Measurement of the shape of objects by the
interferometry with two wavelengths
Pavel Pavlicek
Joint Laboratory of Optics of Palacky University and Institute of Physics
of the Academy of Sciences of the Czech Republic
Tr. 17. listopadu 50a, 772 07 Olomouc
Czech Republic
Gerd Häusler
Institute of Optics, Information and Photonics, Max Planck Research
Group, University of Erlangen-Nuremberg
Staudtstrasse 7, D-91 058 Erlangen
Germany
1 Introduction
White-light interferometry is an established method for the measurement
of geometrical shape of object with smooth or rough surface [1]. One of
the disadvantages of white-light interferometry is that the required
broadband light sources suffer from a low luminance. This shows up when
the shape of object with a weakly reflecting surface is measured or when
the measured area is large. One way to overcome this disadvantage is to
replace the broadband light source by two (or more) lasers with various
wavelengths.
If the broadband light source is replaced by two lasers with various
wavelengths, a typical beat pattern arises at the output of the
interferometer instead of white-light interferogram. The beat pattern can be
used for the determination of the position of the object's surface in a
similar way as white-light interferogram. Unlike to white-light
interferogram, the beat pattern is periodic and therefore the unambiguity
range is limited.
2 Theory
If a Michelson interferometer is illuminated by a laser with wavelength λ1,
the output intensity as the function of the object's position z is given by
W. Osten, M. Kujawinska (eds.), Fringe 2009, DOI 10.1007/978-3-642-03051-2_55,
© Springer-Verlag Berlin Heidelberg 2009
340
Application Enhanced Technologies
I1 ( z ) = I 0 [1 + V cos(2k1 z )],
(1)
where I0 is the mean intensity, V is the visibility a k1 = 2π/λ1 is wave
number. A similar expression results for a laser with wavelength λ2. If the
interferometer is illuminated by two lasers with equal intensities and
wavelengths λ1 and λ2, the output intensity as the function of z is given by
I ( z ) = 2 I 0 {1 + V cos[(k1 − k 2 )z ]cos[(k1 + k 2 )z ]},
(2)
which is the known expression describing the beat pattern. The envelope
of the beat pattern is equal to
⎛ 2π
E ( z ) = V cos[(k1 − k 2 )z ] = V cos⎜
⎝ Λ
⎞
z ⎟,
⎠
(3)
where
Λ=
λλ
2π
= 1 2
k1 − k 2
λ1 − λ 2
(4)
is synthetic wavelength. It follows from Eq. 3 that the unambiguity range
of two-wavelengths interferometry is given by ±Λ/4 [2].
The expression for the output intensity as the function of z given by
Eq. 1 is valid only for objects with smooth surface. If the surface of the
measured object is rough, the reflected light wave results from
superpositions of large numbers of scattered waves with random phases
and random amplitudes [3]. Consequently an additional phase shift ϕ1
arises in the argument of cosinus in the expression in Eq. 1
I 1 ( z ) = I 0 [1 + V cos(2k1 z + ϕ 1 )].
(5)
If we take into account phase shift ϕ2 for the light with wavelength λ2,
the envelope of the beat pattern takes the form
1
⎡ 2π
⎤
E ( z ) = V cos ⎢ z + (ϕ 1 − ϕ 2 )⎥.
2
⎣Λ
⎦
(6)
It follows from Eq. 6 that the phase shift gives rise to error ∆z, which is
equal to
∆z =
1
Λ(ϕ 1 − ϕ 2 ).
4π
(7)
Application Enhanced Technologies
341
The corresponding measurement uncertainty is expressed as the standard
deviation of error ∆z
δz =
1
Λσ ϕ ,
4π
(8)
where σϕ denotes the standard deviation of phase shift difference (ϕ1 - ϕ2).
