A New Algorithm for Holographic Interferometry Zhang Jianqiang Computation and Technology Center Kunming university of science and technology Kunming, 650000,China
Zhang Yaping, Wang Peng
Computation and Technology Center Kunming university of science and technology Kunming, 650000,China II. THEORY
Abstract—As there are many noise influences such as speckle noise in the reconstructed object light field, how to obtain a high
A. The Principle of Holographic Interferometry
quality wrapped phase image is one of the most important part
In holographic interferometry, CCD is used to record two
in holographic interferometry. The principle of holographic
holograms of a standard component respectively. The first is
interferometry is introduced and a new algorithm for
the hologram that the standard component is without load and
holographic interferometry is put forward. And it is also verified
the second is with load. Then the reconstruction object wave
by experiments. The experiment result shows that the new
field of the two holograms is got by means of using computer.
algorithm is not only simpler than the traditional algorithm but
Comparing with the phase differences, we can achieve the
also can obtain a higher quality wrapped phase image with less
variation of the standard component with load and without
noise. It is convenient for the subsequent phase-unwrapping
load. Then we will do some quantitative analysis on the method
work. Keywords: digital holography; holographic interferometry;
above. First, the situation that the hologram is got by without load is considered. Assuming O~( x, y) is emanated from the
phase-unwrapping
object plane, it travels to from the optical axis to the I.
INTRODUCTION
holography recorded plane. The optical field distribution of
The principle of holography has been developing very fast and applied to many fields. It has become one of the most important tools used as a result of its unique properties. Holography is widely used and the most important application
reference light is R ( x, y ) . Then the intensity distribution of the hologram is given by I1 ( x, y) =| R ( x, y) |2 + | O1 ( x, y) |2 + R * ( x, y)O1 ( x, y) + R ( x, y)O1* ( x, y)
areas are: holographic interferometry, holographic optical element and holography display. Holographic interferometry is
(1)
an important field in holography application. The foundation
Similary, when the component is with load, the object
of holographic interferometry is the comparison of wavefront.
wave field recorded by CCD is O 2 ( x, y) , then the intensity
Holography is the only method that can record and reconstruct the wavefront. The biggest disadvantage of the traditional algorithm is that the phase map has significant noise and it will
recorded by CCD is: I 2 ( x, y ) =| R ( x, y ) |2 + | O 2 ( x, y ) |2 + R * ( x, y )O 2 ( x, y) + R ( x, y)O 2* ( x, y)
cause some trouble for the next phase-unwrapping work. A new algorithm for holographic interferomentry is put forward
(2)
in this paper. The new algorithm is not only simpler than the
The reconstructed object light field can be got by Fresnel
traditional algorithm but also can obtain the higher quality
diffraction calculation after the hologram is illuminated by the
wrapped phase image with little noise. It is advantage for the
complex conjugate light of reference light. Thus the
subsequent phase-unwrapping work.
reconstructed object light field when the component is without
Attain National Natural Science Foundation of China (Grant No. 61007061) support Corresponding author: Zhang Yaping (
[email protected])
978-1-4577-0911-1/12/$31.00 ©2012 IEEE
load can be expressed as:
The reconstructed object wave field can be expressed as:
U1 = A1 ( x, y ) exp( jϕ1 ( x, y )) U = A ( x, y ) exp( jϕ ( x, y ))
exp( jkz ) jπ U1 ( xi , yi , z ) = exp[ ( xi2 + yi2 )] jλ z λz
2
(3) Similarly, the reconstructed object light field when the exp( jkz) jπ U 2 ( xi , yi , z ) = exp[ ( xi2 + yi2 )] jλ z λz
=
jπ ⎧ ⎫ ×F ⎨ R * ( x, y) I 2 ( x, y) exp[ ( x2 + y 2 )]⎬ λz ⎩ ⎭
=
A2 ( x, y ) exp( jϕ2 ( x, y )) A1 ( x, y ) exp( jϕ1 ( x, y ))
A2 ( x, y ) *exp( j (ϕ2 ( x, y ) − ϕ1 ( x, y )) A1 ( x, y )
(10)
ΔU = exp( j (ϕ2 ( x, y ) − ϕ1 ( x, y ))
(11)
Then the phase information can be expressed as:
ΔΦ′ = arctan(ΔU ) = exp( j (ϕ2 ( x, y ) − ϕ1 ( x, y )) (12)
(5)
Then the phase information of the reconstructed object light field when the component is with load can be written as:
The distribution of interfere fringes is
I′ = cos(ΔΦ′)
(13)
