TERAHERTZ DIGITAL HOLOGRAPHY
Yan Zhang, Weihui Zhou, Xinke Wang, Ye Cui and Wenfeng Sun
Department of Physics, Capital Normal University
Xisanhuan Beilu 105
Beijing 100037, China
ABSTRACT
The terahertz (THz) technology is combined with the digital holography for THz imaging. The characteristics of the propagation
behavior of the THz pulse in the free space are investigated by using numerical simulations. The algorithm is based on the
angular spectrum theory. The spatiotemporal coupling of the THz pulse during propagation results in a significantly time
dependent beam diameter and wave front. The two dimensional dynamic evolution of the THz pulse passing through an
aperture is obtained. The diffraction is time dependent as the pulse travels through the object, which can be clearly observed
in simulations. The simulation algorithm and result have been used to reconstruct the original object with the spatiotemporal
amplitude recorded by using CCD. The implementation of the THz digital holography is presented and the corresponding
experimental result is given.
Introduction
In recent years, the terahertz (THz) radiation sandwiched between the microwave and infrared radiation has attached a lot of
attentions and interest. THz emission is typically referred as electromagnetic wave with the frequencies from 100GHz to
10THz, and the corresponding wavelengths from 3mm to 30μm. It has three major advantages: The THz wave are transmitted
through many materials that block light. The photon in the THz range has very low power which is relatively to the human.
Many important materials such as drugs and explosives have obvious spectra in the THz range. Therefore, the THz
technology is applied in information and communications technology, biology and medical sciences, non-destructive evaluation,
homeland security, quality control of food and agricultural products, global environmental monitoring, and many scopes.
THz spectroscopy and imaging technology are two very important methods which can be used in practical applications firstly.
Since many opaque materials are translucent in the THz range, the THz imaging is much useable and can give higher
resolution than microwaves. In the past decade, various THz imaging technologies have been proposed. The beam-scan
imaging technology takes the advantage of the time-domain spectroscopy and gets the image point by point. At each point,
this technology can achieve a time-domain spectrum which contains much information about the material. However, due to the
image is achieved point by point, this technology is time consuming. The THz imaging system based on the CCD camera has
also been used to study the property of the object. The THz field distribution passing through the object can be displayed on
the computer screen in real-time. Three dimensional (3D) THz wave imaging has also been demonstrated based on the
computer tomography technology. However, all of these technologies are based on the focus plane imaging; the diffraction
and dispersion of the THz in the free space have not been considered. Since the wavelength of the THz radiation is quite
longer than the visible light, the diffraction effect is more serious. Furthermore, the bandwidth is close and even wider than the
center frequency of the THz pulsed wave, the propagation of the near circle pulsed wave will bring some new phenomena
such as space-dependent pulse time delay, pulse broadening, distortion, and red-shifting of the spectrum [1-12]. Therefore, in
order to resolve the exact information about the object, the diffraction and propagation characteristics of THz radiation should
be taken into account.
Digital holography is a new kind of technology for exacting full information of the original object. In this work, the THz imaging
technology is combined with the digital holography method for getting the information of the object in real time. The basic idea
of the THz digital holography is as follows: The CCD camera is used to record time domain waveforms of the two-dimensional
distribution of the THz field after it pass through or is reflected form the object, then the reconstruction algorithm is used to
construct the field just after the object, thus the exact information about the object can be extracted form the reconstructed
time domain waveform. In the reconstruction, the frequency component of the THz pulse is drawn by using the Fourier
transform, and the Angle Spectrum theory is used to retrieve the original distribution for each component, at last the Fourier
transform is used again to get the time domain waveform. Thus the information about the object such as absorb spectrum,
index distribution can be achieved. Since the CCD records the all of the information including the phase information of the field
in time domain waveform, it can be also called digital holography.
In this presentation, the characteristics of the propagation behavior of the THz pulse in the free space which is quite important
in the reconstruction are investigated. Then the algorithm and simulation result are used in the reconstruction of the object.
The experiment setup for digital holography and the reconstruction object from the experimentally recorded data are also
presented.
