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Fourier Holography

Sahand NoorizadehSchool of Electrical and Computer Engineering

Portland State University Portland, Oregon 97201

Email: sahand.noorizadeh@gmail.com

AbstractIn this paper the limitations of direct imaging andthe need for holography is explained. The fundamental conceptof holography is presented and two types of Fourier holographytechniques with and without lens are discussed in detail withmathematical derivations. Furthermore, a brief comparison ofthe available recording media and their limitations are outlined.Lastly, an application of the Fourier holography in the biomedicalmicroscopy is explored.

I. INTRODUCTION

All the available optical recording media such as film or

CCD are only capable of recording the intensity of the light.The intensity being a time-averaged quantity does not carry

any phase information. Rather it is proportional to the power of

the optical wave incident on the recording medium from which

only the amplitude can be obtained. A direct recorded image

of an object only has information about the amplitude of the

light wave received from the object and the phase information

is lost. In most practical applications of optics, it is the phase

information that is of interest. The phase of a traveling wave

with wavelength is proportional to the distance traveled bythe wave. As shown in Figure 1, two identical waves originated

from the same point, traveling in two different directions, and

observed at the distance x have two different amplitudes. If

multiple observations of the amplitude of the two waves atpoint x are made over a long period of time as the waves traveland then these values are averaged, the resulting amplitude

values for both points on the observation line would be the

same. That is the reason why the phase information is lost in

the intensity measurements.

x

o

Fig. 1. Phase vs. traveling distance of a wave.

Holography is a method of recording optical interference

of light from an object with a reference light to be able

to reconstruct the image of the recorded object. Until the

invention of coherent light sources such as laser, holography

was not entirely feasible and practical because in order to be

able to form well-defined and measurable fringe patterns by

the means of interference, the wavelength of the light source

needs to be stable and coherent.

Assuming two waves a(x, y) and A(x, y) represented bytheir phasor, their interference (superposition) is given by Eq.

1 and the intensity of their interference is given by Eq. 2

which is the magnitude squared of the amplitude interference

function.

B(x, y) = |a(x, y)| ei(x,y) + |A(x, y)| ei(x,y) (1)

I(B)=|a(x, y)|2 + |A(x, y)|2 + 2|a(x, y)||A(x, y)|

cos[(x, y) (x, y)] (2)

The interference allows the phase difference of the waves to be

preserved. If the phase of one of the interfered waves is known,

the phase of the other wave can be found. In holography, the

behavior of one of the two waves is known (the reference

wave) and the other is the scattered wave from a subject whose

bahavior will be measured.

The options of arrangement of the recording setup (i.e.

position of the object with respect to the recording medium

and the reference wave) has led to a wide range of classes

of holography. Fore example, depending on the distance of

the object from the recording medium, the propagation of

light waves from the object could be best characterized by

the Fresnel (near-field) propagation law or by the Fraunhofer

(far-field) propagation law. The different effects of each of

the preceding arrangements on the interference pattern at the

recording plane has been the cause of different classification

of holography systems. Another type of holography is defined

by the angle of illumination: on-axis and off-axis. One other

type of holography that is the subject of the remaining of this

paper is Fourier holography.

I I . FOURIER HOLOGRAPHY

In Fourier holography, the Fourier transform (FT) of the

objects amplitude transmittance is recorded. To achieve this,

there are two methods used: a) Fourier Transform Holography

with Lens and b) Lens-Less Fourier Holography.

A. Fourier Transform Hologram with Lens

In the first method, a lens is used to place the object and

the reference wave at the back focal plane of the lens and

record the FT of the interference of the the reference and the

objects transmittance function. Figure 2 shows a setup of a

FT hologram where the object transmittance function O(x, y)

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is illuminated by a coherent plane wave which is also incident

on a smaller lens, L0, separated by its focal length from theobject plane to convert the incident plane wave into a point

source, (x a, y b), that is located at point (a, b) on the(x, y) plane. The lens L1 performs the FT operation. The field

Recording

Medium

Illumination

ff

Fourier Plane

L1

L0

z

Fig. 2. Fourier holography with lens.

distribution on the (, ) plane is given by Eq. 3.

