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Spatially incoherent single channel digital

Fourier holography

Roy Kelner* and Joseph Rosen

Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel

*Corresponding author: kelnerr@post.bgu.ac.il

Received June 14, 2012; accepted July 16, 2012;

posted July 20, 2012 (Doc. ID 170643); published August 31, 2012

We present a new method for recording digital Fourier holograms under incoherent illumination. A single ex-

posure recorded by a digital camera is sufficient to record a real-valued hologram that encodes the complete

three-dimensional properties of an object.© 2012 Optical Society of America

OCIS codes:090.0090, 090.1995, 110.6880, 100.3010, 070.6120.

In recent years we have witnessed noteworthy achieve-

ments of incoherent holography techniques such as op-

tical scanning holography [1] and Fresnel incoherent

correlation holography (FINCH) [2,3]. The latter has fun-

damental system robustness since it is based on a single

channel incoherent interferometer and because it does

not require any scanning or mechanical movement. How-

ever, though a single FINCH image contains the complete

three-dimensional (3D) information of an object, at least

three images are required to solve the twin image pro-

blem [4]. In this Letter, we present a new method for

recording digital Fourier holograms under incoherent il-

lumination. Fourier holograms [5,6] possess some advan-

tages over Fresnel holograms including, but not limited

to, increased space–bandwidth product performance

[7,8] and the relatively easy ability to process and manip-

ulate the hologram, since it is captured in the spatial fre-

quency domain. In addition, the hologram is more robust

to information loss, as each object point is distributed

over the entire hologram plane. Moreover, by recording

a Fourier hologram of a half plane (or space), the twin

image problem is avoided and the object can be recon-

structed from a single exposure. Still, the proposed meth-

od maintains many other advantageous characteristics of

FINCH [9]. We coin our method Fourier incoherent single

channel holography (FISCH).

The FISCH system is shown in Fig. 1. A white-light

source illuminates a 3D object. Light scattered from

the object passes through a band pass filter (BPF), be-

comes partially temporally coherent, and continues to

propagate through a single channel incoherent interfe-

rometer. Eventually, interference patterns are captured

by the CCD. Consider a point source object of complex

amplitude Aspositioned at the coordinate ?xs;ys;zs?, a

distance zsfrom the lens L0. A tilted diverging spherical

wave of the formT?x;y; ⃗rs;zs? ? Asc?⃗rs;zs?L?−⃗rs∕

zs?Q?1∕zs? is induced over the L0

⃗rs? ?xs;ys?, c?⃗rs;zs? is a complex valued constant de-

pendent on the position of the point source, and L?⃗s? ?

exp?i2πλ−1?sxx ? syy?? and Q?s? ? exp?iπsλ−1?x2? y2??

are the linear and the quadratic phase functions, respec-

tively, in which λ is defined as the central wavelength. A

diffractive optical element of the form Q?−1∕f1? is dis-

played on the spatial light modulator (SLM), positioned

at a small angle relative to the optical axis. The SLM is

polarization sensitive, so by introducing the linear polar-

izer P1 the single channel optical apparatus is effectively

plane, where

split into two beams (see [10] for a detailed explanation).

With one beam the SLM functions as a converging diffrac-

tive lens with a focal length f1, while in the other it acts

as plane mirror. The second polarizer P2 rejoins the

two beams.

The unmodulated wave at the SLM plane, right after

passing through it for the second time, is

C1?x;y; ⃗rs;zs??T?x;y; ⃗rs;zs?·Q

?−1

f0

?

?Q

?

1

d0?2f1

?

; (1)

while the modulated wave, at the same position, is

C2?x;y; ⃗rs;zs? ? T?x;y; ⃗rs;zs? · Q

?−1

?

f0

?

? Q

?−1

?1

?

d0

?

· Q

?−1

f1

?

? Q

?1

2f1

· Q

f1

;(2)

where ?Q?1∕zd? denotes a Fresnel propagation of a wave

to a distance of zd[mathematically, ? denotes a two-

dimensional (2D) convolution] and Q?−1∕f? denotes

the influence of a converging lens of focal length f upon

a wave propagating through it. Upon arrival to the CCD

plane the waves are

Fig. 1.

solution chart; BPF, bandpass filter; BS, beam splitter; SLM,

spatial light modulator; M, mirror; CCD, charge-coupled device;

P1 and P2, polarizers.

(Color online) Schematic of a FISCH recorder: RC, re-

September 1, 2012 / Vol. 37, No. 17 / OPTICS LETTERS3723

0146-9592/12/173723-03$15.00/0 © 2012 Optical Society of America

Page 2

Dk?x;y; ⃗rs;zs? ? Ck?x;y; ⃗rs;zs? ? Q

?1

?

f2

?

· Q

?−1

?

f2

?

