J Vis (2015) 18:409–412
DOI 10.1007/s12650-014-0249-7

SHORT PAPER

Nao Ninomiya • Yamato Kubo • Daisuke Barada • Tomohiro Kiire

Direct measurement of longitudinal velocity
by Doppler phase-shifting holography

Received: 14 October 2014 / Revised: 5 November 2014 / Accepted: 6 November 2014 / Published online: 6 December 2014
! The Visualization Society of Japan 2014

1 Introduction
According to the development of computer and imaging devices, the new measuring techniques based onflow visualization have been developed and utilized in many kinds of field. The typical examples are socalled PIV, i.e., particle image velocimetry, and PTV, i.e., particle tracking velocimetry. The principal setupof PIV utilizes one high-resolution camera and a laser light sheet, which illuminates the tracer particles
suspended in the fluid flow. By calculating the displacement of particles, the two-dimensional distribution of
the two-dimensional velocities can be obtained. Recently, by implementing several cameras and the carefulcalibration of their three-dimensional orientations, three-dimensional measurement of the three-dimensional
velocities can also be achieved by so-called Tomographic PIV.
In order to understand the details of the flow field in the micro- and nano-fluidic devices, it is necessary
to measure the three-dimensional velocity under a microscopy. Thus, the development of a new measuring
technique for the three-dimensional velocity by a single camera observation is strongly needed. One solution
is the use of holography (Schnars and Jueptner 2005), but it has been known that the resolution in the depth
direction is very poor for the commonly used in-line holography. Presently, the Doppler phase-shiftingholography (Kikuchi et al. 2010) has been used for the three-dimensional measurement of an object. Thismethod extracts the signal of a fixed frequency caused by the Doppler beat between the object light and thereference light. The frequency of the beat is determined by the velocity difference between the object lightand the reference light. This implies that the velocity of an object can be measured by the beat frequency.
This implies that the longitudinal velocity, which cannot be obtained by the conventional measurement, canbe measured together with the two-dimensional in-plane velocities that can be easily measured by PIV orPTV. If this method is applied to the particles motions, three-dimensional velocities can be measured by asingle camera.
2 Doppler phase-shifting holography
Digital holography observes the interference of the observed light scattered at the surface of an object andthe reference light. Then, the three-dimensional shape of an object can be reproduced by calculating thediffraction by the computer. But, in the commonly used in-line holography, the diffraction is contaminated
N. Ninomiya (&) ! Y. Kubo ! D. Barada ! T. Kiire
Graduate School of Advanced Interdisciplinary Science, Utsunomiya University, 7-1-2 Yoto,
Utsunomiya 321-8585, Japan
E-mail: nino@utmu.jp
Tel.: ?81-28-689-6030

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by the 0th order object light and the accuracy in the depth direction is deteriorated. In order to improve thisdrawback, several phase-shifting methods have been proposed, which can calculate only the 1st orderdiffraction and thus the longitudinal accuracy is guaranteed. Presently, Doppler phase-shifting holography ischosen because it is capable of unsteady measurement and does not need multiple cameras. Figure 1 showsthe experimental setup of a Doppler phase-shifting holography using Michelson Interferometry.
The complex intensities of the object light and of the reference light are expressed as follows:
EO ðx; y; tÞ ¼ aO ðx; yÞ expfi½/O ðx; yÞ % xO ðtÞt'gER ðx; y; tÞ ¼ aR ðx; yÞ expfi½/R ðx; yÞ % xR ðtÞt'g

ð1Þ

ð2Þ

where aO and aR represent the amplitudes, /O and /R are the phase angles and xO and xR are the angularvelocities of the object light and the reference light. If the object travels at velocity tO and the reference
mirror at tR ; the angular velocities are shifted by the Doppler effect of light as:sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2"#1 þ tO ðtÞ=c2tO ðtÞ
ð3Þ
xO ðtÞ ¼ x0
ffi x0 1 þ
1 % tO ðtÞ=cc
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2"#
1 þ tR ðtÞ=c2tR ðtÞxR ðtÞ ¼ x0
ffi x0 1 þð4Þ
1 % tR ðtÞ=cc
where x0 is the angular frequency of the light source and c is the speed of light. The Doppler effects aredoubled by the mirror and the object when they receive the light and when they reflect the light.
Consequently, the superposition intensity of the holograms detected by the image sensor is expressed asfollows:I ðx; y; tÞ ¼ jEO ðx; y; tÞ þ ER ðx; y; tÞj2

