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Contouring of Diffused Objects by using Digital Holography
Chandra Shakher, Dalip Singh Mehta, Md. Mosarraf Hossain and Gyanendra Sheoran
Laser Applications and Holography Laboratory
Instrument Design Development Center
Indian Institute of Technology, Delhi
New Delhi – 110016, India
Email: [email protected], [email protected]
ABSTRACT
A method for contouring of diffused objects using lens-less Fourier transform digital holography has been presented. The method
proposed here adopts dual index immersion way of contour generation. Depth contour interval up to a minimum of 0.12 mm
could be achieved without making the interference phase fringes over crowded.
1. INTRODUCTION
Three dimensional shape measurements by non-contact optical methods have been extensively studied because of its
importance in automated manufacturing, quality control of components, robotics and modeling application. Several optical
surface contouring methods interferometry, light scattering, speckle photography, moiré, Talbot interferometry and holography
etc. have been developed [1-9]. Optical surface contouring method using interferometry fulfills the condition of non-contacting
and whole field measurement of surface shape with high resolution. Fizeau and Twyman-Green interferometers can be used
only for continuous mirror surfaces, on the other hand low coherence interferometry [10] and wavelength scanning interferometry
are applicable for stepped and diffusely reflecting surfaces [11]. These methods are not suitable for small and complicated
components such as MEMS and MOEMS. Therefore, there is need to develop a contouring method that can provide larger depth
measurement and higher lateral resolution. These requirements can be well satisfied by digital holography (DH).
Nowadays, with the availability of new generation of high-speed microprocessors and high-resolution opto-electronic sensors,
digital holography [12, 13] became feasible, where holograms are recorded using opto-electronic sensors and the digitized
holographic data are processed numerically to reconstruct the images using computer programs which makes the entire process
faster, reliable and almost real time. The process is fast, robust and can be easily adopted in industrial environment by using
proper instrumentation as compared to conventional holography. The process is also quite flexible as focusing can be adjusted
numerically. In addition, phase distribution can be quantitatively derived [12]. In digital holographic interferometry (DHI), the
holograms of the two states of the object, which are recorded electronically and stored in digital image processing systems in
computers, can be processed in two ways, first the two holograms can be superimposed and it’s reconstruction gives the
intensity interferogram superimposed on the image of the object [14-17], second the individual phases of two numerically
reconstructed wavefronts are calculated and can be compared to yield the desired interference phase [12].
Contour generation is the formation of image of an object on which contours of constant elevation with respect to some plane are
superimposed. Such an image and contour maps can be easily interpreted and provide full field display of three-dimensional
surface shapes. Several methods for contouring have been investigated by using conventional holography, by using the
refractive index variation, changing the wavelength, varied illumination direction and light-in-flight recording. In digital holography,
contouring by light-in-flight measurement [9] and by varying the direction of object illumination and phase shifting digital
holography [18] have been reported.
In this paper, a method of contouring for diffused objects, which exploits the dual index immersion way of contour generation, is
presented by using lensless Fourier transform digital holography.
2. BASIC THEORY
Fig. 1 show schematic of a typical experimental set-up to record lensless Fourier transform digital hologram (LFTDH). In lensless
Fourier transform holography, the cross section of interest for the object and the point reference source are kept in the same
plane and equidistant from the hologram plane. The reconstruction algorithm for lensless Fourier transform digital holography is
originally based on Fresnel reconstruction method [12]. Fig. 2 show planes of (X O , YO ), (X H , YH ) and (X I , Y I )
cartesian coordinate system as the object, hologram and image p lanes respectively. Fresnel reconstruction algorithm, which is
expressed by Huygens-Fresnel principle with Fresnel approximation [12, 19], is given by
Laser: Helium-Neon; Mi: mirrors; BS: beam splitter; MO i: microscope objectives
and Pinholes; L1: collimating lens; R: reference waves; O: object waves;
CMOS: CMOS sensor.
Fig. 1- Schematic of the setup for lens-less Fourier transform digital holography.
O(X I , YI )=
(
exp( ikd )
È Ê k ˆ 2
2
expÍ- iÁ ˜ X I + YI
ild
2
d
Ë
¯
Î
)˘˙ F
˚
ld
(
Ï
È Êk ˆ
2
2
ÌH (X H , YH )R(X H , YH )expÍ- iÁ ˜ X H + YH
2
d
Ë
¯
Î
Ó
)˘˙¸˝
(1)
˚˛
where, O (X I , YI ) is the reconstructed wave field in the image plane, which is at a distance of ‘d’ from the hologram plane,
H(XH,YH) is the hologram, R(X H , YH ) is reference wave and Fld indicates the two-dimensional Fourier transformation scaled by
a factor of 1
ld
.
