INTRODUCTION
This study presents a new nonlinearity compensator for error reduction in threelongitudinalmode interferometers. The nonlinearity reduction in the threelongitudinalmode interferometer can effectively increase the nanometric displacement measurement accuracy. Laser interferometers are being widely used as instrument systems for small displacement measurement with subnanometric uncertainty. Twocolor interferometers are commercially used for increasing the range of unambiguity. But the displacement measurement accuracy can be improved by replacing the twocolor source by threelongitudinalmode laser. A subnanometer heterodyne interferometric system using threelongitudinalmode HeNe laser is presented by Yokoyama et al. (2001) and then by utilizing a proper configuration, the maximum measurable velocity is enhanced (Yokoyama et al., 2005).
The displacement accuracy is limited by many factors such as laser frequency instability, uncertainty of the refractive index determination, phase detection error and thermal instabilities (Yim et al., 2000; Kim and Kim, 2004; Olyaee and Mohammad Nejad, 2007a). The optical setup can also effectively decrease the displacement accuracy. The optical deviations can be divided into two main groups' namely imperfect alignment and nonideal polarized beams. These phenomena produce the periodic nonlinearity error that is modeled by Cosijns et al. (2002) for twomode interferometer. However, imperfect alignment of optical components and nonideal polarized light emerging from the laser head can limit the displacement measurement accuracy of the laser interferometers.
The modeling of polarization properties can be done by plane wave or matrix methods. Based on the Jones calculus, matrix methods are suited to modeling of polarization sensitive systems. In fact the Jones calculus is a powerful and simple method for modeling optical systems involving polarized modes.
In the present study, we characterized and mathematically modeled the most important optical deviations including ellipticity of polarized modes and rotation angle of polarizing beam splitter in threelongitudinalmode heterodyne interferometer by using the Jones matrices of the optical components. In addition, we improved the system performance by designing an optical setup and a signal conditioner circuit in the measurement arm. The results confirm that the periodic nonlinearities are strongly reduced.
MATERIALS AND METHODS
A threelongitudinalmode heterodyne interferometer similar to current commercial
interferometric systems consists of two arms namely base and measurement (Olyaee
and Mohammad Nejad, 2007b). Also measurement arm includes reference and target
paths. The measurement beam is split into target and reference beams by the
polarizing beam splitter. Ideally, the side and central modes are directed towards
the reference and target corner cube prisms, respectively. When a rotation angle
of polarizing beam splitter with respect to the laser head or elliptically polarized
modes appears, there will be leakage of modes.
According to Fig. 1, the ellipticity of side and central
modes are denoted by respectively.
Therefore, the electrical fields of three optical frequencies are respectively
given by:
where, x_{i} = 2πv_{i}t+φ_{0i}, i = 1, 2,
3 are the optical frequencies and φ_{0i} are the initial phases.
By Jones calculus, the Jones vector of the reference electrical field is obtained
as
where, PBS_{r} and CCP_{r} are the Jones matrices of the polarizing
beam splitter and reference corner cube prism for reference beam, respectively,
shown in Fig. 2. It is assumed that the optical axis of polarizing
beam splitter is aligned properly compared to the laser head. These matrices
can be described as:
Similar to the reference path, the Jones vector of the target electrical field
is given by:
where:

Fig. 1: 
Schematic representation of (a) the ellipticity of orthogonal
polarized modes and (b) the position of threelongitudinalmode in the gain
profile 

Fig. 2: 
The optical setup of the threelongitudinalmode laser interferometer 
where, ΔΦ = 4nπΔd/λ, n is the refractive index, λ
is the central wavelength and Δd is the optical path difference between
two paths. The total electric field can be obtained by summing the reference
and target fields.
The Jones vector of the electrical field incident on the ideal polarizer with
45° orientation is given as:

Fig. 3: 
The schematic diagram of analog signal conditioner for
base and measurement arms 
The wave intensity can be expressed as the product of the electrical field
and its complex conjugate. As a result, the photocurrent of the measurement
avalanche photodiode, ,
is
where, is
the complex conjugate of the electrical field. Substitution of Eq.
11 into Eq. 12 gives a quite complex equation. But the
optical frequencies are eliminated by the measurement avalanche photodiode (APDm).
Furthermore, in the signal conditioner circuit a Band Pass Filter (BPF) is used,
as shown in Fig. 3. DC component and high order frequencies
(e.g., v_{1}+v_{2}, 2v_{2}, v_{2}+2v_{3})
are eliminated and primary beat frequencies including v_{2}v_{1}
and v_{3}v_{2} remain. The measurement photocurrent described
in Eq. 12 is converted to voltage signal that can be simplified
as:
where, x_{L} = 2π(v_{2}v_{1})t+(φ_{02}φ_{01})
and x_{H} = 2π(v_{3}v_{2})t+(φ_{03}φ_{02})
are corresponding to the lower and higher intermode beat frequencies, respectively.
The constant factors in Eq. 13 are obtained as
The signal is led to a DoubleBalanced Mixer (DBM) and then is filtered. Therefore
the secondary beat frequency, x_{0} = x_{H} x_{L},
is extracted. The normalized measurement signal is:
On the other hand, a similar signal conditioner is used in the base arm. Hence,
the normalized base signal is given by:
For extraction of the phase nonlinearity, Eq. 17 must be
rewritten as:
where:
Also interfering of three modes on the base avalanche photodiode produces offset
phase error. This error can be obtained by substitution of zero for Δφ
in Eq. 20 as:
For characterization of rotation angle of polarizing beam splitter, we assume
that the polarization of modes is perfectly linear, δ_{εt}
= δ_{εr} = 0 therefore
and instead of that a rotation angle of polarizing beam splitter with respect
to the laser axis is considered and denoted by α. In this case the polarizing
beam splitter matrices for target and reference beams are respectively taken
as
and for the polarizer matrix,
The reference and target electrical fields are calculated by substitution of
above matrices in Eq. 4 and 7 as:
Based on the comprehensive study on the nonlinearities which has been characterized
and simulated in the Optoelectronic and Laser Laboratory of IUST, the phase
nonlinearity resulting from rotation angle of PBS with respect to the laser
head can be given by
RESULTS AND DISCUSSION
Earlier, the phase nonlinearity resulting from ellipticity of orthogonal polarized
modes and rotation angle of PBS are modeled. Figure 4 shows
the periodic nonlinearity due to the ellipticity of polarized modes in terms
of the nanometric displacement. As shown in Fig. 4, the first
order or second order nonlinearity can be produced.

Fig. 4: 
The periodic nonlinearity resulting from ellipticity of polarized
modes in terms of the nanometric displacement, 

Fig. 5: 
The periodic nonlinearity resulting from oneelliptically
polarized modes in combination with the rotation angle of polarizing beam splitter 

Fig. 6: 
The designed system for nonlinearity compensation. (a) The
optical setup and (b) signal conditioner 
If , the first order nonlinearity (Fig. 4d) and in other cases the second order nonlinearity appears. The rotation angle of polarizing beam splitter effect in combination with the
oneellipticity of polarized modes is shown in Fig. 5. This
deviation causes to increase the peaktopeak of the second order periodic nonlinearity.
Based on the nonlinearity compensation for twomode interferometer presented
by Hou and Wilkening (1992), we design an optical setup and a signal conditioner
for nonlinearity compensation, as shown in Fig. 6. In the
measurement arm, two linear polarizers oriented at +45 and 45° and two
avalanche photodiodes are used. The average signal of two photocurrents is produced
by the signal conditioner circuit. As a result, the peaktopeak nonlinearity
is considerably decreased (Fig. 7). The peaktopeak nonlinearity
of 2.64 nm is result of the two photocurrents which by nonlinearity compensation,
it is reached 26 pm.

Fig. 7: 
The periodic nonlinearity in terms of the nanometric displacement,
(a) initial nonlinearities and (b) compensated nonlinearity, α =
1.7°, 
CONCLUSIONS
In this study, the periodic nonlinearities resulting from ellipticity of orthogonal polarized modes and rotation angle of polarizing beam splitter in the threelongitudinalmode laser interferometer have been modeled. For nonlinearity modeling, the Jones matrices are used. The displacement error depends on the ellipticity and rotation angle can appear as first or second order nonlinearities. Also, an optical configuration and electronic signal conditioner are designed for nonlinearity compensation. Using this system, the periodic nonlinearity considerably decreased.