U(fX , fY)= F{O(x, y) + (x a, y b)}

= F{O(x, y)} + F{(x a, y b)}

= Q(fX , fY) + ei2(afX+bfY) (3)

Where Q(fX , fY) is the FT of O(x, y), fX = /f, fY =/f, and f is the focal length of L1. The recorded intensityis given by Eq. 4.

I(fX , fY)=1 + |Q(fX , fY)|2 + Q(fX , fY) e

i2(afX+bfY)

+ Q(fX , fY) ei2(afX+bfY) (4)

The recording of this intensity (whether on a film or with

a CCD) will produce a transmittance function that can beassumed is linearly proportional to the intensity of Eq. 4.

Therefore, for reconstructing the image of the object, a plane

wave of the same wavelength can be used to illuminate

this transmittance function which will in turn generate a

wavefront, W whose complex amplitude immediately passedthe transparency (the zero propagation length) is the same

as the transmittance function. In the Fourier transform of W(done either numerically using a computer or with a lens) the

first two terms in Eq. 4 will produce zero-order (DC) terms

and the last two terms will reproduce two inverted images

of the original object centered at (a,b) and (a, b). Theimages are inverted because a double FT had to be performed

and the Fourier transform of the Fourier transform of a

function returns the inverted-domain version of that function:

F{F{f(x)}} = f(x).

B. Lensless Fourier Hologram

In the previous section, the Fourier transforming properties

of the lens were exploited to perform the FT operation.

However it is possible to FT the transmittance function of an

object without employing a lens. Figure 3 shows a holography

system in which the object is illuminated with a plane wave

and a reference point source is located on the same plane d unitdistance away from the recording medium at point (a, b) ofthe object plane as the object. It is necessary that the reference

wave and the object be on the same plane.

Recording

MediumIllumination

z

Point Source d

Fig. 3. Lensless Fourier hologram.

Since the objects illumination is a plane wave, the ampli-

tude of the light distribution to the immediate right hand side

of the object is simply the transmittance function of the object.

The distance d is chosen so that the propagation of the lightfrom the object can be expressed by Fresnel diffraction given

by Eq 6. Where the first term is constant phase factor, the

second term is a quadratic phase exponential, and the integral

is the FT of the product of the object transmittance function

O(x, y) and a quadratic phase exponential.

Ui(, )=eikd

ideik

2d(2+2)

O(x, y)e

ik

2d(x2+y2) eik/d(x+y)dxdy (5)

= C eik

2d(2+2) Foe(fX , fY); (6)

Eq. 6 is the compact form of Eq. 5 where Foe(fX , fY) =F{O(x, y) eik

2d(x2+y2)}, fX = /d, and fY = /d.

From the reference point source a spherical wave propagates

towards the (, ) plane. The propagation of this wave is givenby Eq. 7.

Ur(, )= eik

2d(2+2) eik/d(a+b)

= eik

2d(2+2) ei2(fXa+fYb) (7)

The field distribution at the holograms plane is the super-

position of the diffracted object field and the reference wave

and the intensity of this superposition is given by Eq. 8.

I(fX , fY)=ADC + C Foe e

i2(fXa+fYb)

+ C Foe ei2(fXa+fYb) (8)

Where ADC is the sum of all zero-order terms. The quadraticphase factor at the hologram plane e

ik

2d(2+2), that was

common in both Ur and Ui, was cancelled in the intensity.Eq. 8 is very similar to Eq. 4 except that there is an additional

constant phase factor which can be dropped and the image

contains a quadratic phase exponential.

In reconstructing the original object, a plane wave can be

used as in the Fourier hologram with lens but a lens will be

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required to remove the quadratic phase exponential. A more

common way is to reconstruct with a point source similar to

the one used for the reference in the recording process which

will make the virtual reconstructed image coincide with the

object.

III. RECORDING MEDIUM

The recording medium mentioned so far was a generic term.

