IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 23, NO. 4, JULY/AUGUST 2017

8400505

Interferometric Terahertz Wavefront Analysis
Emmanuel Abraham, Takayuki Ogawa, Mathilde Brossard, and Takeshi Yasui

Abstract—Wavefront characterization of terahertz beams is useful for various applications such as terahertz spectroscopy andimaging. In this paper, we report on the aberration measurementof a terahertz beam issued from a quantum cascade laser. By using
a terahertz camera and a two-wave noncommon path interferometer, we measured the wavefront distortions. As an example, we
evaluated the Zernike coefficients giving the aberrations of spherical wavefronts induced by a converging lens. Associated with adeformable mirror, the sensor will open the route to terahertz
adaptive optics.
Index Terms—Terahertz imaging, wavefront, optical aberrations, quantum cascade laser, interferometer.

I. INTRODUCTIONAWAVEFRONT sensor is an optical beam analyzer that
determines the surface of equi-phase of an electromagnetic wave. For instance, the well-known Hartmann sensor wasinvented more than a century ago [1]. It makes it possible tolocally measure the wavefront slopes of an incoming opticalradiation by using a mask composed of a holes array placed justin front of a 2D sensor. At present, many other wavefront sensors are commercially available for visible and infrared light,
based on intensity measurement or interference fringe analysis[2]–[4]. These systems are essential since they can provide thewavefront characterization for adaptive optics and a wide variety of applications including astronomy, ophthalmology andmicroscopy. These sensors are widely used to measure opticalaberrations such as astigmatism, coma, spherical aberration, etc.
In the terahertz (THz) spectral domain, it is still challenging
to fully measure the spatial profile and wavefront of a THz beam
due to the lack of effective THz cameras. A few previous studies were oriented towards the measurement of THz wavefronts
for both continuous wave and pulsed sources. In 2008, Bitzer
et al. determined the beam profiles of THz pulses after passing
through a hyperhemispherical silicon lens [5]. They observed
an asymmetric spatio-temporal field dynamic in the focus of thelens attributed to a distortion of the incident THz wavefront.

Manuscript received September 29, 2016; revised November 10, 2016; accepted December 7, 2016. This work was supported in part by the CNRS PICS
Project no. 6507.
E. Abraham is with the Laboratoire Ondes et Matière d’Aquitaine, CNRS
UMR 5798, Bordeaux University, Bordeaux Cedex 33000, France (e-mail:emmanuel.abraham@u-bordeaux.fr).
T. Ogawa and T. Yasui are with the Graduate School of Advanced Technology and Science, Tokushima University, Tokushima 770-8501, Japan (e-mail:ogawa@femto.me.tokushima-u.ac.jp; yasui.takeshi@tokushima-u.ac.jp).
M. Brossard is with NeTHIS – New TeraHertz Imaging Systems, Mérignac
33700, France (e-mail: mathilde.brossard@nethis-thz.com).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JSTQE.2016.2638540

However, they did not measure the THz wavefront but only
the THz beam profile and their indirect and time-consuming
method used a XY-scanning system in order to determine pointby-point the beam profile. Another aperture scanning device wasproposed in 2013 to characterize the THz beam propagation ina time-domain spectrometer [6]. Associated with a Hartmannmask, it has been used to produce 2D topographical image of
the THz wavefront. In 2012, Richter et al. proposed the firstTHz wavefront measurement with a Hartmann sensor and microbolometer camera [7]. The performance of the system wasdemonstrated by characterizing the wavefront of a THz beamemitted by a quantum cascade laser. Cui et al. analyzed theTHz wavefront of a quantum cascade laser and a gas laser byusing a Hartmann mask. However, they also used an indirect
XY-scanning method to detect the radiation with a pyroelectric
sensor [8], [9]. More recently, we reported on the wavefront
measurement of THz pulses using a Hartmann sensor associated with a 2D electro-optic imaging system composed of a
ZnTe crystal and a CMOS camera. We quantitatively determined
the frequency-resolved deformations of planar and converging
spherical wavefronts using the modal Zernike reconstruction
least-squares method [10].
In this paper, we propose a simple interferometric method tomeasure the wavefront of a THz continuous wave. The measurement can be useful to control the optical alignment of a THz
imaging system for which an incident planar THz wavefront
is required in order to optimize, for instance, the focalizationof the THz beam at the sample position. Basically, the system
is based on a modified Mach Zehnder interferometer. It only
requires a sensitive THz camera which is able to record theinterferometric image generated by the superposition of two incoming THz waves. Then, using a simple fringe analysis, we canobtain the THz wavefront and its decomposition along the wellknown Zernike coefficients. As a demonstration, the method has
been applied to measure the wavefronts and optical aberrationsof convergent and divergent THz beams. Associated with deformable mirrors, the sensor will also open the route to THz
adaptive optics.

