1. Tilt angle dependence of the modulated interference effects in Photo-elastic Modulators.
Md. Abdul Ahad Talukder1, Wilhelmus J. Geerts1
1Department of Physics, Texas State University, San Marcos, TX 78666, U.S.A.

2. The effect of the PEM tilt angle and incident polarization on intensity modulation effects observed
in Magneto-Optical Kerr setup is studied for a single axis photo-elastic. The dc, 1, and 2
components of the detector signal vary periodically as a function of PEM tilt angle. Although it is
possible to adjust the PEM tilt angle to null the 1 or 2 detector signal at small tilt angles, it is not
possible to null both of them simultaneously for light linearly polarized parallel or perpendicular to
the PEM’s modulation direction. The ac detector signals can be minimized though by adjusting the
polarization angle of the light incident on the PEM at small relative retardation. Direct observations
of the time dependence of the detector signal indicates that the effects of refraction index and
thickness variations on the intensity modulations are opposite.

3. I. INTRODUCTION
Photo-Elastic modulators (PEMs)1 are often used to measure the magneto-optical Kerr effects of
thin films and multilayers2,3. The incident or reflected beam’s state of polarization are modulated
by a standing sound wave in the optical head4,5,6 of the PEM and converted to an intensity variation
using polarizers. This allows for a determination of the Kerr rotation and ellipticity with a S/N ratio
limited by the shot noise of the light source. For single axis modulators the resonance condition is
only fulfilled for one axis resulting in a time dependent refraction index for light linearly polarized
parallel to this axis. Because of Poisson’s ratio, periodic strain variations are also expected
perpendicular to this modulation direction resulting in a weak modulation of the refraction index
for the other two axis of the optical head. In addition, the modulation of the strain parallel to the
optical axis will result in a time dependent thickness of the crystal, resulting in a time dependent of
the optical path length. When using the modulator with a coherent light source this effect causes a
time dependent interference of the laser beam in the crystal which can result in intensity variations
several orders of magnitude larger than the intensity variations caused by the MO Kerr effect of the
sample7. The interference effect can be avoided or suppressed by using incoherent light, by coating
the PEM with anti-reflection coatings, by tilting the PEM with respect to the optical axis of the
setup, or by using a special optical head design so input and output surface of the modulator are no
longer exactly parallel7,8. Polnau et al. showed that the interference effects also takes place in
double axis modulators although those modulators do not have a time varying thickness. The
modulation of the refraction index is sufficient to induce intensity variations in the odd harmonics
of the modulator frequency. They concluded this from measurements of the 1 component versus

4. the polarizer angle and from the time dependence of the detector signal9. In this paper we
investigate in more detail the PEM interference effect for a single axis modulator in particularly the
dependence on PEM tilt angle and polarizer angles are investigated and the consequences for the
MO Kerr technique are discussed.
II. EXPERIMENTAL PROCEDURE
A Melles Griot intensity stabilized HeNe laser (05 STP901) without beam shaping optics is used for
the light source (633 nm, rms of the intensity < 1%). The optical components of the setup are a
plastic quarter wave-plate, a polarizer, a photo-elastic modulator and a silicon photo-detector all
mounted on top of a vibration isolation table. A Glan-Taylor prisms (MGTYS15, Karl Labrecht)
mounted in a Newport servo motor stage that can be controlled by a computer (0.5 mdeg.) is used
for the polarizer. The orientation of the fast axis of the quarter wave plate is at approximately 45
degrees with the horizontal so the linearly polarized laser light is converted into circularly polarized
light just before the polarizer. Light reflected from the polarizer or PEM will one more time pass the
quarter wave plate and be vertically linearly polarized when heading back to the laser preventing it
from entering the laser cavity and destabilizing its intensity control. The HINDS PEM-90 is mounted
horizontally on a non-magnetic optical post that can be rotated by a computer controllable Melles
Griot micro-encoder rotation stage. This allows us to change the angle between the laser beam and
the optical axis of the modulator with a resolution of 0.21 mdeg. A PDA50 Thorlabs photodetector
that includes a pre-amplifier is used to convert the light into an electric signal which is monitored
by a Tektronix scope. The AC and DC components of the signal are measured by an HP3457
multimeter and two SR830 lock-in amplifiers.