Numerical simulations show that the standard deviation σϕ depends on
the local intensity Iobj of the speckle pattern (the subscript obj means the
intensity at the output of the interferometer with shut reference arm),
synthetic wavelength Λ, and rms surface roughness σh. If the amplitudes of
light with λ1 and λ2 are large (large intensity Iobj) and the two wavelengths
are close (large synthetic wavelength Λ), the standard devitation σϕ of
phase shift will be small [3]. Finally the numerical solution provide
following expression for measurement uncertainty
δz =
1
2
I obj
I obj
σh,
(9)
where 〈Iobj〉 denotes the mean value of Iobj.The expression in Eq. 9 has the
same form as that derived by Dresel for white-light interferometry in
quasi-monochromatic approximation [4, 5]. Thus the measurement
uncertainty is directly proportional to rms roughness and inversely
proportional to the squared root of the local intensity in speckle pattern.
3 Measurement
The schematic of experimental setup is shown in Fig. 1. Michelson
interferometer is illuminated by two laser diodes with wavelengths λ1 and
λ2. The temperature and operating current of the laser diodes are
controlled to minimize intensity fluctuations and maintain wavelength
stability.
342
Application Enhanced Technologies
Fig. 1. Schematic of the interferometer with two wavelengths
The light from the lasers is transmitted by optical fiber, which ensures
that the light beams of both wavelengths are parallel. The measured object
is shifted in the longitudinal direction as indicated by the arrow. During the
shifting, the intensity at the output of the interferometer is captured by a
CCD camera. From the recorded interferograms for each pixel of the
camera, the shape of the measured object is determined.
First, a flat stainless steel plate with rough surface is used as the
measured object. Rms roughness of the plate is σh = 0.45 µm. The
measurement is performed with wavelengths λ1 = 830 nm and λ2 = 839
nm and the results are depicted in Fig. 2.
Fig. 2. (a) Measured interferogram, (b) measured height profile
Application Enhanced Technologies
343
Figure 2(a) shows the recorded interferogram, which has the typical
form of the beat pattern. According to Eq. 4, the synthetic wavelength for
given λ1 and λ2 is Λ = 78 µm. In Fig. 2(b), the measured height profile is
plotted.
A disadvantage of the described method is that the measurement range
is limited to ±Λ/4. Therefore the synthetic wavelength must be chosen long
enough. On the other hand a wide interferogram is difficult to evaluate.
This disadvantage can be overcome by overlay measurement. Figure 3(a)
shows the measured grey-scale coded height profile of the earthball on the
1 euro cent coin. The height profile was measured with wavelengths λ1 =
830 nm and λ2 = 836 nm (Λ = 116 µm). After this measurement, the height
profile was measured with wavelengths λ1 = 830 nm and λ2 = 785 nm (Λ =
14.4 µm). The task of the second measurement is to improve the results
obtained from the first measurement. The cross section of the height
profile along the vertical line is depicted in Fig. 3(b).
Fig. 3. (a) Grey-scale coded height profile of the part of 1 euro cent coin, (b) cross section
of the height profile along the vertical line (from up to down)
344
Application Enhanced Technologies
4 Conclusions
The numerical calculations have shown that measurement uncertainty of
two-wavelength interferometry depends on the surface roughness and the
local intensity in the speckle pattern. The measurement uncertainty obeys
the same equation as the measurement uncertainty of white-light
interferometry.
Interferometry with two wavelengths can be used for the measurement
shape of objects. However the measurement range is limited by the used
synthetic wavelength. The results of the measurement can be improved by
overlay measurement when three or more wavelengths are used.
5 Acknowledgements
This research was supported financially by the projects MSM 6198959213
and 1M06002 of Ministry of Education, Youth and Sports of the Czech
Republic.
6 References
1. Dresel, T, Häusler, G, Venzke, H (1992) Three-dimensional sensing of
rough surfaces by coherence radar. Appl. Opt. 31: 919–925
2. de Groot, P, Kishner, S (1991) Synthetic wavelength stabilization for
two-color laser-diode interferometry. Appl. Opt. 30: 4026-4033
3. Fercher, A, Hu, H, Vry, U (1985) Rough surface interferometry with a
two-wavelength heterodyne speckle interferometer. Appl. Opt. 24:
2181-2188
4. Dresel, T (1991) Grundlagen und Grenzen der 3D-Datengewinnung
mit dem Kohärenzradar. Master's thesis, University ErlangenNuremberg
5. Pavlicek, P, Hybl, O (2008) White-light interferometry on rough
surfaces - measurement uncertainty caused by surface roughness.
Appl. Opt. 47: 2941-2949