YO
YH
XO
YI
XH
Actual wave
propagation
XI
Numerical wave
propagation
d
d
Object & Point
Reference
Source Plane
Hologram
Plane
Real Image
Plane
Fig. 2- Schematic diagram of the geometry used for recording and reconstruction of holograms.
In the specific geometry of lensless Fourier transform holography, the effect of the spherical phase factor associated with the
Fresnel diffraction pattern of the object is eliminated by use of a spherical reference wave R (X H , Y H ) with the same average
curvature [13]
(
)
ÈÊ k ˆ
2
2 ˘
R(X H , YH ) = (const1) expÍiÁ ˜ X H + YH ˙
Î Ë 2d ¯
˚
(2)
Substituting equation (2) into (1) results in a simpler algorithm for lensless Fourier transform digital holo graphy:
O (X I , YI ) = (const 2 )
(
exp(ikd )
È Ê k ˆ 2
2
exp Í- iÁ ˜ X I + YI
il d
2
d
Ë
¯
Î
)˘˙ F
˚
ld
{H (X H , YH )}
(3)
Fig. 3 shows the schematic arrangement for contouring the diffused objects by dual Index immersion method in lensless Fourier
transform digit al holography.
Cell Window
Emersion Cell
Illumination
beam
Object Beam

Reference
Source
Entrance aperture
of CMOS
Sensor chip
Reference
Beam
Fig. 3- Schematic of the arrangement for contouring diffused objects by dual Index immersion method in lensless Fourier
transform digital holography.
In this method, a lensless Fourier transform digital double exposure hologram (LFTDDEH) (h1 + h2 )(m, n ) is formed from two
recorded component LFTDH’s h1 (m, n ) and h 2 (m, n ) respectively corresponding to two different refractive indices n1 and n2.
Numerical reconstruction of the LFTDDEH gives an interferogram superimposed on the reconstructed image of the object.
Phase distributions corresponding to two LFTDH’s
f (k , l )= arctan
f1 (k, l ) and f 2 (k, l ) are given by the following formula
Im [O (k , l )]
Re [O (k , l )]
where, Im [O (k , l )] and Re [O (k , l )] represent the imaginary and real parts of the complex reconstructed wave field. The
interference phase is determined by modulo 2 p subtractions by using the following relation
ÏÔf1 (k , l )- f 2 (k , l )
D f (k , l )= Ì
ÔÓf1 (k , l )- f 2 (k , l )+ 2 p
if f1 (k , l ) ≥ f2 (k , l )
if f1 (k , l )< f2 (k , l )
The interference phase provides quantitative information regarding the shape of an object or any irregularity within it through the
depth contour interval, Dh , which represents the change in object depth for one fringe spacing and is expressed as [20]
Dh =
l
[(1 + cos b 1 )n1 - (1 + cos b 2 )n 2 ]
(4)
È
˘
where, b = arcsin ÊÁ sin q ˆ˜
ÍÁ
1
˜˙
ÎË n1 ¯˚
and b 2 = arcsin ÈÍÊÁ sin q ˆ˜˘˙
Á
˜
ÎË n2 ¯˚
are the angles of refraction when cell contains liquid of refractive
indices n1 and n2 (n2>n1) respectively, q is the angle of incidence of the object illuminating beam.
3. EXPERIMENTAL
Fig. 1 show the schematic diagram of a lensless Fourier transform digital holography set-up. A He-Ne laser having output power
≈30 mW is used as a light source. Light from the laser is divided into two parts by using a beam splitter. In th e object beam, a
spatial filter along with a collimating lens produces collimated beam, which illuminates the object. The angle of illumination of
o
object is ≈43.5 . Another spatial filter in the reference beam produces spherical reference wave. The interference pattern
formed by the superposition of the reference beam and the light scattered from the object is recorded by a CMOS sensor,image
grabber card and stored inside a personal computer (PC). The pixel size on the CMOS sensor is 6.7 ¥ 6.7 m m and the total
number of pixels are 1280 ¥ 1022. The sensor chip dimensions are 8.6 mm ¥ 6.9 mm. Distance of the sensor chip of the sensor
is 50 cm from the plane containing the object and the reference source. An immersion cell of dimensions of 6 cm¥ 3 cm ¥ 6.5 cm
having optically flat window with flatness of the order of ~ l 6 is used in the experiment. The thickness of the window was
8.5mm. The plane of the cell window is kept normal to the mean direction of propagation of object waves reaching the sensor
chip such that, even when the refractive index is altered, the angle of incidence of the object beam onto the hologram rema ins
practically unchanged. Also, since the wavelength and the angle of incidence of the reference beam also do not change, there
happens no lateral displacement or change of scale of the reconstructed image. This results in the localization of the interfer
ence
fringes on the surface of the object.