5. III. MEASUREMENT RESULTS
Fig. 1 shows the measured time dependence of the intensity for various orientations of the PEM’s
tilt angle. Note that the intensity variations decrease with the PEM tilt angle similar to the result
reported by Oakberg8. Note that the modulation depth for vertical linearly polarized light is larger
than for horizontal linearly polarized light. This suggests that the effect of the refraction index and
thickness variations are opposite since refraction index modulations are largest for horizontal
linearly polarized light.
Fig. 1. Intensity as a function of the PEM orientation at 0.25 wavelength retardation depth for
vertically and horizontally polarized light for different tilt angles (graphs for different tilt angles
are shifted vertically).
Fig. 2 below shows the DC, 1, and 2 signal as a function of the PEM angle. The phase of the lock-
in amplifiers was adjusted at perpendicular incidence using the auto-phase button on the SR830

6. resulting in a positive signal. The presence of even harmonics of the modulation frequency in our
detector signal indicates that the optical path length variations are partly caused by the PEM’s
thickness oscillations 7,8,9. All three signals have an extreme at perpendicular incidence and are
periodic with the PEM-angle: amplitude and period decreases for larger angles. The 1 and 2
signals appear to be phase shifted by 90 degrees with respect to each other. Although an intensity
stabilized laser was used, drifts were observed in the DC signal. The different values of the average
DC component for both polarization directions are caused by a slight misalignment of the
broadband plastic quarter wave-plate which caused the light on the polarizer to be elliptically
polarized (Iv/Ih=0.8).
Fig.2: DC, 1 and 2 components as a function PEM tilt angle.
Fig. 3 shows the 1, and 2 components of the detector signal as a function of the polarizer angle
for perpendicular incidence. Both signals are periodic as a function of the polarizer angle. Note
that both the 1 and 2 zero at 51 degrees. This is off from the 45 degrees observed by Polnau et
al. for a 2 axis PEMand indicates that the thickness of the modulator varies with the time9.

7. Fig. 3: DC, 1, and 2 components of the detector signal as a function of the polarizer angle (-90
and 90 degrees are vertically polarized).
IV. DATA ANALYSIS
We used the approach of Hecht10 to derive an expression for the transmission of the PEM that
includes the interference effect.






)(2
2
2
)(2
0
)(2
)12(
2
1
1
tnd
i
ga
tnd
i
agga
k
tnd
ki
k
gagaagi
ii
t
er
e
tterttE
EE
E
T

 



[1]
Where tag (tga) is the amplitude transmission coefficient for the air-glass (glass-air) interface, rga is
the amplitude reflection coefficients for the glass-air interrface,  is the laser wavelength, t is the
time, and nd(t) the optical path length upon one pass of the laser beam through the optical head.
Since both the refraction index and the thickness of the optical head are modulated, the optical
path length is described by the product of two periodic functions:

8.      
   tbatnddndntdn
tnddndndntddtnntnd
oooo
oooooo


sin)sin()(2cos
2
1
)sin()(
2
1
sinsin)(


[2]
Where no is the refraction index of the optical head, i.e. fused silica (1.4569 at 633 nm),  is the
modulation frequency of the PEM, do is the thickness of the optical head (6.35 mm), n the
amplitude of the refraction index variations induced by the periodic strain, and d the modulation
of the thickness of the optical head. For a single axis modulator 1
2
1
 dn and can the 2nd dc
term and the 2wt term in equation (2) can be neglected. The transmitted intensity can be found
from squaring equation (1) followed by a Taylor approximation, substituting equation (2), and then
rewriting the cos(sin) and sin(sin) terms as Fourier series of Bessel functions Jn:
      2
212
4
)2cos()cos(2sin)()sin(2)cos(
2
1
io EHHtbJatbJabJa
r
r
I 







  [3]
Where HH indicate the higher harmonics terms beyond 2t. Note that although we do not see a
cos(2t) term in equation [2], the intensity shows a 2t component originating from the sin(sin)
term of the denominator of equation (1). Strictly speaking equation (3) is only valid for
perpendicular incident as equation (2) only provides the optical path length at perpendicular
incident. For non zero tilt angles, nd(t) can be approximated by replacing a and b in equations (2)
and (3) by a/cos(no) and b/cos(no), where  is the polarizer tilt angle. This approximation
ignores the effect of the tilt angle on the effective refraction index. As a>>1, all three terms will
oscillate as a function of the tilt angle. Ignoring the effect of the tilt on b and using a small angle

9. approximation for cos(/no) we find for the tilt angle dependence of the intensity ( up to the 2nd
harmonic):
       2
12
2
22
2
2
4
sin)(
2
sin2)2cos(2
2
cos
2
1
i
o
o
o
EtbJ
n
atbJbJ
n
a
r
r
I


























 



 [4]
The DC and 2t term will oscillate in phase as they both have a cosine dependence on f, and the 1
term will oscillate 90 degrees out of phase in agreement with our observations reported in Fig. 2.
V. CONCLUSIONS
Intensity variation caused by the PEM interference effect can be minimized at small PEM tilt angles
by adjusting the PEM tilt angle although not simultaneously for the 1 and 2 components. For a
0.25 wave retardation depth, the 1 and 2 signal can be minimized simultaneously by adjusting
the polarizer angle. By applying both adjustment one can avoid the large fluctuations caused by the
PEM interference effect at small PEM angles.
ACKNOWLEDGEMENT
This work was supported by a DOD grant (HBCU/MI grant W911NF-15-1-0394). MT
acknowledges financial support from the Graduate College of Texas State University.

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