Chapter 2
Principles of Holography
Contents
2.1 Introduction . . . . 7
2.2 Light waves . . . . 7
2.3 Interference . . . . 9
2.4 Coherence . . . 10
2.4.1 Temporal coherence . . . . 10
2.4.2 Spatial coherence . . . . 12
2.5 Diraction. . . 14
2.6 Holography . . . 15
2.6.1 Hologram recording and reconstruction . . . . 16
2.6.2 Optical reconstruction of a wave eld . . . . 16
2.1 Introduction
In this chapter, the physical basis of holography is discussed. The main phenomena constituting holography are interference and diraction, due to the wave nature of light. A description of the wave theory of light as far as it is required to understand the recording and reconstruction of holograms is presented.
2.2 Light waves
Light is a transverse, electromagnetic wave characterized by time-varying electric and magnetic elds. Since electromagnetic waves obey the Maxwell equations, the propagation of light is described by the wave equation which follows from the Maxwell equations. The wave equation for propagation of light in vacuum is [1]:
∇2E − 1 c2
∂2E
∂t2 = 0 (2.1)
whereE is the electric eld. An identical equation is satised for the magnetic eld.
∇2 is the Laplace operator dened as:
∇2= ∂2
∂x2 + ∂2
∂y2 + ∂2
∂z2 (2.2)
where(x, y, z) are the spatial coordinates, tis the time andc is the speed of light.
Transverse waves vibrate perpendicularly to the direction of propagation and so they are described in vector notation. Such property is related to the polar- ization of light. For most applications, it is not necessary to use the full vectorial description of the elds, so we can assume a wave vibrating mainly in a single direction. Such a wave is called linear polarized light. For a one component linear polarized wave eld, the scalar wave equation is sucient:
∂2
∂z2E− 1 c2
∂2E
∂t2 = 0 (2.3)
It is easy to verify that a linearly polarized harmonic plane wave is the solution of the wave equation:
E(x, y, z, t) =acos(ωt+kr+ϕ0) (2.4) where E(x, y, z, t) is the electrical eld vector at point r = (x, y, z) at time t and a is the amplitude of the wave. The propagation of the wave is described by the vector k = kn where n is a unit vector in propagation direction and k is the modulus ofk which is equal tok≡k= 2π/λ. The angular frequencyω is related to the frequency f of the light wave by ω = 2πf. The frequency f is connected to the wavelength λ by the speed of light c through the relation c =λ/f. The term kr+ϕ0, varying spatially, is the phaseϕ, withϕ0 being a constant phase.
The only directly measurable quantity1 is the intensity I, dened as the time average of the square of the electrical eld:
I =ε0c E2
t=ε0c lim
T→∞
1 2T
+T
−T
E2dt (2.5)
where t means the time average over many light periods, the factor ε0c comes from the Maxwell equations, withε0 being the vacuum permittivity. The intensity of the plane wave of Eq.(2.4) is:
I =ε0c a2
cos2(ωt+kr+ϕ0)
t= ε0c a2
2 (2.6)
given that the intensity is computed by the square of the amplitude. In complex form, Eq.(2.4) is rewritten as:
E(x, y, z, t) =a Re
e−j(ωt+kr+ϕ0) (2.7)
1Indeed, as the wavelengths of visible light are in the range of400nm (violet) to780nm(deep red), the corresponding frequency range is7.5×1014Hz to3.8×1014Hz. These high frequencies cannot be detected by light sensors (the human eye, photodiodes or CCD's) for a variety of technical and physical reasons.
2.3. Interference 9 wherej =√
−1andRe represents the real part of the complex function. This com- plex representation allows the separation of the spatial and temporal components2: E(x, y, z, t) =ae−jϕ(x,y,z) e−jωt (2.8) By substituting Eq.(2.8) in Eq .(2.3), the following time-independent equation (known as the Helmholtz equation) is obtained:
(∇2+k2)u(x, y, z) = 0 (2.9) wherek= 2π/λis the wave number andu(x, y, z)is named complex amplitude and is dened as :
u(x, y, z) =ae−jϕ(x,y,z) (2.10) Eq.(2.8) and Eq.(2.10) are valid for plane waves but also for three-dimensional waves whose amplitude aand phase ϕcan be a function ofx, y and z.
The intensity is now calculated in complex notation as the square of the complex amplitude:
I =|u|2 =u∗u=a2 (2.11)
where ∗ denotes the complex conjugate, where the constant factor ε0c/2 can be neglected in many practical calculations and where the absolute value ofI is not of interest.
