# Phase Velocity Estimation of a Microstrip Line in  a Stoichiometric Periodically Domain-InvertedLiTaO3  Modulator Using Electro-Optic Sampling Technique

Authors: Shintaro Hisatake; Akira Kaino; Tomoaki Yasuda; Tadao Nagatsuma
Journal: Advances in OptoElectronics
 (2008)

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## Abstract

We estimate the phase velocity of a modulation microwave in a
quasi-velocity-matched (QVM) electro-optic (EO) phase modulator
(QVM-EOM) using EO sampling which is accurate and the most reliable
technique for measuring voltage waveforms at an electrode. The
substrate of the measured QVM-EOM is a stoichiometric periodically
domain-invertedLiTaO3 crystal. The electric field of a standing wave
in a resonant microstrip line (width: 0.5 mm, height: 0.5 mm) is measured by
employing a CdTe crystal as an EO sensor. The wavelength of the traveling
microwave at 16.0801 GHz is determined as 3.33 mm by fitting the theoretical
curve to the measured electric field distribution. The phase velocity is
estimated as  vm=5.35×107 m/s,  though there exists
about 5% systematic error due to the perturbation by the EO sensor.
Relative dielectric constant of εr=41.5 is led as the maximum
likelihood value that derives the estimated phase velocity.


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## Body

## 1. Introduction

Wide expansion of an optical spectrum by a deep phase
modulation at a high-modulation frequency is the essential technique for many
electro-optic (EO) light controlling fields, such as ultrashort pulse
generation [1], comb
generation [2],
time-to-space mapping of an optical signal [3], and so on.Quasi-velocity-matching (QVM) with periodic domain
inversion of an LiTaO3 (LT) or LiNbO3 (LN) crystal in a traveling-wave EOM is one of the promising
techniques to widely spread the optical spectrum. Especially, periodically
domain-inverted stoichiometric LT (SLT) crystals are attractive QVM-EOMs
because of their stronger EO effect and lower coercive fields than
congruent-melt LT crystals. On the basis of the QVM technique, Morimoto et al.
spread a 3-THz-wide modulation sidebands (comb) at a 16.25-GHz modulation
frequency [4]. The QVM
technique compensates for velocity mismatching between an optical group
velocity vg and a microwave phase velocity vm in the EO crystal by periodically inverting
the sign of the EO coefficient of the crystal. In the ideal QVM-EOM, the
modulation index is almost proportional to the interaction length and as a
result, large modulation index at the high modulation frequency can be achieved
with long device.Accurate values ofvg and vm are needed in designing the QVM-EOM, because
uncertainty of these values results in degradation of the modulation
efficiency, which is achievable modulation index per unit interaction length
per unit modulation electric field strength. The group velocity of the optical
wave can be calculated with group refractive index ng(λ) of the EO crystal. Likewise, the phase
velocity of the modulation microwave is calculated as vm=c/ϵeff,
where ϵeff is the effective relative dielectric constant
which is determined by the geometry of the transmission line and the relative
dielectric constant ϵr of the substrate. The group refractive index
of SLT crystals is estimated accurately using Sellmeier equation with constants
listed in [5]. In
contrast with the accurate value of ng(λ),
very little data of ϵr for LT crystal are available, which is in the
range 38.9≤ϵr≤44.5 at microwave frequencies [6, 7]. To our knowledge, no
accurate data of ϵr for SLT crystals at microwave frequencies are
available.In this paper, we estimate the phase velocity of the
modulation microwave in SLT-based QVM-EOM on the basis of a direct observation
of an amplitude of a standing-wave in a resonant microstrip line using EO sampling
technique [8]. Because
the EO sampling is accurate and most reliable technique for measuring voltage
waveforms at an electrode, the phase velocity can be estimated accurately from
the relation of vm=λmfm,
where λm is the wavelength of the traveling microwave
and fm is the resonant frequency. In Section 2,
the principle of the QVM is summarized. Section 3 discusses influence that the
estimation error of the phase velocity gives to the modulation efficiency. In
Section 4, experimental results are presented.