? Q

?1

2f2

?

· Q

?−1

f2

? Q

?1

d2

; (3)

where k ? 1, 2 represent the unmodulated and modu-

lated beams,respectively.

justification presented in [11], the recorded intensity

over the CCD plane, I?x;y; ⃗rs;zs? ? jD1?x;y; ⃗rs;zs??

D2?x;y; ⃗rs;zs?j2, is equal to

I?x;y; ⃗rs;zs? ? Is?jb1j2? jb2j2?

?

?

where Is? As· A?

andb2

are constants,

fe? d0− f2? d2and c.c. is the complex conjugate of

the left term inside the square brackets. Since each point

source is only spatially coherent to itself, the recorded

hologram due to many point sources is simply a summa-

tion over all point source contributions. That is, the

recorded hologram is

ZZZ

Consider the special case of a point source object po-

sitioned at the front focal plane of the lens L0(i.e.,

zs? f0). In that case, Eq. (4) is easily reduced to

I?x;y; ⃗rs;f0? ? Is?jb1j2? jb2?j2?

?

Note that according to the third term of Eq. (6), each

source point at ⃗rsis mapped at the ?x;y? plane to a linear

phase with ⃗rsdependent inclination and with amplitude

dependent on Is. Such mapping exactly defines a 2D

Fourier transform (FT) of the object points. Hence, we

have a digital Fourier hologram. Points positioned at

the front focal plane of the lens L0can be reconstructed

by a simple calculation of the inverse FT of H?x;y?.

Points located on other planes may be out of focus,

and can be reconstructed by applying additional Fresnel

propagation by the reconstruction distance zr. The re-

construction procedure (performed either digitally or op-

tically), depicted in Fig. 2 where Lr is a Fourier

transforming lens of focal length fr, gives

?

where F−1is the inverse FT and v is the scaling operator,

which states that v?a?f?x? ? f?ax?. Based on Eqs. (4) and

(7), the reconstruction distance is

Followingmathematical

?

b1b?

2Is· L

?2⃗rsfe

?

zs

·

f3

f2

3− 4f2

1

?

· Q

−4f1

f2

3− 4f2

1

? c:c:

?

; (4)

sis the intensity of the point source, b1

fe? f0zs∕?f0− zs?,f3?

H?x;y? ?

I?x;y; ⃗rs;zs?dxsdysdzs:(5)

?

b1b?

2Is· L

?2⃗rs

f0

?

? c:c:

?

:(6)

s?x;y;zr? ?

v

?1

λfr

?

F−1fH?x;y?g

?

? Q?1∕zr?;(7)

zr? ?4f1f2

r∕?f2

3− 4f2

1?:(8)

According to Eq. (6) and Fig. 2, the reconstruction plane

of the point source ?⃗rs;zs? f0? would contain the point

source image and its twin, located at ??2⃗rsfr∕f0;zr? 0?,

and an additional zeroth order term, located at its origin.

Thus, by recording a Fourier hologram the twin image

problem is avoided if the object is properly positioned

inside a half plane (e.g., ys> 0).

The system shown in Fig. 1 was implemented using a

Holoeye PLUTO SLM (1920 × 1080 pixels, 8 μm pixel

pitch, phase only modulation) and a PixelFly CCD

(1280 × 1024 pixels, 6.7 μm pixel pitch, monochrome).

Other parameters in the system were f0? 30 cm,

f1?17.5cm, f2?20cm, d0?13cm, and d2?12cm.

The resolution test chart (Edmund Optics Negative

NBS 1963A) was set at two different zs positions

(30 cm and 25 cm) and was illuminated using a 150 W

EKE Halogen lamp light filtered through a 650 ?

20 nm BPF. The intersection angle of the optical axes

at the SLM plane was 17°.

We recorded holograms of a single exposure, two ex-

posures, and three exposures. The two latter versions

use the phase-shifting method [2,4] to eliminate most

of the bias term and, in the three exposures version, also

the twin image. Note that the single exposure and two

exposures holograms are real-valued, whereas the three

exposures hologram is complex-valued. The resulting ho-

lograms were digitally reconstructed based on Eq. (7).

The real-valued two exposures holograms were also op-

tically reconstructed using an apparatus based on Fig. 2,

in which the holograms were displayed on a Holoeye

LC2002 SLM (800 × 600 pixels, 32 μm pixel pitch, set

to amplitude modulation through a proper configuration

of an input and output polarizers). No special processing

was applied to the holograms before displayed on the

SLM, besides resizing with accordance to the SLM reso-

lution and linear mapping of the gray levels to discrete

values from 0 to 255 (8 bits). This mapping naturally

requires the introduction of a bias term, since the two

exposure holograms contain both negative and positive

real values.