The common options are film and charged coupled device

sensors (CCD). Films require to be developed to be used as

transparency for reconstruction of the image of the object. The

process of preparing the film for reconstruction is often tedious

and time-consuming. This can be specially a disadvantage if

an application requires multiple and fast exposures. Also the

reconstruction process is a manual and analog process which

complicates data processing. The advantage of films are their

very high angular resolution compared to CCDs.

A CCD is a two-dimensional array of NM square sensorsand they also can only record the intensity. The resolution of

CCD sensors is a function of the array size and the pixel

dimension. Each pixel samples the intensity in its coveragearea. For a fixed boundary, the more pixels the higher the

sampling rate. Therefore, smaller pixels are desirable. The

limiting resolution of the CCD camera is determined by its

Nyquist limit. This is defined as being one half of the sampling

frequency (i.e. # pixels/mm).

The angle between the reference beam and the object beam

in the holographic setup is limited because the holographic

fringe structures in the hologram plane need to be sampled

by the CCD sensor. The sampling theorem requires that the

angle between the object beam and the reference beam at any

point of the CCD sensor be limited in such a way that the

microinterference fringe spacing is larger than double the pixel

size [1].The use of CCD allows the numerical reconstruction using

computers. This provides easy data processing capability such

as filtering. Also, the holograms of different object states in

holographic interferometry can be reconstructed with different

wavelengths and still interfere numerically. This is of particular

interest for multiple-wavelength techniques that are used for

holographic contouring [2].

IV. AN APPLICATION OF FOURIER HOLOGRAPHY

The digital Fourier hologram shown in Figure 4 can be used

to measure the angular spectrum of the elastically scattered

light at many spatial locations covering a large field of view

based on a single capture or a few image captures [3].In this hologram, the beam splitter B1 splits the laser beam

and polarizes it using P1 for a uniform illumination that isapplied to the sample at an angle in as reflected by the mirrorM1. Since the samples are placed inside a medium with adifferent index of refraction, there will be a change in the

angles of entrance and exit beam from the sample container.

Lens L1 is a focal length (of L1) away from the sample toFourier transform the backscattered light from the sample and

L2 and L3 transfer the image of this spectrum to the CCD

Fig. 4. Schematic of the setup for the spatially resolved Fourier holographiclight scattering angular spectroscopy[3].

sensor. The other split beam from B1 is expanded by thetelescopic system T and routed by the mirror M3 to interferewith the image of the spectrum of backscattered light from

the sample at B2 to form a holographic interference patternat the CCD plane. This reference beam is superimposed with

the image of the spectrum at an angle of 2.3 so that the twin

images become separable during reconstruction.

The complex spectrum obtained by lens L1 is proportionalto the size of the scatterers and the refractive index of

their container. In the Fourier plane there is a one-to-one

correspondence between spatial position and scattering angle

[4]. The holographic technique provides a reference beam

to interfere with this spectrum so that it is the interference

that is recorded not the spectrum itself. This way, very smallscattering angles beyond the spatial resolution limits of the

CCD can be encoded into the interference pattern and then

numerically be reconstructed. The analysis of the spectrum

of the backscattered light waves can reveal information about

the features of the sample and this method has been used

on biological samples to deduce morphological information at

all points in the field of view. Combining the Mie or other

scattering theories will extract scatterer sizes and refractive

index contrasts [3].

REFERENCES

[1] U. Schnars, Direct phase determination in hologram interferometry with

use of digitally recorded holograms, J. Opt. Soc. Am. A 11, 20112015(1994).

[2] Christoph Wagner, Snke Seebacher, Wolfgang Osten, and Werner Jptner,Digital Recording and Numerical Reconstruction of Lensless FourierHolograms in Optical Metrology, Appl. Opt. 38, 4812-4820 (1999).

[3] Sergey A. Alexandrov, Timothy R. Hillman, and David D. Sampson,Spatially resolved Fourier holographic light scattering angular spec-troscopy, Opt. Lett. 30, 3305-3307 (2005).

[4] M. T. Valentine, A. K. Popp, D. A. Weitz, and P. D. Kaplan, Microscope-based static light-scattering instrument, Opt. Lett. 26, 890-892 (2001)