] [the hole corresponds to the none-black pixels in Fig. 1(c)].
NLPare complex arrays with constant amplitude (|NL(k,l)| = 1 ∀k,l); the shape of the lens is defined by a
standard polynomial model [18] (other models are possible, as Zernike polynomials [8]):

λ
NL(k,l) = exp
−i2π
α+β=o
?
α=β=0
Pαβ· kαlβ

,
(3)
where Pαβare the NL parameters and o is the polynomial order. These parameters can be computed automatically
by fitting the assumed to be flat areas in the reconstructed phase image with the NL model [8,18] or by using a con-
jugated reference hologram [17].
Furthermore, NLs can be used for magnification. Indeed, NL can be defined by a thin lens transmittance [21].
To be able to have different magnification in horizontal (Mk) or vertical (Ml) direction, this magnification lens is
written:
NLH,M(k,l) = exp
λ
From Eqs. 3 and 4, the standard polynomial parameters of the numerical magnification lens (NPLH,M) are writ-
ten: PH,M
20
= ∆x2/(2fk) and PH,M
02
= ∆x2/(2fl). From lens equation and knowing the initial reconstruction
distance, it is easy to determine the NPLH,Mfrom the wanted magnification Mkand Ml[8].
The use of this magnification NL in the hologram plane changes the focus reconstruction distance as presented
in Refs. [8, 19]. To use two different magnifications, two reconstruction distances are needed in the reconstruction
propagation. This is done by changing the kernel of the propagation:
exp?−iπλd(ν2
This modification was already presented in different papers in order to compensate for anamorphism and astima-
tism [11,13].
?
i2π
?k2∆x2
2fk
+l2∆y2
2fl
??
.
(4)
k+ ν2
l)?−→ exp?−iπλ(dkν2
k+ dlν2
l)?.
(5)
BS
M1
M2
CCD
Object
CL
BS
OC
RL
Source
R
O
(a)
z
R
O
θ
x
(b)(c)
Fig. 1. (a) Digital holographic microscope: O, reference wave; R, the object wave; M1, M2, mirrors, BS, beam splitter; OC, condenser; RL,
reference lens; CL, cylindrical lens. Inset presents the off-axis geometry. (b) Digital hologram recorded with a USAF test target object. (c) Filtered
spectrum of (b), FM has value 0 in the dark area and 1 elsewhere.
3. Results
Figure 2 presents different reconstructions obtained from the same hologram [Fig. 1(a)] filtered with the mask FM
presented in Fig. 1(c). Fig. 2(a,b) present respectively the amplitude and phase image with the parameters PH
−3.96426E − 8, PH
(AP = 1). We note that the amplitude image suffers from astigmatism (horizontal edge of steps are not in focus)
and the phase is disturbed by residual phase aberration. In Fig. 2(c,d), numerical apodization is applied and the astig-
matism is compensated by adjusting P02 = 4.22365E − 10. Furthermore the NPL in the image plane is adjusted
automatically [18] to compensate for phase aberrations. In Fig. 2(e,f), we adjust PH
the distortion of the image [see insets in Figs. 2(d,e)]. Finally two different magnifications are applied in Fig. 2(g,h)
(Ml = 1,22 and Mk = 1.01) to provides the correct quotient between the length and the width (5/1) given by the
constructor of the USAF test target.
10=
01= −1.40369E − 7, P20 = P02 = 4.08365E − 10 and without apodization compensation
11= 4.1E − 11 to compensate for

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(a)(b)(c)(d)
(h)(g) (f) (e)
Fig. 2. Different steps of the reconstruction of amplitude and phase images of the USAF test target. (a,b) "normal" reconstruction; (b,c)
apodization, astigmatism and phase aberration compensation; (c,d) distortion compensation; (e,f) magnification Ml= 1,22 and Mk= 1.01 to
provide correct length and width quotient (5/1).
4.Conclusion
We demonstrate in this paper the powerful of numerical optics applied in digital holographic microscopy. Physical
optics such as pinholes, apodizing optics or lenses can be replaced advantageously by numerical optics. They can
have any shape and can be placed at any position avoiding obstruction problems to provide aberration- and distortion-
free amplitude and phase images.
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