## 2. Quasi-Velocity-Matching

Figures1(a) and 1(b) show a schematic of a normal traveling-wave
EOM and a QVM-EOM, respectively. To avoid unnecessary complexities, we employ a
one-dimensional analysis for our devices. For a traveling-wave EOM, if the y axis is the direction in which the optical and
modulation wave propagate in a virgin (single-domain) EO crystal, the variation
of the refractive index induced by an electric field Emsin⁡(ωmt) is obtained at y as(1)Δn(y;t0)=Δnmsin⁡(ωmt0−πL0y),where Δnm=(1/2)ne3γ33Em is the amplitude of index changes, ne is the extraordinary refractive index of the
crystal, γ33 is the EO coefficient of the crystal, and ωm=2πfm.
A half-period L0 of the domain inversion is given
by (2)L0=12fm(1/vm−1/vg), where vm is the phase velocity of the modulation wave,
and vg is the group velocity of the optical wave.
Here, we assume that the optical wave arrives at point y=0 at time t=t0.(a) Traveling-wave EOM, (b) QVM-EOM with periodic domain inversion.
(a)
Traveling-wave EOM(b)
QVM-EOMForvg>vm,
the phase retardation of optical wave at the position y is  expressed as(3)θ(y;t0)=2πλ∫0yΔn(y;t0)dy=Δϕsin⁡(ωmt0−π2L0y),where(4)Δϕ=4L0λΔnmsin⁡(π2L0y).If there is the
so-called velocity mismatching between the modulation wave and the optical
wave, the modulation index |Δϕ| becomes a periodical function of y with period 2L0.When a traveling-wave EOM has a suitable
domain-inverted half-period ofL0,
QVM occurs and accordingly a large modulation index is achieved. In such a
situation, phase retardation θ(l;t0) given to the optical wave passing through the
length of l, qL0≤l<(q+1)L0,
in a periodically domain-inverted crystal as shown in Figure 1(b) is expressed
as(5)θ(l;t0)=2πλ∫0L0Δn(y;t0)dy−2πλ∫L02L0Δn(y;t0)dy⋯+(−1)q2πλ∫qL0lΔn(y;t0)dy=ΔϕQVMcos⁡(ωmt0−Φ).The modulation indexΔϕQVM can be expressed by(6)ΔϕQVM={4L0Δnmλq2+(2q+1)sin⁡2(πl2L0)(q:even),4L0Δnmλq2+(2q+1)cos⁡2(πl2L0)⁢(q:odd).Figure2 shows modulation indices of (a) the QVM-EOM,
and (b) the typical traveling-wave EOM (velocity mismatching). The solid line
of Figure 2(a) is calculated using (6). The modulation index achieved by
QVM-EOM is almost proportional to the interaction length though it is lower
than perfect velocity-matched condition by a factor of 2/π.
The upper limit of the modulation index for non-domain-inverted EOM is Δϕ0=8L0Δnm/λ.Figure 2
Modulation indices of EOMs. (a) QVM-EOM with periodic domain inversion, and (b) typical
traveling-wave EOM (velocity mismatching).