Computer Simulations
For the THz radiation, a unique feature is that its spectral bandwidth is comparable to the carrier frequency, or even be well
greater than the central frequency, so the propagation theory describing the quasi-monochromatic light no longer suits for the
THz pulse. Here in order to get the evolution of the diffraction pattern when the THz pulse propagates from an incident plane to
a parallel observation plane in the free space, the propagation is dealt in the following way [13]: First, the input pulse beam is
decomposed into a continuous sum of the monochromatic waves which can be treated independently by using the Fourier
transform. Then the propagation of each monochromatic component is treated by the angular spectrum method. Lastly, the
pulse shape in observation plane is obtained via the inverse Fourier transform.
Assume that the emission and observation planes are in a same coordinate system ( x, y, z ) and the emission plane is located
at z = 0 . The input pulsed complex distribution u ( x, y,0; t ) is firstly decomposed into a continuous sum of the monochromatic
waves via the Fourier transform [14]
+∞
u ( x, y,0; ω ) =
∫ u( x, y,0; t ) exp(−i2πωt )dt.
(1)
−∞
Then each monochromatic component of the pulsed THz beam propagates from the input plane to a coaxial parallel output
plane. Since the complex wave function u ( x, y,0, ω ) on the emission plane is associated with a physically realizable wave field,
it possesses a Fourier transform in term of angular frequencies ξ and η ,
+∞ +∞
U 0 (ξ ,η ; ω ) =
∫ ∫ u( x, y,0;ω ) exp(−i2π (ξx + ηy))dxdy.
(2)
−∞ −∞
Consequently, u ( x, y,0; ω ) can also be expressed as the inverse transform of U 0 (ξ , η ; ω ) :
+∞+∞
u ( x, y,0; ω ) =
∫ ∫U
0 (ξ , η ; ω ) exp(i 2π (ξx + ηy ))dξdη .
(3)
− ∞− ∞
In the similar manner, the complex wave function on the observation plane can be written as
+∞ +∞
u ( x, y , z ; ω ) =
∫ ∫U
z (ξ ,η ; ω ) exp(i 2π (ξx
+ ηy ))dξdη .
(4)
− ∞ −∞
The relationship between the functions U 0 (ξ , η ; ω ) and U z (ξ , η ; ω ) can be obtained by solving the Helmholtz equation with
the propagated complex wave function u ( x, y,0; ω ) [14]:
U z (ξ ,η ; ω ) = U 0 (ξ ,η ; ω ) exp{i
2πz
λ
1
[1 − (λξ ) 2 − (λη ) 2 ] 2 }.
(5)
Eq. (5) describes the phase difference between two corresponding plane waves caused by the propagation between two
parallel planes,
H = exp{i
2πz
λ
1
[1 − (λξ ) 2 − (λη ) 2 ] 2 }
(6)
is the propagator and ω = 2πc / λ .
Therefore, the function of each monochromatic wave after propagation can be achieved with the following 2D inverse Fourier
transform
+∞ +∞
u ( x, y , z ; ω ) =
∫ ∫ U (ξ ,η; ω ) exp(i2π (ξx + ηy))dξdη.
(7)
z
− ∞− ∞
Finally, an inverse Fourier transform is performed on the above expression, and the spatiotemporal representation of the
transmitted pulsed beam is obtained by
H = exp{i
-20
0
20
40
-60
λ
1
[1 − (λξ ) 2 − (λη ) 2 ] 2 }
-40
-20
0
20
40
-20
0
20
40
-50
0
50
Distance from the axis(mm)
-50
0
50
Distance from the axis(mm)
-50
0
50
Distance from the axis(mm)
t=0.03ps
-40
60
60
60
(8)
-60
t=0.00ps
Distance from the axis(mm)
t=-0.03ps
-40
Distance from the axis(mm)
Distance from the axis(mm)
-60
2πz
(b)
(c)
Figure1. The evolution image of the capital letters CNU in the observation plane,
(a)-(c) corresponds to the times t = -0.03, 0.00, and 0.03ps, respectively.
