Direct observation of electro-optic modulation in a single
split-ring resonator
Dmitry Yu. Shchegolkov1, Matthew T. Reiten1, John F. O’Hara2, and Abul K. Azad1*
1
2
Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
School of Electrical and Computer Engineering, Oklahoma State University, Stillwater, OK
74078, USA
Supplementary Materials:
Theory of polarization rotation with zinc-blende type crystals.
For observation of the electro-optic nonlinearity we used [1,1,0] cut ZnTe, the crystal which has
zinc-blende lattice type and one of the strongest quadratic (Pockels type) nonlinearities [1].
U2
Eo
[0,0,1]
U1
β
Erf

α
[-1,1,0]
Fig. 1. Axes assignment in a [1,1,0] plane of a zinc-blende crystal. Erf – direction of applied
quasi-static field, Eo – direction of electric field in linearly polarized optical beam. U1 and U2 –
axes along crystal's eigen polarizations.
Applying electric field to ZnTe crystal induces birefringence, so that in [1,1,0] cut crystal the two
normal polarizations, which are linear in this case, can be characterized by an angle  between
the crystal axis [-1,1,0] and the direction of the electric field in one of them, U1, the other one,
U2, being orthogonal (Fig. 1). The angle depends on the direction of the applied in-plane quasi1
static electric field 𝐸𝑟𝑓 , which we will characterize by an angle α with the same axis [-1,1,0]. We
will be doing sampling of the induced birefringence with a linearly polarized laser beam which
electric field Eo has an angle β with [-1,1,0] crystal direction.
The dependence of normal polarizations on the direction of applied quasi-static electric field Erf
is as following [2]:
sinα
𝑐𝑜𝑠2 = √1+3cos2 .
(1)
α
The relative phase shift between two polarizations after passing through the crystal is
𝛤=
𝜔0 𝑑
𝑐
(𝑛1 − 𝑛2 ) =
𝜋𝑑
𝜆0
𝑛03 𝑟41 𝐸𝑟𝑓 √1 + 3𝑐𝑜𝑠 2 𝛼,
(2)
where d is the crystal thickness, 𝑛0 is the refractive index, 𝑟41 is the electro-optic coefficient, 𝜆0
is the probe beam's wavelength in vacuum.
In the coordinate system associated with normal polarizations (U1, U2) the incident field
amplitude can be written as 𝑬𝑂 = 𝐸0 ∙ (𝑐𝑜𝑠(𝛽 − ), 𝑠𝑖𝑛(𝛽 − )). After passing through the
crystal it will turn into 𝑬1 = 𝐸0 ∙ (𝑐𝑜𝑠(𝛽 − )𝑒 𝑖𝛤⁄2 ,
𝑠𝑖𝑛(𝛽 − )𝑒 −𝑖𝛤⁄2 ). The main linear in Γ
effect here is appearance of an orthogonal polarization, 90 degrees out of phase to the wave of
initial polarization. In order to transform it into effective polarization rotation, a quarter wave
plate oriented in accordance with Eo direction is used, so that after a quarter wave plate the
orthogonal polarization oscillates in phase with the main component and this is used for further
detection. The amplitude of the orthogonal polarization is |𝑬1⊥ | = 𝐸0 𝛤 |𝑠𝑖𝑛2(𝛽 − )|⁄2.
After Wollaston prism oriented to separate the wave of initial polarization into two beams of
equal power we get those beams amplitudes as following:
2
𝐸0 ∙ (
1
√2
±
𝛤
2√2
|𝑠𝑖𝑛2(𝛽 − )|).
Using two balanced square-law detectors, that results in voltage difference signal 𝐼 =
𝜌𝐸02 𝛤|𝑠𝑖𝑛2(𝛽 − )|, where constant 𝜌 characterizes detector sensitivity and is determined by its
internal properties.
For β=90⁰ we get
𝜋𝑑
𝐼 = ρ𝐸02 𝛤|𝑠𝑖𝑛2| = 2 𝜆 𝑛03 𝑟41 𝐸𝑟𝑓 𝐸02 𝑐𝑜𝑠𝛼,
0
(3)
where formulas (1) and (2) were used. The maximum signal corresponds to 𝛼 = 0.
In our experiment we had [1,1,0] ZnTe crystal sample cut at 45 degrees to its axes and β=135⁰,
that results in
𝐼 = 𝜌𝐸02 𝛤|𝑐𝑜𝑠2| =
𝜋𝑑
𝜆0
𝑛03 𝑟41 𝐸𝑟𝑓 𝐸02 𝑠𝑖𝑛𝛼.
(4)
Provided that the RF field in the SRR gap in our experiment had an angle α=45⁰, that gives 2√2
times lower EO signal compared to the previous case (3) and α=0.
References:
[1]
B. Clough, D. H. Hurley, P. Han, J. Liao, R. Huang, and X.-C. Zhang, Sens. Imaging 10,
55 (2009).
[2]
S. Casalbuoni, H. Schlarb, B. Schmidt, P. Schmuser, B. Steffen, and A. Winter, TESLA
Report (2005-01)
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