2.3 Interference
Interferences happen with the superposition of two or more waves in space. This forms the basis of holography. Thanks to the linearity of the wave equation, if the single waveui(x, y, z) is a solution of Eq.(2.1), the superposition of N waves is also a solution:
u(x, y, z) =N
i=1
ui(x, y, z) (2.12)
is also a solution.
Consider the interference of two waves having the same wavelength and the same frequencies. The complex amplitudes of the waves are:
u1(x, y, z) =a1e−jϕ1(x,y,z)
u2(x, y, z) =a2e−jϕ2(x,y,z) (2.13)
2where theResymbol is omitted
By adding these two individual amplitudes, the resulting complex amplitude is cal- culated as u=u1+u2. The intensity is computed thanks to Eq.(2.11):
I =|u1+u2|2
=a21+a22+ 2a1a2cos(ϕ1−ϕ2)
=I1+I2+ 2
I1I2cos Δϕ (2.14)
with I1 and I2, being the intensities of the two waves and Δϕ, the dierence between the two phases. Consequently, the resulting intensity is given by the intensity of the two beams plus an interference term. The intensity reaches its maximum when Δϕ = 2nπ (constructive interference)and its minimum when Δϕ = (2n+ 1)π (destructive interference)for n = 0,1,2, . . .. An interference pattern consists then of dark and bright fringes as a result of the constructive and destructive interference. The periodic structure of the intensity in space is usually referred to as interference fringes.
The fringe visibility of an interference pattern is dened as a contrast mea- sure by:
V = Imax−Imin
Imax+Imin (2.15)
whereImaxandIminare two neighboring intensity maxima and minima, correspond- ing to constructive and destructive interferences.
2.4 Coherence
To generate interference fringes, the phase of the individual waves must be correlated in a special way. This correlation property is named coherence and is investigated in this section. Coherence is the ability of light to interfere [2]. The two aspects of coherence are the temporal and the spatial coherence. In this section, a simplied description of these two coherences is outlined. Temporal coherence describes the correlation of a wave with itself at dierent instants while spatial coherence is related to the mutual correlation between two points of the same wavefront.
2.4.1 Temporal coherence
Temporal coherence describes the correlation of a wave with itself considering its behaviour at dierent instants. Michelson introduced a technique to measure the temporal coherence. The instrument, called the Michelson interferometer, is described in this section to illustrate the temporal coherence. Consider the path of rays in the Michelson interferometer shown in Fig.(2.1). The light emitted from the light source is divided into two beams by a beam splitter (BS). The two resulting beams are back-reected by the mirrors M1 and M2, and superimposed on the screen after passing again through the beam-splitter. The optical path length from
2.4. Coherence 11
BS to M1 and back to BS is S1 and the optical path length from BS to M2 and back to BS is S2.
Experiments demonstrate that the fringe visibility decreases with increasing the dierence |S1 −S2|. This decrease, which depends on the source type, can be characterized by a typical dierence in path length L, that is called temporal coherence length. Its corresponding emission time is the coherence time computed as τ = L/c. The coherence length L is a measure of the spectral width Δf as it is approximately inversely proportional to the spectral width. Light having a long coherence length is named highly monochromatic. The length of coherence of light bulbs is of the order of micrometers while some lasers have coherence light of a few millimeters to several hundred meters.
The fringe visibility, dened in Eq.(2.15), can be calculated by inserting in
Fig. 2.1: The Michelson's interferometer Eq.(2.14) Δϕ= 0 andΔϕ=π, for Imax andImin, respectively:
V = 2√ I1I2
I1+I2 (2.16)
with I1 and I2, being the intensities of the two interfering beams. This resulting visibility is thus the one of an innite coherence length (monochromatic wave).
To consider the eect of a nite coherence length, the complex self coherence function Γ is introduced :
Γ(τ) =E(t+τ)E∗(t)t= lim
T→∞
1 2T
+T
−T
E(t+τ)E∗(t)dt (2.17)
where E(t) and E(t+τ) are the electrical eld at the time t and the one delayed by a time τ, respectively. The degree of coherence can be dened by the normal- ized quantity γ(τ) = Γ(τ)/Γ(0). Through simple calculations, for cases of nite coherence length, Eq.(2.16) becomes:
V = 2√ I1I2
I1+I2 |γ| (2.18)
which leads to V = |γ| for two waves having the same intensities. The degree of coherence is then equal to the fringe visibility, and so to the ability of the two beams to interfere. The following cases are distinguished :
|γ(τ)| = 1 completely coherent 0 < |γ(τ)| < 1 partially coherent
|γ(τ)| = 0 completely incoherent (2.19) 2.4.2 Spatial coherence
The mutual correlation between the complex amplitude at two points of the same wavefront describes the spatial coherence. One of the best ways to introduce spatial coherence is through the double-slit of the Young experiment, illustrated in Fig.(2.2). The light source is extended to emit light from dierent points. In Fig.(2.2), two light rays from a point source are shown in red. An aperture with two transparent holes is mounted between the light source and the screen. Under certain conditions, interference between the beams coming from the upper and the lower hole are visible on the screen. If the distance a between the holes exceeds a critical limit, the interference pattern vanishes. This limit is called coherence distance. This is due to the fact that the waves emitted by dierent source points are superimposed on the screen. As a consequence, it may happen that at a certain point of the screen, an maximum interference is resulting from a source point and at the same point of the screen, a minimum interference is obtained by another source point. This is due to the dierent optical path lengths for light rays emerging from dierent source points. Generally, the contributions from all source points compensate themselves and the contrast vanishes.