-40
-20
0
20
40
60
-60
t=0.00ps
Distance from the axis(mm)
-60
t=-0.03ps
Distance from the axis(mm)
Distance from the axis(mm)
-60
-40
-20
0
20
40
60
-50
0
50
Distance from the axis(mm)
(a)
t=0.03ps
-40
-20
0
20
40
60
-50
0
50
Distance from the axis(mm)
(b)
-50
0
50
Distance from the axis(mm)
(c)
Figure 2. Reconstructed image of the capital letters CNU, (a)-(c) corresponds to
times t = -0.03, 0.00, and 0.03ps, respectively.
The computer simulation results are presented here to demonstrate the basic idea of the THz digital holography. The object
used is a sample with three capital letters “CNU”. The evolution pattern of a pulsed plane THz beam at z=50cm after passing
though the sample are shown in Fig. 1. Only three field distributions at different time are shown. It can be seen that due to the
diffraction and propagation, the evolution pattern obtained on the observation plane cannot reveal the real information of the
picture. The original object is blurred at arbitrary time. Only for the time of t=0.00s, the object is clear, however, it can still find
the original object from this picture. Therefore, it is necessary to reconstruct the original field distribution from these pictures.
By using the method mentioned above, the original complex field distribution can be completely retrieved. The reconstructed
images are show in Fig. 2 for three different times. It can be found that the original sample has been well reconstructed and
the sharp boundary can been clearly seen at different times. This simulation demonstrates the valid of the THz digital
holography well.
Experimental result
The experiment setup is schematically shown in Fig. 3. The laser pulse generated from the femtosecond laser was separated
into two parts by the beam splitter. The intensity ratio two beams, pump beam and probe beam, can be controlled by the half
wave plate located before the beam splitter. The THz wave is generated by pumping the delayed pump beam onto the THz
emitter and collimated by the parabolic mirror. The probe beam is expanded and combined with the pump beam by the ITO.
The THz detector transfers the THz intensity to the polarization of the probe beam, then polarizer P2 is used to detect the
change of the polarization and CCD is used to achieve field distribution. By adjusting the delay stage, the complex amplitude
distribution of the THz pulse at different time can be recorded by the CCD into the computer. The chopper is used for achieve
two images with and without THz illuminating on the THz detector. Thus the signal-to-noise ration of can be improved by
subtraction of these two images.
Figure 3 Schematic of the optical implementation of the THz digital holography.
Figure 4 gives a typical three dimensional THz digital hologram achieved by the system. Fig. 4(a) describes the two
dimensional spatial distribution at a fix time, there are four points marked in the image. Fig. 4(b) presents the time domain
distribution of these four points. The amplitudes at each point vary with time. By using the method described in Sect. 2, the
original object is reconstructed and one image is shown in Fig. 5. This is an intensity distribution of the polyethylene slice.
More reconstructed image will be presented in conference.
(a)
1500
Relative Amplitude
1000
500
0
-500
0
0.1
0.2
0.3
0.4
0.5
0.6
Time delay (s)
0.7
0.8
0.9
1
-11
x 10
Figure 4 Spatiotemporal l distribution of a THz hologram. (a) Two-dimensional distribution and (b) Time dependent distribution
at four points marked by “*”.
Conclusions
In summary, we have combined the THz technology with the digital holography to develop a new THz digital holography for
achieve the full information about the object. The angular spectrum theory has been used to study the spatial and temporal
properties of the THz pulse propagating in the free space. The different diffraction loss of the lower-frequency and the higherfrequency components lead to the change of the frequency composition of the pulse at different position. The characters of the
propagation of the THz pulse in the free space is investigated and used in the reconstruction of the THz digital holography.
The optical implementation for the THz digital holography is proposed. Both simulation and experimental results demonstrate
the valid of this new THz imaging method.
Figure 5 A typical reconstructed image.
Acknowledgments
This research was supported by the Beijing Science Nova Program (Grant 2004B35) and the National Natural Science
Foundation of China (Grant 10390160).
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