II. EXPERIMENTAL SETUP
The experimental setup of the wavefront sensor is presented
in Fig. 1. In our study, the source is a Quantum Cascade
Laser (QCL, Longwave) providing a continuous wave at 3 THz(λ = 100 µm) with an output power of 1.7 mW. As many THz
sources, the output beam is highly divergent and requires a
specific beam shaping before any further utilization for THz
imaging applications for instance. In our case, we used two

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8400505

IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 23, NO. 4, JULY/AUGUST 2017

Fig. 1. General experimental setup for the analysis of planar THz wavefront.
L1 and L2: plano-convex lens, focal distance of 50 mm. Pinhole diameter D =3.2 mm.

plano-convex lenses (L1 and L2 ) and a spatial filtering to provide a planar beam.
The type of interferometer has been selected regarding thesensitivity of the THz camera. We used a NEC microbolometer array (320 × 240 pixels, 23.5 µm pixel size) with a sensitivity of 1 nW. Therefore, to get a sufficient signal-to-noise
ratio, it is better that the initial beam passes only once througha beamsplitter. For this reason, we selected a modified Mach
Zehnder interferometer where the reference and sample beams
travel along different paths before interfering. This shearing interferometer will be able to measure small aberrations since thefringe lateral shifting must be smaller to 2π to avoid any phaseambiguity [11].
The interferometer is simply composed by a HFRZ-Si beamsplitter (reflection = 54%, transmission = 46% for the verticalpolarization) and two mirrors. The incoming THz beam is divided by the beamsplitter in order to form sample and reference
beams. Both beams are recombined spatially in the plane of apinhole whose diameter is set to D = 3.2 mm in the present
study. The angle between the two interfering beams is noted α.
The intensity distribution in the plane of the pinhole is imaged
with a magnification factor equal to one by the THz camera
equipped with a 34 mm objective lens. In the present study, interference pattern is easily obtained owing to the long coherence
length of the THz source (tens of centimeters).

Fig. 2. Optical aberrations delivered by the interferometer. (a) Interferogram.
White circle: image selection for FFT analysis. (b) Fourier domain image.
(c) Reconstructed wavefront. (d) Amplitude of the Zernike coefficients. Dottedblack lines: limits of the measurement.

III. RESULTS AND DISCUSSION
A. Optical Distortions Delivered by the Interferometer
First, before characterizing the signal THz wavefront, weneed to quantify the optical aberrations delivered by the interferometer itself, attributed to the beamsplitter and the two
metallic mirrors.
Fig. 2(a) shows the interferogram corresponding to the experimental configuration presented in Fig. 1, with α = 18◦ . Owing
to the nearly equal amplitude of the electromagnetic waves andthe sensitivity of the THz camera, the interference contrast is
excellent (estimated to 85%). First, we changed the value of theangle α in order to decide the best experimental configurations
for the THz wavefront analysis. The idea is to obtain at least 10
fringes in the image. As shown in Fig. 3, the experimental fringe
spacing i is in excellent agreement with the theoretical relation

Fig. 3. Evolution of the fringe spacing as a function of the angle α between
the two interfering waves. In the rest of our study, the angle α was fixed to 18°,
indicated by the red square experimental point.

given by i = λ/sinα. In the rest of our study, the angle α was
fixed to 18°, indicated by the red square experimental point in
the figure.
Then, the analysis of the interference image is performed
with the OpenFringe software as follows. First, we need to select the beam area in which the wavefront reconstruction will
be calculated. In our case, this circular zone corresponds to theaperture of the pinhole (D = 3.2 mm diameter), indicated by
the white dotted circle in Fig. 2(a). Then, as shown in Fig. 2(b),ABRAHAM et al.: INTERFEROMETRIC TERAHERTZ WAVEFRONT ANALYSIS

8400505

TABLE IZERNIKE POLYNOMIALSi

Aberration

Z i (ρ, θ )

