This paper presents a new method for measuring the halfwave voltage
V
_{π}
of an electrooptic phase modulator based on a phasemodulated photonic link with interferometric demodulation. By using this method, the
V
_{π}
can be obtained with the RF voltage amplitude input required to achieve 1dB gain compression of link and the differential delay of a MachZehnder interferometer. We measure the
V
_{π}
of a commercial phase modulator by using the presented method and the carrier/the first sideband intensity ratio method. Furthermore, we compare the two measurements with the typical value provided by the manufacturer. The experiment shows that this novel measurement method is feasible, straightforward, and accurate.
I. INTRODUCTION
The use of photonic approaches to transmit and process microwave signals has been an interesting research area for several decades, due to its advantages of large bandwidth, low loss, and immunity to electromagnetic interference (EMI)
[1]
. Owing to the more linear conversion of input voltage to optical phase and requiring no bias controller, the LiNbO
_{3}
electrooptic phase modulator (PM) attracts a great deal of attention and has a host of applications
[2
,
3]
, such as microwave photonic links
[4]
, optoelectronic oscillators
[5
,
6]
, optical comb generators, and alloptical microwave filters
[7]
. The halfwave voltage (
V
_{π}
) is one of the most significant parameters to characterize a PM, which is the voltage required to produce π radians optical phase shift and represents the modulation efficiency of the PM. Therefore, the measurement of
V
_{π}
is important for PM manufacturers and end users.
Recently, several different methods for measuring the
V
_{π}
of a PM in the optical domain using an optical spectrum analyzer (OSA) have been proposed, such as the carrier nulling method
[8]
, sideband nulling method, Carrier/the first sideband (FSB) intensity equalization method and Carrier/FSB intensity ratio method
[9]
. Nevertheless, most of these methods require high driving power with peaktopeak voltage significantly higher than
V
_{π}
, which may change optoelectronic properties of PM and damage the device under test. And it is inconvenient to finetune the RF power manually at each frequency point to meet the critical measurement condition. Moreover, the resolution of commercial OSA is generally larger than 2 GHz, so that it is hard to measure the
V
_{π}
of PM in the lower frequency range by using these methods.
The method for measuring
V
_{π}
of PM in the electric domain based on the gain of phasemodulated link is also demonstrated
[10]
. Although this method eliminates the manual adjustment of RF power at each frequency point and does not require high driving power, lots of factors that have to be calibrated beforehand may degrade the measurement accuracy, such as the optical power of laser, optical insertion loss and responsivity of photodiode (PD). Even worse, the responsivity of PD is also frequencydependent and it is hard to calibrate as well.
In this paper, we propose a method for measuring the
V
_{π}
of a PM based on a phasemodulated MachZehnder interferometric demodulation (PMMZI) photonic link. In the measurement, the
V
_{π}
only depends on the RF voltage amplitude input required to achieve 1dB gain compression of the link (
V_{RF,in,1dB}
) and the differential delay of MachZehnder interferometer (MZI). And the
V_{RF,in,1dB}
can be automatically swept using a network analyzer (NA), eliminating the manual adjustment of RF power at every frequency point. The
V_{RF,in,1dB}
of the link can be reached at low driving power, which ensures the safe operation of the PM under test. Moreover,
V
_{π}
can be measured in the low frequency range. Additionally, this method does not depend on the frequency response of the PD.
II. PRINCIPLE
The
Fig. 1
shows a PMMZI link which comprises a laser, an external PM, an asymmetric MZI and a PD. Firstly, the signal is modulated on the phase of optical carrier by the PM. Then, the optical carrier is fed into the MZI which converts phase modulation into intensity modulation. And finally, the RF signal can be directly recovered by a PD. The photocurrent output of the PD is given as follows with the MZI set at quadrature
[11]
:
where
I_{dc}＝αP_{0}ℜ/2, α
is the total optical insertion loss of the link,
P_{0}
is the optical power output of the laser, ℜ is the PD responsivity,
τ
is the differential delay of the MZI, and
φ(t)＝πV_{RF}sin(ωt)/V_{π}
with
V_{RF}sin(ωt)
as the driving signal.
The schematic diagram of halfwave voltage measurement.
The photocurrent given in the Eq. (1) can be expanded using the Bessel function of the first kind
[12]
and it is given by
where
J_{2n1}
is the (2n1)th order Bessel function of the first kind. According to the Eq. (2), the RF power output at the fundamental frequency (
P_{RF,out}
) of the link can be derived with n＝1. And it is given by
where
Z_{L}
is the output impedance. Owing to the nonlinearity of the first order Bessel function, the
P_{RF,out}
will be compressed when the RF power input is large enough. Generally, the small signal approximation of the RF power output at the fundamental frequency (
P_{RF,out,ss}
) can be obtained using the linear approximation of the 1st order Bessel functions
J_{1}(x)＝x/2
. And it is given as follows:
With
and Eq. (4), the smallsignal RF gain of the PMMZI link is
As can be seen from Eq. (5), the PMMZI link exhibits a frequencydependent gain dictated by the differential delay of MZI. And the free spectral range (FSR) of the gain is equal to the reciprocal of the differential delay of MZI (
FSR＝1/τ
).
With the quotient (4) over (3) set to
10^{0.1}
(1 dB), Eq. (6) can be obtained. In this case, the
V_{RF}
equals the RF voltage amplitude input required to achieve 1dB gain compression (
V_{RF,in,1dB}
) of the link.
In order to find the relational expression of
V
_{π}
, we expand the first order Bessel function of the first kind to the ninthorder polynomial based on the following formula
[12]
Consequently, an eighthdegree polynomial equation with the unknown of
V
_{π}
is obtained. The relational expression of
V
_{π}
can be found by numerically solving this eighthdegree polynomial equation and is given by
As can be seen from Eq. (8), the
V
_{π}
of the PM can be calculated directly with the
V_{RF,in,1dB}
and the differential delay of the MZI. Importantly, the measurement of
V
_{π}
is independent of the responsivity of PD, the optical power as well as the optical insertion loss.
The RF voltage amplitude input required to drive the link into 1dB compression is related to
V
_{π}
by
Obviously, the smallest peaktopeak voltage swings 30% of
V
_{π}
when 
sin(ωτ/2)