The experimental results are shown in Fig. 3. The first

hologram was recorded with the resolution test chart

positionedatzs? f0? 30 cm.

[Figs. 3(a)–3(d)] are simply the inverse FT of the holo-

gram (discrete transform, in the digital case). Figure 3(a)

demonstrates that a single exposure is sufficient for re-

taining most of the target information. Yet, the zeroth or-

der, clearly visible in the center of the image, dominates

the hologram and reduces the quality of the image. Using

two exposures [Fig. 3(b)] greatly reduces the bias term,

increases the dynamic range of the hologram, and there-

by enhances the quality of the image. A third exposure

[Fig. 3(c)] removes the twin image, but does not enhance

the quality of the image any further, at least not visibly.

Itsreconstructions

Fig. 2.

apparatus.

(Color online) Schematic of the reconstruction

3724OPTICS LETTERS / Vol. 37, No. 17 / September 1, 2012

Page 3

Finally, Fig. 3(d) is the optical counterpart of 3(b), and

demonstrates the possibility of optical reconstruction

of the resulting real-valued hologram using a simple

apparatus, as previously described.

To demonstrate the system capability of maintaining

3D information, another hologram was recorded; this

time with zs? 25 cm < f0. Part of the recorded holo-

gram (using two exposures) is shown in Fig. 3(e). A

digital reconstruction at the best plane of focus is shown

in Fig. 3(f), where zrwas calculated according to Eq. (8).

The optical reconstructions in Figs. 3(g) and 3(h) clearly

demonstrate the refocusing capability of the system,

where the image and its twin are in and out of focus, in-

terchangeably. This was simply achieved via back and

forth movement of the CCD within the z-axis.

A simplified schematic of FISCH is presented in Fig. 4.

We note that in the actual system (Fig. 1) the beams pro-

pagate twice through the same SLM, here represented by

SLM1 and SLM2. In addition, the CCD is now positioned

right after SLM2. This configuration may be considered

to be equivalent to Fig. 1 with d2? f2, so that the waves

reaching the CCD are simply a 180° rotated version of the

waves right after exiting the SLM the second time. Here

we consider the case of zs? f0. We further assume that

the recorded hologram is limited by a clear disc aperture

P?RH? of radius RH, which is determined by the overlap

area of the two interfering beams on the CCD plane, and

that the system is only aperture limited by the radius of

SLM1, R0. In this case, RHcan geometrically be shown to

equal R0, regardless of the source distance from the op-

tical axis, RS. The recorded hologram is then equal to

Eq. (6) multiplied by P?R0?. It can be reconstructed as

shown in Fig. 2 so that

h?⃗r? ? v

?1

λfr

?

F−1fI?x;y; ⃗rs;f0?P?R0?g;(9)

where ⃗r ? ?x;y?. Equation (9) results with three terms

attributed to the bias, the point source image and its twin,

which are assumed to be properly separated, once RSis

big enough. The point source image is then proportional

to

hF?⃗r? ∝ Jinc

?2πR0

λfr

?????????????????????????????????????????????????????????

?x−MTxs?2??y−MTys?2

q

?

; (10)

where Jinc?r? ? J1?r?∕r, J1?r? is the Bessel function of

the first kind andof order one, andMT? ?2⃗rsfr∕f0?∕ ⃗rs?

2fr∕f0is the transverse magnification of the presented

FISCH configuration. A comparison of Eq. (10) to

Eq. (11) of Ref [9] demonstrates that, under the above

conditions, both FISCH and FINCH possess the same

point spread function (PSF). Therefore, just like FINCH

[9], FISCH has improved resolution beyond the Rayleigh

limit when compared to conventional imaging.

In this Letter we have presented FISCH, a new method

for recording spatially incoherent digital Fourier holo-

grams. The viability of FISCH and its ability to maintain

the complete 3D information of an object were demon-

strated. We believe that the combination of FISCH inher-

ent robustness with its capability of recording single

exposure holograms under incoherent illumination holds

great potential for many possible applications of FISCH.

We thank Barak Katz for sharing his valuable knowl-

edge and experience. This work was supported by the

Israeli Ministry of Science and Technology (MOST).

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Fig. 3.

zs? 30 cm from a single exposure, two and three exposures

holograms, respectively; (d) is the optically reconstructed

equivalent of (b). (e) is a two exposures hologram (shown par-

tially to reveal details, recorded with zs? 25 cm) and (f) is its

digitally reconstructed image at the best plane of focus. (g) and

(h) are optical reconstructions of (e) at the best plane of focus

of one of the images and its twin, respectively.

(a), (b), and (c) are digitally reconstructed images for

Fig. 4.

recorder.

(Color online) Simplified schematic of a FISCH

September 1, 2012 / Vol. 37, No. 17 / OPTICS LETTERS3725