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

PistonX tiltY tiltDefocus
X astigmatism
Y astigmatism
X comaY comaSpherical
X trefoil
Y trefoil
X 2nd astigmatism
Y 2nd astigmatismX 2nd coma
Y 2nd coma
2nd spherical
X tetrafoil
Y tetrafoil
X 2nd trefoil
Y 2nd trefoil1
ρ cos θρ sin θ2ρ 2 − 1ρ 2 cos 2θρ 2 sin 2θ
(3ρ 3 − 2ρ) cos θ(3ρ 3 − 2ρ) sin θ6ρ 4 − 6ρ 2 + 1
ρ 3 cos 3θρ 3 sin 3θ
(4ρ 4 − 3ρ 2 ) cos 2θ
(4ρ 4 − 3ρ 2 ) sin 2θ
(10ρ 5 − 12ρ 3 + 3ρ) cos θ(10ρ 5 − 12ρ 3 + 3ρ) sin θ20ρ 6 − 30ρ 4 + 12ρ 2 − 1
ρ 4 cos 4θρ 4 sin 4θ
(5ρ 5 − 4ρ 3 ) cos 3θ
(5ρ 5 − 4ρ 3 ) sin 3θ

we computed the magnitude of the 2D Fast Fourier Transformation (FFT) of the interferogram. This is the usual procedure to
perform a spatial frequency analysis of the interferogram. The
output of this transformation represents the interference image
in the Fourier or frequency domain where, each pixel represents
a particular frequency contained in the spatial domain image.
The magnitude of the Fourier domain image in Fig. 2(b) presents
three horizontal bright spots since the original interferogram iscomposed of vertical fringes. The central spot can be ignored
since it corresponds to the DC component of the Fourier domain
image. The rest of the image, including the two remaining complex conjugated bright spots contain all the required information
to calculate the phases shift between the sample and reference
waves at the pinhole position.
The wavefront presented in Fig. 2(c) is almost planar even
if some aberrations are visible in the edge of the beam, with amaximum wavefront distortion of about 100 µm near the beamedge. This indicates that the optical elements of the interferometer, i.e. the Si beamsplitter and both mirrors, does not distort
the wavefront of the incoming THz beam.
To quantify these aberrations delivered by the interferometer,
we decomposed the wavefront W into a linear combination of
the Zernike polynomials [12]:!
ai Zi (ρ, θ)(1)W (ρ, θ) =i

where (ρ, θ) are the normalized polar coordinates (ρ ≤ 1) andthe coefficient ai represents the amplitude of the aberration attributed to the polynomial Zi , as referenced in Table I. Due tothe non-collinear interaction between the reference and signal
beams, the X tilt term is naturally very large (a2 = 575 µm, calculated with a beam diameter of D = 3.2 mm)). For the wavefront reconstruction, the piston and XY tilt terms have been set
to zero since they did not represent any intrinsic aberrations of
the interferometer. Fig. 2(d) shows the amplitudes of the 20th

Fig. 4. Experimental setup for the analysis of spherical THz wavefront. L1and L2: lens, focal distance of 50 mm. L: lens with f ′ = 100 mm. Pinhole
diameter D = 3.2 mm.

first Zernike coefficients, corresponding to the representation of
the wavefront up to the second-order aberrations. All terms are
smaller than 3 µm, i.e. λ/33. The precision of our measurement
has been estimated to 10%, taking into account the accuracy of
the fringe spacing determination (error of ±1 pixel corresponding to a fringe spacing of (300 ± 25) µm for α = 18◦ ). We also
estimated the minimum amplitude of the Zernike coefficientsthat the system is able to calculate. Considering a minimum tiltof i = 25 D = 80 mm and the relation
a2 =

R
"
# $%
tan arcsin λi(2)

one gets a2 = 2 µm that can be generalized for any given
Zernike coefficient. Therefore, we conclude that the minimum
value of the Zernike coefficients that our system can measureis 2 µm, indicated by the two horizontal dotted black lines inFig. 2(d). This shows the method makes it possible to reveal,
at the limit of our detection limit, the following major intrinsic
aberrations delivered by the interferometer: a11 = 2.6 µm (Ytrefoil), a14 = 2.2 µm(X 2nd coma), a19 = −2.7 µm (X 2nd
trefoil) and a20 = 2.9 µm (Y 2nd trefoil). In the rest of this paper, these small aberrations will be subtracted in order to measure the signal THz wavefront without any perturbation comingfrom the interferometer itself.
B. Wavefront Analysis of a THz Spherical Beam
This section will present the wavefront analysis of THz spherical beams with different radius of curvature. The aim of thestudy consists in comparing the experimental defocus parameter to its theoretical prediction and the determination of all theother distortions.
The previous experimental setup has been modified by inserting, a plano-convex lens L with a focal distance f ′ = 100 mm
on the pathway of the signal beam (Fig. 4). By varying the
distance d between the lens and the pinhole it is possible
to change the beam from converging (d < f’) to diverging(d > f’) wavefronts. Theoretically, assuming an incoming collimated Gaussian beam, the radius of curvature R of the THz