## 3. Efficiency Analysis for the QVM-EOM

The analysis presented in the former section was ideal
case in which there is no error in the period of the domain inversion.
Uncertainties in the value of the phase velocity and group velocity degrade the
modulation efficiency through the length error of the
domain-inversion period for the QVM. In this
section, we will discuss degradation of the modulation efficiency in the
presence of length error in the period of the domain inversion.Figure3 show the refractive index changes seen from
the optical wave in periodically domain-inverted EO crystal. The half period of
the domain inversion in Figures 3(a) and 3(b) is L=L0 and L=L0+ΔL,
where ΔL/L0=10%,
respectively. The QVM occurs in the case of Figure 3(a), and as a result
the traveling optical wave of ωmt0=0 sees the positive refractive index change
throughout the interaction. The case of 10% error in the domain-inversion length is shown
in Figure 3(b). If there is an error in length of domain inversion, the
optical wave of ωmt0=0 sees not only positive but also a negative
refractive index change throughout the interaction.Refractive index change seen from the optical wave in periodically domain-inverted EO
crystals. (a) Perfectly quasi-velocity-matched condition. Half period of the
domain inversion isL=L0.
(b) Half period of the domain inversion is L=L0+ΔL,
where ΔL/L0=10%.
(a)(b)Figure4(a) shows the normalized modulation index
calculated for the interaction length of l=12L0.
The modulation efficiency almost proportionally decreases with the increase of |ΔL|.
The modulation efficiency decreases to about 50% in the case of ΔL/L0=10%.
Figure 4(b) shows the modulation index as a function of the interaction
length. The calculation is carried out for the length error in the half period
of the domain inversion of (i) ΔL/L0=0%,
(ii) ΔL/L0=10%,
and (iii) ΔL/L0=15%.
An upper limit of the modulation index exists if ΔL⁢≠0.
Because longer interaction length is essential for larger modulation index, it
is important to estimate the length of the half period of the domain inversion
in high accuracy.(a) Modulation index as a function ofΔL/L0.
The interaction length is 12L0.
(b) Modulation index as a function of the interaction length. (i) ΔL/L0=0%,
(ii) ΔL/L0=10%,
and (iii) ΔL/L0=15%.
(a)(b)Propagation of uncertainties ofvm and vg can be evaluated by(7)ΔLL0=vgvg−vmΔvmvm≈1.8×Δvmvm,ΔLL0=vmvg−vmΔvgvg≈0.8×Δvgvg.The influence on the modulation efficiency of
uncertainty ofvg is lower than half of the influence on the
modulation efficiency of uncertainty of vm.
Moreover, vg can be estimated more accurately than vm by using Sellmeier equation with constants for
SLT crystal listed in [5]. Using Sellmeier equation, ng=2.41 at λ=514.5 nm is derived. From this reliable
value of ng,
the group velocity is derived as vg=1.24×108 m/s.The phase velocity of the modulation microwave can be
calculated asvm=c/ϵeff.
The effective relative dielectric constant for microstrip line can be
calculated using Kobayashi's formula [9, 10]. Kobayashi's formula is claimed to predict dispersion
better than 0.6% in the range 1≤ϵr≤128 and 0.1≤w/h≤10,
where w and h are width and height of the microstrip line,
respectively. However, uncertainty of the estimated vm is comparatively large, because the reported
value of ϵr for LT crystal is inaccurate and ranging 38.9≤ϵr≤44.5 [7]. If we employ values of 38.9≤ϵr≤44.5,vm of the modulation microwave for 16.25 GHz can
be calculated as 5.14×107 m/s ≤vm≤5.53×107 m/s using Kobayashi's formula. In that case,
the half period of the domain inversion for QVM at 16.25 GHz is calculated as
2.7 mm ≤L0≤ 3.1 mm with the group velocity of vg=1.24×108 m/s. There is 14% of length uncertainty in the half period of
the domain inversion. From Figure 4(b), ΔL/L0=14% results in the degradation of the modulation
efficiency to about 10%.