It can be seen that the new algorithm is not only simpler, but also can eliminate the influences brought by the amplitude
(6)
As a result, the phase differences can be got as:
ΔΦ = Φ1 − Φ2
of the reconstructed wave field. Thus, the wrapped phase map with little noise that can be got and it is convenient for the next
(7)
The interference fringes of the phase differences are:
phase-unwrapping work. It is verified by experiments in the following part. III. EXPERIMENATL RESULTS
(8)
The changes of component can be got based on computation and analysis of the value of I. The range of the phase is limited in [ −π , +π ] when using arctangent in computer programming. We will do phase-unwrapping work so as to get the real phase distribution. However, the wrapped phase map got by the algorithm contains plenty of noise and it
B. Theory of the New Algorithm
U1
thought A1 ( x, y ) ≈ A2 ( x, y ) , so “(10)”can be written as:
In digital holographic interferometry, the most important part is the information of the phase. So the phase information of the reconstructed object light field when the component is without load can be expressed as:
is hard for the next phase-unwrapping work.
U2
As the amplitude changes little after rotation, it can be
(4)
I = cos(ΔΦ )
Then, we can assume
ΔU =
component is with load can be written as:
Im[U 2 ( xi , yi , z )] Φ2 = arctan Re[U 2 ( xi , yi , z )]
2
2
Where U1 is the reconstructed object wave field of the component before rotation and U 2 is after rotation.
jπ ⎧ ⎫ ×F ⎨ R * ( x, y ) I1 ( x, y) exp[ ( x 2 + y 2 )]⎬ λz ⎩ ⎭
Im[U1 ( xi , yi , z )] Φ1 = arctan Re[U1 ( xi , yi , z )]
(9)
The experimental setup is shown as “Fig.1”. The laser becomes a parallel light after travelling through a beam expander, a filter and a collimator. The parallel light is split in two by a beam splitter. One illuminates on the circled ring component and it is called object light. The other is called reference light. The object light and reference light travel through the beam splitter again and the two light meet on the CCD plane and interfere. It is recorded by CCD and CCD is connected to the computer. First the hologram of circled ring component (shown as “Fig.2”) is recorded. When the other conditions don’t change, just do a small rotation of the
component and record the second digital hologram (shown as
In the following part, we will calculate the wrapped
“Fig.3”). The interfere fringes can be got by doing diffraction
phase map and the distribution of unwrapped phase by
computation on the two holograms. The rotation angle of the
means of the traditional algorithm and the new algorithm.
component can be calculated by means of computing the phase
They are shown as “Fig.4”.
change of the light field.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 1. Experimental steup
Figure 2. The hologram recorded before rotation
(g)
(h) Figure 4. Results
Figure 3. The hologram recorded after rotation
“Fig.4(a)” is the wrapped phase map obtained by the
traditional algorithm. It can be easily seen that it has plenty of speckle noise. “Fig.4(b)” is the result of doing 5*5 median
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interferometry for the temperature field
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“Fig.4(h)” is the unwrapped phase by discrete cosine transform(DCT) least square method. Comparing “Fig.4(e)”
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speckle noise. At the same time, our new algorithm also uses less time when doing the phase-unwrapping work. So we can get that the wrapped phase that is obtained by using the traditional algorithm changes at an extremely fast rate, has less pixels and is easy to occur undersampling. It also has plenty of speckle noise and will cause some difficulty in the following phase-unwrapping work. The wrapped phase obtained by the new algorithm, the phase changes slowly, and it has more pixels. So it is not easy to occur undersampling. Also it has less speckle noise. Comparing with the traditional algorithm, it is convenient for the next phase-unwrapping work. IV.CONCLUSION A new algorithm for holographic interferometry is put forward. And it is also verified by experiments. The experiment result shows that the new algorithm is not only simpler than the traditional algorithm but also can obtain the higher quality wrapped phase image with little noise. It is advantage for the subsequent phase-unwrapping work. REFERENCES
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