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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 23, NO. 4, JULY/AUGUST 2017

Fig. 5. THz wavefront (color) and THz beam (grey) as a function of the
distance d between the lens L and the pinhole. The measurements are performedby the THz camera that images the pinhole plane. The distance d = 100 mm
corresponds to the theoretical position of the beam waist.

wave at the pinhole position is given by&'R(d) = (d − f ′ ) 1 +zR
d − f′(2 )(3)

where zR = πw02 /λ is the Rayleigh length, with w0 the radius of
the beam waist. This shows that the beam waist is located at thepinhole position if d = f ′ . Using the knife edge technique [13],we calculated w0 = 0.6 mm, which gives zR = 10.45 mm. Theinterferogram is obtained by the interaction of this sphericalsample beam and the reference beam. As already explained,
the intrinsic optical aberrations of the interferometer will be
removed.
Fig. 5 shows the evolution of the THz wavefront (color) as
a function of the distance d between the lens L and the pinhole
position, from d = 50 mm to d = 160 mm. As expected, we cannotice that the THz wavefront is convergent for 50 mm < d <100 mm and then divergent for 100 mm < d < 160 mm. Aroundd = 100 mm, the wavefront is nearly planar as it is also expected
since this position corresponds to the theoretical beam waist of
the optical setup. However, it is clear that this wavefront suffersfrom severe aberrations that will be discussed later. Fig. 5 also
shows the beam transverse modes (grey color) as a functionof the distance d. These images have been obtained by simply blocking the reference beam. Around d = 100 mm, whichcorresponds to the focus point, we can observe the minimum
beam diameter which has been measured at the 1/e2 maximum
amplitude to 1.1 mm.
First, we analyzed the evolution of the defocus as a functionof the distance d. As previously explained, we decomposed the
wavefronts into a linear combination of the Zernike polynomials(see Eq. (1)). The defocus coefficient a4 is then related to theradius of curvature R by the expression# D $2
ρW (ρ) = 2a4 ρ = 2
2R
2

(4)

Fig. 6. Evolution of the amplitude of the radius of curvature R as a functionof the distance d between the lens L and the pinhole.

where D is the diameter of the pinhole, taken as a reference forthe beam diameter at the position where the wavefront is calculated. The Eq. (4) simply shows that it is possible to calculate
the radius of curvature R from the values of D and a4 . The result
is presented in Fig. 6 for different values of the distance d. It is ingood agreement with theoretical values given by Eq. (3), whichshows that R(d) is proportional to the quantity (d – f’) except
near the beam waist where the radius of curvature diverges since
the wavefront becomes almost planar. For d = 50 mm for instance, the defocus coefficient is a4 = (−13.6 ± 2)mm, whichcorrespond to R = (−47 ± 7)mm. This experiment indicatesthat our wavefront reconstruction is able to properly measure
the beam defocus induced by the presence of the lens in thesample arm of the interferometer.
The final analysis concerns the ability of the interferometer to properly determine the other THz beam distortions as
a function of the distance d. This measurement is particularly
important for many applications since it is well-known that a
distorted wavefront will not allow a strong focalization of theTHz beam. Fig. 7 shows the amplitude of the Zernike coefficients a5 (X astigmatism), a6 (Y astigmatism), a7 (X coma),a10 (X trefoil), a11 (Y trefoil) and a18 (Y tetrafoil) for different
values of the distance d between the lens L and the pinhole. Outof the Rayleigh length zR = 10.45 mm, i.e. roughly 90 mm <d < 110 mm, we can notice that all the Zernike coefficients arearound 5 µm, near the detection limit of our system that has been
estimated to 2 µm. It means that out of the Rayleigh range, theconvergent or divergent beam does not suffer from measureable
aberrations. However, within the Rayleigh range, all the precedent Zernike coefficients increase up to ±25 µm, indicating thatnear the beam waist the THz beam experiences severe distortions such as astigmatism, trefoil, coma and tetrafoil. This result
is important since it opens the route to THz adaptive optics thathas never been demonstrated yet. By using a deformable mirror on the THz signal beam pathway and by measuring the THz

ABRAHAM et al.: INTERFEROMETRIC TERAHERTZ WAVEFRONT ANALYSIS

8400505[9] M. Cui et al., “Beam and phase distributions of a terahertz quantum
cascade wire laser,” Appl. Phys. Lett., vol. 102, 2013, Art. no. 111113.
[10] E. Abraham, H. Cahyadi, M. Brossard, J. Degert, E. Freysz, and T. Yasui, “Development of a wavefront sensor for THz pulses,” Opt. Express,
vol. 24, pp. 5203–5211, 2016.
[11] D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping:
Theory, Algorithms, and Software. Hoboken, NJ, USA: Wiley, 1998.
[12] W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Amer., vol. 70, pp. 998–1006, 1980.
[13] A. E. Siegman, M. W. Sasnett, and T. F. Johnston Jr., “Choice of clip
levels for beam width measurements using knife-edge techniques,” IEEE
J. Quantum Electron., vol. 27, no. 4, pp. 1098–1104, Apr. 1991.