## 4. Estimation of the Phase Velocity of the Modulation Wave Using EO Sampling Technique

Figure5 shows a schematic of our QVM-EOM. An SLT
crystal is used for an EO substrate. The modulation electrode structure is a
microstrip line. The width of the microstrip line is w=0.5 mm and the height of the substrate is h=0.5 mm. The strip line is open-terminated for
resonance.Figure 5
A schematic of
our QVM-EOM. The substrate of the QVM-EOM is the stoichiometric LiTaO3 crystal. The domain inversion is performed with the half period of L.
The modulation electrode structure is the microstrip line. The width of the
microstrip line is 0.5 mm. The height of the substrate is 0.5 mm.Figure6(a) shows an experimental setup. A pulsed
fiber laser (repetition frequency: frep=40 MHz) was used for probe pulses. The
repetition frequency of the pulsed laser and the frequency of the microwave
signal source are synchronized with each other. The frequency of the microwave
should be set to be fm=N×frep+Δf,
where N is an integer and Δf is an offset frequency. In the experiment, the
frequency of the microwave is set to be fm=16.0801 GHz, where N=402 and Δf=100 kHz.(a) Experimental setup. (b) Cross-section of the microstrip line under
the measurement.
(a)(b)Figure6(b) shows a cross-section of the microstrip
line under the measurement. A CdTe crystal is used for the EO sensor. The
aperture size of the EO sensor is 3 mm × 3 mm. The probe beam is focused on the CdTe
crystal by the object lens (×10). The spot size is about 20μm. The probe beam is reflected by a dielectric
mirror attached to the EO sensor. Polarization of the reflected beam is
modulated by the electric field of the microwave. The polarization-modulated
beam propagates along the same path of the incident beam path. Two orthogonal
polarization components of the modulated beam are differentially detected by
two photodiodes. The detected signal component of Acos⁡(2πΔft+ϕ) passes through a bandpass filter whose center
frequency is 100 kHz. The amplitude of the signal, which is proportional to the
amplitude of the electric field of the standing wave, is measured by a spectrum
analyzer.In the experiment, we scanned the probe beam in they-direction with 50μm step and measured electric field profile of
the resonant standing wave. Figure 7 shows an experimental result. The
theoretical curve of a+20Log[V(y;γ)] (solid line) is fitted to the experimental
data. Fitting parameters are offset power a and γ=α+jβ.
From the least square fitting, a,α,
and β are estimated as a=−64.7 dBm, α=−0.0057,
and β=1.888,
respectively. As a result, the wavelength of the traveling microwave is
estimated as λm=2π/β=3.33 mm. The phase velocity at the modulation
frequency of fm=16.0801 GHz is estimated as fm×λm=5.35×107 m/s. From this phase velocity, relative
dielectric constant of ϵr=41.5 is derived using Kobayashi's formula. The half
period of the domain inversion at the modulation frequency of near 16 GHz is
derived as L=2.94 mm with the phase velocity of vm=5.35×107 m/s and the group velocity of 1.24 ×109 m/s.Figure 7
Magnitude of the voltage
standing wave on an open-terminated microstrip line
atfm=16.0801 GHz.The estimated phase velocity is slower than the
real-phase velocity because of the perturbation of the EO sensor. In our case,
this systematic error is estimated by numerical analysis (moment method) to be
5% or less. The 5% error in the phase velocity corresponds to the
domain-inversion length error ofΔL/L0=9% which results in the degradation of the
modulation efficiency to about 50%.The systematic error depends mainly on (1) the
difference of the dielectric constant between the device under test (DUT) and
the EO sensor, (2) the distance between the DUT and the EO sensor, and (3) the
thickness of the EO sensor. Using low-dielectric
constant materials such as a polymer [11] as a sensor, the perturbation can be reduced. By
loading a low-dielectric material between the DUT and the sensor to keep a
distance, the perturbation can also be reduced, though the sensitivity is
reduced at the same time. The use of a thinner sensor is an adequate plan to
reduce the perturbation, however it also reduces the sensitivity. For a
specific case, the tradeoff relationship between the perturbation strength and
the sensitivity has been analyzed in the literature [12].By using SLT substrate of the EOM as the EO sensor, it
is also possible to measure the standing wave without using an external
superstrate sensor [13]. In this case, special geometry should be constructed
to transmit the probe beam between the microstrip line and the ground plane.
This type of EOS system is now under construction and the results including the
comparison with current technique will appear elsewhere.


## 5. Conclusion

We have estimated the phase velocity of the modulation
microwave of 16 GHz in the periodically domain-inverted SLT phase modulator.
The theoretical curve of the amplitude distribution of the standing wave in the
microstrip line was fitted to the experimental data measured by EO sampling
technique. From the curve fitting, the phase velocity ofvm=5.35×107 m/s was estimated though there is about 5% systematic error due to the perturbation by
the EO sensor. Relative dielectric constant of ϵr=41.5 was led as the maximum likelihood value that
derives the estimated phase velocity.



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*Source: 752847-2008-10-30.xml*