Abstract

This paper presents a detailed investigation of the physical mechanisms underlying the disruption of a lithium niobate electrooptic modulator by RF pulses. It is shown that short-term modulator disruption is a direct consequence of resistive heating within the metal conductor of the coplanar waveguide electrode, which leads to a thermo-optic optical phase shift in the waveguides of the modulator. Resistive heating is also shown to contribute to permanent modulator damage at higher RF power. These results indicate that short-term RF disruption, and possibly RF damage, can be mitigated through improved thermal management. They also predict that short-term photonic link disruption can be reduced, if not eliminated, by use of a phase modulated photonic link.

© 2009 OSA

1. Introduction

Photonic links are commonly utilized in electromagnetic protected communications links due to the non-conducting nature of fiber-optic cable [1,2]. In such systems, both electrooptic (EO) and optoelectronic conversion takes place within shielded compartments, which minimizes interference or damage from electromagnetic pickup. This method is commonly utilized for damage protection from lightning and high-power microwave (HPM) threats, such an electromagnetic pulse (EMP) [2].

In wireless systems employing open-air antennas, however, it is not possible to entirely prevent unwanted electromagnetic (EM) pickup. In such systems, circuit protection can be achieved by placing a voltage limiter between the antenna and the transmit/receive circuitry. However, protection against high-power, fast rise-time pulses continues to pose a challenge. Conventional diode limiters are inherently susceptible to damage from high-power inputs [35], while high-power gas discharge tubes do not protect against short pulses (<100 ns) [4,5]. Plasma limiters on the other hand will provide high-power, shorter-pulse protection [4,5], but require high-power to engage and can therefore suffer significant leakage. For such reasons, there is continued interest in developing limiters capable of delivering high-power protection and fast response.

A potential solution is to use a photonic link as an electronic limiter [6,7], as shown in Fig. 1 . Nonlinear EO conversion in the modulator and the non-conductive nature of optical fiber both function to limit the voltage transmitted from antenna to the end of the photonic link. One advantage over existing limiters is that there is no mechanism for high-voltage leakage prior to “turn-on.” Another is that extremely fast recovery should be possible due to the high-speed of EO modulation [8]. Widespread use of photonic links in wireless communications systems also offers the potential for wide-scale, infrastructure power protection, without the addition of dedicated limiter components. The key to a photonic link functioning as a limiter, however, is that its modulator must be able to survive a high-voltage input without lingering change to its operation.

 

Fig. 1 Schematic of an antenna-fed photonic link. Electromagnetic shielding and nonconductive fiber-optic cables isolate the laser and receiver electronics from high-power pickup at the antenna. Only the modulator is exposed to external electromagnetic sources.

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In a recent work, we reported the performance of a microwave photonic link employing a commercial lithium niobate (LN) EO modulator as a high-power, high-speed electronic limiter [7]. The modulator was able to survive received radio-frequency (RF) pulse powers up to 200 W (though not optimized for this purpose), and thus provide effective moderate-power protection. However, sub-millisecond transients in the modulator transmission were also observed following each pulse. In order to develop a true high-power, fast recovery EO limiter, it is therefore necessary that both permanent damage and temporary disruption of the modulator be addressed.

In this paper, we investigate the physical mechanisms underlying LN modulator disruption by RF pulses. In addition, solutions are proposed to mitigate RF disruption and damage in EO modulators. Section II begins with a theoretical description photonic link high-voltage response, which forms the basis for interpreting experimental results of pulsed-RF tests presented in Section III. Based on the findings, Section IV explores potential physical mechanisms that could lead to modulator disruption by microwave pulses. The underlying physical process is determined in Section V, through a comparison of simulation and experiment. The significance of these results for mitigating both RF disruption and RF damage of EO modulators, and thereby improving EO limiter speed and power handling, are discussed in Section VI.

2. Theoretical high-voltage response

In order to establish the physical mechanisms underlying transient phenomena in a photonic link, it is important to first consider link response from the system standpoint. This section discusses how general mechanisms such as changes in optical power or phase can lead to transient behavior. These results provide a basis for interpreting experimental data later in Section III, and will be expanded upon in terms of their underlying physical processes in Sections IV and V.