Nevertheless, this compensation is avoided if the following condition is fullled:
r2−r1 < λ
2 (2.20)
2.4. Coherence 13 wherer1 =
R2+ ((a−h)/2)2 and r2 =
R2+ ((a+h)/2)2, withh the width of the light source.
By using the assumptions thataRandhR, the following relation is obtained:
r2−r1 ≈ a h 2R < λ
2 (2.21)
Therefore, the coherence distanceac is:
ach 2R = λ
2 (2.22)
As seen through Eq.(2.22), the spatial coherence depends not only on the properties
Fig. 2.2: The Young's interferometer
of the light source, but also on the geometry of the interferometer.
The autocorrelation function dened in Eq.(2.17) is extended to consider spatial coherence:
Γ(r1, r2, τ) =E(r1, t+τ)E∗(r2, τ)t
= lim
T→∞
1 2T
+T
−T
E(r1, t+τ)E∗(r2, τ)dt (2.23)
withr1 andr2 being the spatial vectors of the two slits. The normalized function is then:
γ(r1, r2, τ) = Γ(r1, r2, τ)
Γ(r1, r1,0) Γ(r2, r2,0) (2.24) with Γ(r1, r1,0) and Γ(r2, r2,0) the intensity at r1 and at r2, respectively. The functionγ(r1, r2, τ), called complex degree of coherence, measures the degree of cor- relation between the light eld atr1 at timet+τ and the light eld atr2 at time t.
This function measures both the temporal and spatial coherence. By takingτ = 0,
a measure of the correlation between the two elds at the same time is possible.
The following cases are then distinguished as for the temporal coherence case:
|γ(r1, r2,0)| = 1 completely coherent 0 < |γ(r1, r2,0)| < 1 partially coherent
|γ(r1, r2,0)| = 0 completely incoherent
(2.25)
2.5 Diraction
This phenomenon of diraction occurs when a light wave hits an obstacle. The term diraction has been dened by Sommerfeld as "any deviation from rectilinear paths which cannot be interpreted as reection or refraction" [3]. The mechanisms of diraction are of fundamental importance for understanding the principle of holog- raphy. In this section, the main results of scalar diraction theory are presented.
From a qualitative point of view, the theory of diraction can be explained by Huygens's principle, illustrated in Fig.(2.3):
Every point of a wavefront can be considered as a source point for sec- ondary spherical waves. The wavefront at any other place is the coherent superposition of these secondary waves.
Fig. 2.3: Huygens's principle
The diraction theory is given by the Fresnel-Kirchho diraction formula [1,3]
computing the eld in the diraction plane (=observation plane):
u(ξ, η, d) = 1 jλ
+∞
−∞
+∞
−∞
u(x, y,0)ejkρ ρ
cosθ+ cosθ
2 dx dy
withρ =
(x−ξ)2+ (y−η)2+d2 (2.26) The used coordinate system is described in Fig.(2.4). The light source S, with coordinates (ξ, η) in the source plane, radiates spherical waves. The light is then
2.6. Holography 15 propagated and u(x, y,0) is its complex amplitude in the aperture plane (z = 0).
Consider rst an opaque aperture with only one hole at the position (x, y,0). By Huygens's principle, this hole becomes now the source for secondary waves. The eld at the position (ξ, η, d) in the diraction plane is proportional to the eld at the entrance side of the aperture u(x, y,0) and to the propagated eld of the secondary spherical waves emerging from(x, y,0), described byejkρ/ρ.
The generalization is done by regarding the aperture plane as a plane con- sisting of many sources for secondary wave. The resulting eld in the diraction plane is then the integral over all secondary spherical waves, emerging from the aperture plane. According to Huygens's principle, the secondary waves can also propagate back into the direction of the source. Since this eventuality does not occur in practice, the inclination factor (cosθ + cosθ)/2 is introduced in the Fresnel-Kirchho integral, where θ is the angle between the incident ray from the source and the unit vector normal to the aperture plane, andθ is the angle between the diracted ray and the normal unit vector.