¼ a2O ðx; yÞ þ a2R þ 2aO ðx; yÞaR cos½D/ðx; yÞ % DxðtÞt'Fig. 1 Experimental setup for Doppler phase-shifting digital holography

ð5Þ

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here,D/ðx; yÞ ¼ /O ðx; y; tÞ % /R ðx; y; tÞDxðtÞ ¼ jxO ðtÞ % xR ðtÞj ffi 2x0 jtO ðtÞ % tR ðtÞj=cThe Fourier transformation of Eq. (5) in the time domain is$%F t ðI Þðx; y; xb Þ ¼ a2O ðx; yÞ þ a2R dðxb Þþ aO ðx; yÞaR A1 ðxb Þ expfi½D/ðx; yÞ þ B1 ðxb Þ'g

þ aO ðx; yÞaR A2 ðxb Þ expf%i½D/ðx; yÞ þ B2 ðxb Þ'g

ð6Þ

ð7Þ

where A1, A2, B1 and B2 are the arbitrary functions of the beat frequency xb. If the angular frequency of theobject light is higher than that of the reference light, the ?1st order diffraction is obtained in the secondterm. If the spectrum of each term can be split, the phase difference D/(x, y) between the object light and thereference light is easily obtained.
The discrete Fourier transformation of the image recorded by the image sensor of the finite size isobtained as:&' N%1&'PI ðx; y; tn Þ exp %i2ptn mb;mDFT½I ' x; y; mb;m ¼
n¼0

tn ¼ nDt; mb;m ¼mðm ¼ 0; 1; 2; ! ! ! ; N % 1ÞNDtð8Þ

If the ?1st order diffraction spectrum of the beat frequency mb only is extracted from Eq. (8), the complex
amplitude of an object can be calculated very accurately from the second term in Eq. (7).
3 Measurement of shape and velocity
Basically the Doppler phase-shifting is invented to measure the object profile accurately. But, as thefrequency of the Doppler beat is proportional to the velocity difference between the object light and thereference light, the longitudinal velocity of an object can be measured by the beat frequency. Thus, thelongitudinal velocity of a concave mirror travelling at a constant speed is directly measured by a Doppler
phase-shifting holography in this study.
Ar-ion laser of 514 nm is used as the light source, which is introduced to an object lens and then shapedby a spatial filter and finally made parallel by a collimator lens of f = 100 mm. A concave mirror of theradius of curvature is R = 30 m is traversed at tO = 100 lm/s by the voice coil motor and the reference
mirror is at a rest, tR = 0 lm/s. The interference images are recorded by a high-speed camera whoseresolution is 1024 9 1024 and its frame rate is 2000 fps with a shutter speed of 1/70,000 s. Figure 2a shows
the time series intensity fluctuation of the central pixel of the 512 images recorded by a high-speed camera
and the Fig. 2b is its spectrum. Doppler beat peaks can clearly be seen at a frequency of tb = xb/
2p = 386.7 Hz, which corresponds to tO = 99.4 lm/s by Eq. (6) with an error of 0.6 % of the given speed.

Fig. 2 a Intensity of a pixel at the image center and b Spectrum obtained from 512 images

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Fig. 3 a Measured shape of concave mirror and b Central cut by the y-axis and a curve fit

Now it has been proved that a Doppler phase-shifting holography can be used for the measurement of thelongitudinal velocity. Moreover, the Fig. 3 shows the 3D and 2D plots of the surface profile of the concave
mirror reproduced by digital holography and the rms error to the R = 30 m profile is only 3.27 nm. This is
the evidence that the present technique is capable of measuring the three-dimensional position and thelongitudinal velocity simultaneously.
4 Conclusion
Presently, a new three-dimensional velocity measuring technique based on a Doppler phase-shiftingholography has been developed. It can measure the longitudinal velocity of an object very accurately by thesingle camera observation. This method can be adapted to the particle flows and then the three-dimensional
particle velocities can be measured simultaneously. It will sure to contribute to the development of microand nano-fluidic devices.
Acknowledgments A part of this work has been supported by the Center for Optical Research and Education (CORE) at
Utsunomiya University, the Project for Bio-imaging and Sensing at Utsunomiya University and the JSPS KAKENHI Grant
Number 25420147. All supports are greatly acknowledged.

References
Kikuchi Y,Barada D,Kiire T,Yatagai T (2010) Doppler phase-shifting digital holography and its application to surface shapemeasurement. Opt Lett, 35(10)Schnars U, Jueptner W (2005) Digital holography digital hologram recording, numerical reconstruction, and related techniques.
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