Fig. 7. Evolution of the amplitude of the Zernike coefficients as a function of
the distance d between the lens L and the pinhole.

wavefront error after it has been corrected by the interferometer,
we can form a closed loop wavefront sensor able to fully correct
the beam distortion at the focus position, where a sample or a
detector can be placed.
IV. CONCLUSION
The paper concerns the characterization of THz wavefronts
issued from a QCL source. With a modified Mach Zehnder interferometer, we determined the optical aberrations of the THz signal beam. Especially, important distortions have been measured
near the focal spot of the THz beam, which can be problematic
for future applications in spectroscopy or imaging for whicha planar terahertz wavefront is required in order to optimize
the focalization of the terahertz beam at the sample or detector
position. Further, we believe that the interferometer could be
associated with a deformable mirror and a realtime controller toopen the route to THz adaptive optics.
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Emmanuel Abraham was born in Reims, France, 1970. He received the Ph.D.
degree in physics from Bordeaux University, France, in 1997.
In 1997–1998, he joined the National Research Laboratory of Metrology,
AIST, Tsukuba, Japan, as a Postdoctoral Research Fellow, and the FemtoSecond
Technology project to study industrial applications of femtosecond lasers, including ultrafast optical Kerr gate. He has published more than 50 peer-reviewed
journals, four patents, four book chapters, and more than 100 conference proceedings. From 1998 to 2013, he was an Assistant Professor in the Laboratoire
Ondes et Matière d’Aquitaine, Bordeaux University, where he is currently aProfessor. He was invited in Osaka University as an Invited Researcher in 2007
and the University of Tokushima as a Short-Term Invited Fellow Researcher
of Japan Society for the Promotion of Science in 2010. His research interests
include femtosecond time-resolved spectroscopy and nonlinear optics. He also
has investigated the potential of terahertz (THz) radiation for the analysis ofobjects related to cultural heritage such as paintings, sculptures, ceramics, etc.,
by using pulsed and continuous millimeter-wave sources for 2-D imaging and
3-D THz computed tomography. He developed intense THz sources and studied their applications for nonlinear spectroscopy. Thanks to the international
collaboration with Tokushima University, he also developed innovative instrumentation for THz science and technology.

Takayuki Ogawa received the Bachelor’s degree in mechanical engineeringfrom Tokushima University, Japan, in 2015, where he is currently working
toward the Master’s degree in the Graduate School of Advanced Technology
and Science. His areas of research are THz digital holography.
Mr. Ogawa is a member of the Optical Society of America and the Laser
Society of Japan.

Mathilde Brossard was born in St Etienne, France, 1992. She received the Engineering degree from the Institut d’Optique Graduate School, France, in 2015.
She started working toward the Ph.D. degree on THz wavefront measurementin Nethis Company in collaboration with Bordeaux University in 2016.

Takeshi Yasui received the Ph.D. degree in mechanical engineering from
Tokushima University, Japan, in 1997.
From 1997 to 1999, he was a Postdoctoral Research Fellow in the National
Research Laboratory of Metrology, Japan. He was with the Graduate Schoolof Engineering Science, Osaka University, Japan, from 1999 to 2010, and wasbriefly with the University of Bordeaux I in 2007 and the University of LittoralCôte d’Opale in 2010 as an Invited Professor. He is currently a Professor in
the Graduate School of Science and Technology, Tokushima University, andan Invited Professor in the Graduate School of Engineering Science, Osaka
University. His research interests include THz instrumentation and metrology,
nonlinear optical microscopy, and optical comb.
Dr. Yasui is a member of the Optical Society of America, the Japan Society of Applied Physics, the Optical Society of Japan, the Laser Society ofJapan, the Japan Society of Medical Electronics and Biological Engineering,and the Japan Society of Mechanical Engineers. He received the Award for the
Most Promising Young Scientist from the Optical Society of Japan in 1998, theSakamoto Award from the Japan Society of Medical Electronics and BiologicalEngineering in 2006, the Optics Paper Award from the Japan Society of Applied
Physics, and the Funai Award from the Japan Society of Mechanical Engineers
in 2009.