Fig. 2.4: Coordinate system
2.6 Holography
In conventional procedures for image acquisition, such as photography, the intensity distribution is recorded but the phase information, i.e. all information about the optical paths to dierent parts of the recorded object, is lost. To overcome the diculty in recording the complete information, a fundamentally dierent approach is necessary, and was suggested by Denis Gabor [4,5] in 1948.
2.6.1 Hologram recording and reconstruction
Usually, holograms are recorded with an optical set-up consisting of a light source (e.g. laser), mirrors and lenses for beam guiding and a recording device, e.g. a photographic plate. Fig.(2.5) illustrates a typical set-up where light with a sucient coherence length is split into two beams by a beam splitter (BM). The rst beam, called the object beam, illuminates the object and is then scattered toward the recording medium. The second beam, named reference beam, illuminates the recording plate directly. The photographic lm records the interference pattern produced by this reference beam and the light waves scattered by the object.
The recorded interference pattern is named hologram. Since any point of this interference pattern also depends on the phase of the object wave, the resulting hologram contains information on the phase as well as on the intensity of the object wave.
If the hologram is illuminated once again with the original reference wave, as shown in Fig.(2.6), the original object wave is reconstructed. Therefore, an observer looking through the hologram sees a three-dimensional image of the object. This virtual image exhibits all the eects of perspective that characterized the original object.
Fig. 2.5: Hologram recording
2.6.2 Optical reconstruction of a wave eld
By using the formalism described in Section 2.3, the mathematical description of the holographic process can be performed. The complex amplitudes of the object
2.6. Holography 17
Fig. 2.6: Hologram reconstruction beam and the reference beam are expressed by:
uo(x, y) =ao e−jϕo(x,y)
ur(x, y) =ar e−jϕr(x,y) (2.27) with ao and ar the amplitude of the object beam and the reference beam, respec- tively, andϕo and ϕr being the phase of the object beam and the reference beam, respectively. The intensity of the two interfering beams is:
I(x, y) =|uo+ur|2
= (uo+ur) (uo+ur)∗
=uou∗o+uru∗r+uou∗r+uru∗o (2.28) On the recording medium (e.g. photographic plate), the amplitude transmission h(x, y) is proportional toI(x, y):
h(x, y) =h0+βτ I(x, y) (2.29) where the constant β is related to the exposure eciency of the plate, τ is the ex- posure time andh0 is the amplitude transmission of the unexposed plate. h(x, y)is named hologram function. For hologram reconstruction, as illustrated in Fig.(2.6), the complex amplitude of the reconstruction wave (reference wave) has to be mul- tiplied by the hologram function:
ur(x, y)h(x, y) =ur(x, y) [h0+βτ I(x, y)]
=
h0+βτ
a2o+a2r
ur+βτ a2ruo+βτ u2ru∗o (2.30)
which reproduces analytically the physical situation of Fig.(2.6). The rst term of Eq.(2.30) is the reference beam, multiplied by a factor and it represents the undiracted wave passing through the hologram, which is in fact the zero diraction order. The second term is the reconstructed object wave that forms the virtual image. The brightness of this virtual image is inuenced by the real factor βτ a2r. The third term generates the real image of the object, distorted by the spatially varying complex factoru2r that modulates the image of the conjugate objectu∗o. To obtain a undistorted real image, the conjugate reference beamu∗r has to be used for the reconstruction:
u∗r(x, y)h(x, y) =u∗r(x, y) [h0+βτ I(x, y)]
=
h0+βτ
a2o+a2r
u∗r+βτ u∗2r uo+βτ a2ru∗o (2.31) A major drawback of the original technique of Gabor was the poor quality of the reconstructed image. It was degraded by the conjugate image, which was superimposed on it, as well as by scattered light from the directly transmitted beam.
The twin-image problem was solved when Leith and Upatnieks [6] developed the o-axis reference beam technique shown schematically in Fig.(2.5) and (2.6).
They used a separate reference wave incident on the photographic plate at an appre- ciable angle to the object wave. Therefore, when the hologram was illuminated with the original reference beam, the two images were separated by large enough angles from the directly transmitted beam, and between each other, to ensure that they did not overlap. Then, with o-axis holography, the three contributions of Eq.(2.31) are spatially separated [6]. The invention of the laser, which provided a powerful source of coherent light, made the development of the o-axis conguration possible, and increased the activities in holography, leading to several important applications [7,8].
In the next chapter, digital holography is introduced as an extension of holography on photographic plates where the two interfering beams are recorded on a Charged Coupled Device (CCD). Combining both, digital Holographic Microscopy (DHM) constitutes our eld of research and is described in the next chapter.