2.1 Photonic link transfer function

The modulator used in this study was an integrated-optic Mach-Zehnder (MZ) modulator, a schematic of which is shown in Fig. 2 . The optical transmission of a MZ modulator will in general vary according to the relation [10]

TMZ=Tavg2(1+ηcosφ),
where ϕ is the relative phase difference at the end of the two interferometric optical paths, Τavg is the mean transmission of the two paths (unity for a lossless modulator), and η is the interferometric visibility, which is related to the relative transmission of the two paths.

 

Fig. 2 Schematic diagram of the integrated-optic, Mach-Zehnder EO modulator used in this study. Light launched into the input optical waveguide of the LN crystal is split equally into two paths at the first y-branch, and then interfered at the second y-branch. Separate RF and bias electrodes apply an electric field to the LN crystal, which in turn modulates the optical phase of the two waveguides via the EO effect.

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The relative phase was modulated via the EO effect by applying both a RF input voltage Vin and a slowly-varying bias voltage Vbias to the modulator, such that

φ=φRF+φSV
φRF=πVinVπ,RF
φSV=πVbiasVπ,bias+φ0

Note that Vπ,RF and Vπ,bias terms used here refer to the applied voltages on each electrode required to shift ϕ by π radians via the fast, sub-nanosecond optical-phonon and electronic components of the EO response [8]. The πV/Vπ terms therefore immediately track changes in applied voltage. The slower, photo-elastic component of the EO response [11] has instead been included in the phase term ϕ0, which accounts for all delayed effects. Dissipative phase fluctuations, originating from energy being dissipated in the modulator, have also been included in the phase term ϕ0. Examples include phase variations due to RF heating, charge generation or structural modification. Due to their relative speed, all such fluctuations have been included in the slowly-varying component of the relative phase, ϕSV.

In a photonic link the MZ modulator converts an input voltage to a modulated optical intensity. At the end of the link a photodiode then converts this signal back to an output voltage Vout, according to the relation (derived from [12])

Vout(t)=(PlaserTfiberZout1+ZT/Zout)TMZ(t),
where Plaser is the input laser power, Τfiber the optical fiber transmission, the photodiode responsivity, Zout the output (load) impedance, and ZT the internal termination impedance of the photodiode. Due to the electromagnetic shielding in Fig. 1, it is assumed that only the properties of modulator, and thus ΤMZ, will vary temporally in response to an input electromagnetic wave.

2.2 Link high-voltage response

In addition to modulating the relative phase ϕRF at radio frequency, a RF pulse applied to a MZ modulator may induce lingering changes to both the optical phase and optical transmission, for example via heating, charge generation or ferroelectric domain inversion [13], which typically vary much more slowly. A complete description of RF pulse response must therefore include not only rapid variations in ϕRF, but also slower variations in ϕ0, Τavg andη.

To this end, it is useful to separate the slowly-varying and RF components of the MZ transmission, through Eqs. (1) and (2)

TMZ=Tavg2[1+η(cosφRFcosφSVsinφRFsinφSV)].

The RF input voltage may be expressed as

Vin(t)=V0(t)sinωRFt,
where V0(t) is a slowly-varying pulse envelope, which is assumed to change gradually with respect to the RF period. It follows that the RF terms in Eq. (6) may be rewritten using familiar Bessel function identities as
cosφRF=J0(πV0Vπ,RF)+2m=1J2m(πV0Vπ,RF)cos2mωRFt
sinφRF=2m=0J2m+1(πV0Vπ,RF)sin(2m+1)ωRFt,
where Jm is a Bessel function of the first kind. Note that the slowly-varying output of the link (that with frequency components much less than ωRF), will not include any of the sinusoidal terms in Eq. (8-9). Thus the slowly-varying component of the modulator transmission may be expressed as

TMZ(SV)=Tavg2[1+ηcosφSVJ0(πV0Vπ)].

This slowly-varying response is easily measured by passing the output of the photonic link through an electrical low-pass filter, as shown in Fig. 3 .

 

Fig. 3 Schematic diagram of the experimental setup. The circuit element after the photodiode represents a low-pass filter.

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For a high-voltage pulse, it is instructive to consider the following limits of Eq. (10)

TMZ(SV)Tavg2(V0>>Vπ)
TMZ(SV)=Tavg2(1+ηcosφSV)(V0=0).