＝1
. The required peaktopeak voltage input based on this method is much smaller than for the previous methods, such as carrier nulling method (
V_{RF,in,pp}≅1.53 V_{π}
)
[8]
, sideband nulling method (
V_{RF,in,pp}
≅2.439
V
_{π}
), and Carrier/FSB intensity equalization method (
V_{RF,in,pp}
≅0.914
V
_{π}
)
[9]
. Therefore, the method presented here with low RF drive voltage ensures the safe operation of the PM.
III. EXPERIMENT
The experimental setup of
V
_{π}
measurement is shown in
Fig. 1
, which is constructed with an Emcore 1772 DFB laser, a Covega LN53 PM, a DSC40SHLPD PD and an asymmetric MZI. We sweep and normalize the gain of the PMMZI link by using a network analyzer (Agilent PNAX N5242A). The FSR of the gain is measured and it is equal to 3.37 GHz, as shown in
Fig. 2
.
The normalized gain of the PMMZI link with FSR＝3.37 GHz.
The Eq. (8) shows that the
V
_{π}
can be calculated directly from the
V_{RF,in,1dB}
and the differential delay of MZI. According to
FSR＝1/τ
, the differential delay of MZI can be calculated and it is equal to 297 ps. Therefore, the
V_{RF,in,1dB}
is required to be measured. The N5242A network analyzer could automatically track the 1dB gain compression point and the RF power input required to achieve 1dB gain compression (
P_{RF,in,1dB}
). With the measured
P_{RF,in,1dB}
, the
V_{RF,in,1dB}
can be calculated by using
.
We measure the
P_{RF,in,1dB}
below 16 GHz with different optical power outputs of laser (
P_{0}
＝60.1 mW, 40.2 mW and 20.8 mW). The error bar chart of
P_{RF,in,1dB}
is shown in
Fig. 3
.
The error bar chart of P_{RF,in,1dB}, which is measured below 16 GHz with different optical power outputs of the laser (P_{0}＝60.1 mW, 40.2 mW and 20.8 mW).
As shown in
Fig. 3
, the
P_{RF,in,1dB}
of the PMMZI link is frequencydependent and does not rely on the optical power output of the laser, which agrees well with Eq. (9). The frequencydependent
P_{RF,in,1dB}
is dictated by the differential delay of MZI. Usually, with small MZI differential delay, the
P_{RF,in,1dB}
varies gently with frequency due to the large FSR. But
P_{RF,in,1dB}
can not be measured at low frequency because it is so large that it exceeds the maximum output power of the network analyzer. Whereas, with large MZI differential delay, the
P_{RF,in,1dB}
swings frequently with frequency because of small FSR. And more frequency points meet 
sin(ωτ/2)

＝1
. Therefore, the MZI with large differential delay is recommended for our
V
_{π}
measurement method.
According to the measured
P_{RF,in,1dB}
in
Fig. 3
, the
V_{RF,in,1dB}
can be obtained. Then, the
V
_{π}
of PM can be calculated with
V_{RF,in,1dB}
and the differential delay of MZI
τ
based on Eq. (8). Furthermore, we also measure the
V
_{π}
of the PM using OSA (YOKAGAWA AQ6370) based on the carrier/FSB intensity ratio method
[9]
. The
V
_{π}
measurements based on the novel method (circles) and the carrier/FSB intensity ratio method (triangles) are shown in
Fig. 4
. As can be seen, the two
V
_{π}
measurements and the typical value of
V
_{π}
provided by the manufacturer increase with frequency and vary between 4 V and 8 V below 16 GHz. Overall, the two
V
_{π}
measurements agree well with the manufacturer’s typical value. The measurement error of carrier/FSB intensity ratio method is less than
0.4 V
. However, the measurement error of the proposed method is not more than
0.2 V
at the frequency that 
sin(ωτ/2)
 is close to 1, and less than
0.4 V
at the frequency that 
sin(ωτ/2)
 is away from 1. Therefore, compared with the carrier/FSB intensity ratio method, this method exhibits high accuracy, especially when the measurement is implemented with a large MZI differential delay because more frequency points meet 
sin(ωτ/2)

＝1
.
The measured Vπ of phase modulator and the typical Vπ provided by the manufacturer.
IV. CONCLUSION
In this paper, we report a novel and simple method for measuring the
V
_{π}
of a PM based on PMMZI photonic link. The theory shows that the
V
_{π}
of a PM can be calculated directly with
V_{RF,in,1dB}
and the differential delay of MZI. In the experiment, we measure the
V
_{π}
of a PM using the proposed method and the carrier/FSB intensity ratio method, respectively. And the two
V
_{π}
measurements agree well with the typical value provided by the manufacturer.
Compared with the previously reported methods, this novel method avoids the process of manual adjustment of RF power and eliminates the dependence on responsivity of PD, optical power and optical insertion loss. Moreover, Based on this method, the
V
_{π}
of a PM can be measured in the low frequency range and with low lever RF driving amplitude, smaller than 0.5
V
_{π}
, which ensures the safe operation of the device under test. Therefore, it is an attractively alternative method to measure the
V
_{π}
of PM.
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