A LFTDH is recorded by immersing the object (a nearly spherical steel ball of 2 cm diameter) in an immersion cell containing
distilled water. Several snaps were registered with water-ethanol solutions with varying ethanol concentration in water.
Refractive indices of water and water-ethanol solutions are measured by using an Abbe-Refractometer (Abbe ‘60’ refractometer)
made by Bellingham & Stanely Ltd. London. All the measurements are relative to mean sodium D lines (5893 Ao) and can be
taken accurate to 0.0002 by estimation. During the experiment, temperature inside the laboratory was around (28 ± 1)oC. The
refractive indices of water for wavelength 632.8 nm were found out by using the data provided in the handbook of chemistry and
physics, CRC Press, London [21].
Image reconstruction and the relevant image processing were carried out in the MATLAB environment (version 6.5). The zero
order spot is removed from the reconstructed image by high pass filtering in the frequency domain using a Gaussian filter
transfer function. The inherent speckle noise from the reconstructed image is reduced by median filtering.
4. RESULT AND ANALYSIS
Fig. 4(a ) shows the part of a LFTDH recorded when the object is kept in air inside the immersion cell. Fig. 4(b ) shows the
reconstructed primary image from the digital hologram shown in Fig. 4(a ) after the removal of zero order and 4 × 4 median
filtering.
(a )
(b )
Fig. 4 (a) Part of a LFTDH recorded when the object is kept in air inside the immersion cell and (b) The reconstructed primary
image from the digital hologram of Fig . 4 (a) after zero order removal and 4 × 4 median filtering.
Figures 5 (a1 ) and 5 (a 2 ) show the primary interferograms reconstructed from the LFTDDEH’s corresponding to the distilled
(a1 )
(b1 )
(a2 )
(b2 )
Fig. 5 ( a1 ) and (a 2 )- The primary interferograms reconstructed from the LFTDDEH’s corresponding to the distilled waterethanol solutions of different concentrations (different refractive indices) (1.3322, 1.3332) and (1.3322, 1.3347) respectively after
zero order removal and 4 × 4 median filtering and (b1 ) and (b 2 )- The interference phases corresponding to Figures 5 (a1 ) and
5 (a2 ) respectively.
water-ethanol solutions of different concentrations (different refractive indices) (1.3322, 1.3332) and (1.3322, 1.3347)
respectively after removal of zero order and 4 × 4 median filtering. In the formation of all the LFTDDEH’s, first hologram is
recorded by using distilled water of refractive index 1.3322 and the second holograms are recorded by using the solutions of
ethyl alcohol in water having concentrations ethyl alcohol in water by volume 2.0 %, and 5.5 % respectively. Figures 5 (b1 ) and
5(b2 ) show the interference phases corresponding to Figures 5 (a1 ) and 5 (a 2 ) respectively. Figure 6 shows the Fourier
transform spectrum of the phase map of Figure 5 (b1 ).
Fig. 6- The Fourier transform spectrum of the phase map of Figure 5 (b1 ).
Figure 7 shows the one-dimensional interference phase 2 p modulo of Figure 5 (b1 ) drawn along the length of the line AB.
Fig. 7- One-dimensional interference phase modulo 2 p of the phase map of Figure 5
(b3 )drawn along the length of the line AB.
Figure 8 shows the unwrapped phase of the 2p modulo interference phase of Figure 7. Depth contour intervals D h ’s
Fig. 8 - Unwrapped phase of Fig. 7.
corresponding to the Figures 5 (b1 ) and 5(b2 )are 0.29 mm and 0.12 mm respectively. Obviously D h decreases with the
increase of difference in refractive indices. Depth contour interval up to a minimum of 0.12 mm has been achieved in the present
the present set up. The quantitative information regarding depth of an object or any irregularity within it can be extracted from the
value of Dh along with the value of fringe spacing of the interference phase maps of Figure 5.
5. CONCLUSION
A simple method of contouring by using lensless Fourier transform digital holographic interferometry is presented. In this
method, the desired interference phase maps are obtained by modulo 2p subtraction of the phases obtained from the individual
holograms recorded with different pairs of refractive indices of distilled water and ethyl alcohol solutions. This method may find
applications to measure defects of various sizes in complicated diffused objects. Because of poor resolution and limited aperture
of the opto-electronic sensors, the minimum depth contour interval obtained in this method is limited. In near future, with the
advancement in sensor technology, the method may find greater utility.
ACKNOWLEDGEMENTS:
Help provided by Mr. N. Devadasa in preparing immersion cell is gratefully acknowledged.
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