Equation (11) results from the fact that the amplitude of the J0 Bessel function approaches zero for large arguments, and describes the MZ transmission during the application of a high-voltage RF pulse. Equation (12) in turn describes MZ transmission immediately following completion of the pulse. Equation (12) is of immediate concern, because it expresses how any pulse-induced modulator transients must depend on the three properties Τavg, η and ϕSV. Equation (11) provides additional insight into modulator behavior during a pulse. In particular, it shows that the slowly-varying modulator response provides a direct measure of the mean transmission Τavg of the two interferometric paths of the modulator during a high-voltage pulse, regardless of their relative phase ϕ. In other words, it directly measures changes in optical waveguide power loss induced by high-voltage. By comparing the slowly-varying modulator transmission measured during and after a pulse, it is therefore possible to determine whether changes in relative phase or changes in waveguide transmission dominate the post-pulse, transient behavior.

3. Pulsed microwave tests

3.1 Test procedure

The experimental setup for testing photonic link transient response is shown in Fig. 3. RF pulses with 2.0 GHz center frequency were generated by the combination of a function generator, RF analog signal generator, and traveling-wave tube (TWT) amplifier. The RF pulses were then fed directly via high-power coaxial cable into the RF port of an 11 GHz bandwidth, x-cut LN MZ modulator, which had electrodes and internal termination as depicted in Fig. 2. The pulse power incident on the modulator was monitored via a calibrated −30 dB directional coupler tap, RF attenuator and RF spectrum analyzer. Concurrently, a bias voltage was applied to a separate lumped-element electrode of the modulator.

The link transient response was measured by launching light from a 1550 nm wavelength laser through the fiber-packaged modulator, polarized along the LN extraordinary axis, and detecting the signal with an 18 GHz photodiode. The photodiode output signal was passed through a 100 MHz cutoff bias tee in order to remove radio frequency components. The resulting slowly-varying link response was measured on a 20 MHz bandwidth oscilloscope.

In order to rule out the possibility of transients originating in the test equipment rather than the photonic link, single-pulse tests were repeated for a number of variations of the setup in Fig. 3. In one test, a high-pass filter was placed between the RF coupler and EO modulator to rule out spurious signals originating in the amplifier. In another, only the modulator was placed inside the anechoic chamber, to rule out electromagnetic coupling between the modulator and test circuitry. In a third test, the low-pass filter was removed, and link output observed directly on the oscilloscope, to determine whether the filter contributed to the observed transient response. In each case, the measured slowly-varying response was essentially unchanged, which confirmed that the transients originated in the photonic link, and not the test equipment. After completion of all other tests, the termination resistor at the end of the modulator RF electrode was also disconnected, and single-pulse tests repeated, to assure that the resistor did not contribute to the observed transient response.

3.2 Pulse response as a function of relative phase

In order to investigate the importance of relative optical phase on the link transient response, RF pulses were applied to the modulator while its relative phase was independently varied via an applied bias. The bias voltage used in this test was a linear ramp function, shown in Fig. 4(a) . In the absence of RF pulses, this produced the near-sinusoidal variation of modulator transmission displayed as the red waveform in Fig. 4(b). Note that the data have been normalized to their peak value, Τmax. By comparing this near-sinusoidal response to theory [Eq. (12)], it was straightforward to infer the regions of the figure for which -π≤ϕSV≤0 and 0≤ϕSV≤π.

 

Fig. 4 Linear voltage ramp applied to the bias electrode (a), and resulting slowly-varying modulator transmission (b). The red curve in (b) plots the modulator response prior to RF pulsing, while the blue plots the response during an applied RF pulse burst. Each narrow spike on the blue curve coincides with a RF pulse, while the broader after-pulse following each spike is indicative of a lingering, non-instantaneous response.

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A burst of 20 μs width, 54 W peak power (cycle-averaged) RF pulses was then concurrently applied to the modulator, with the resulting transmission displayed as a blue waveform in Fig. 4(b). Each narrow spike in this curve coincides with the application of a single RF pulse, while the broader, weaker “after-pulse” following each spike represents a lingering, transient response.

One noteworthy feature of Fig. 4(b) is that the change in transmission during each after-pulse depends strongly on the relative phase of the MZ modulator. For relative phases in the range -π≤ϕSV≤0 the link output decreased following each pulse, while in the range 0≤ϕSV≤π it increased. That the polarity of the after-pulses depended strongly on the sign ofϕ, although the optical power on the photodiode during each pulse did not (due to the cosϕ dependence of ΤMZ), provides clear evidence that the after-pulses did not originate in the photodiode or thereafter. This validates the treatment of the after-pulse as a variation in modulator transmission. The observed behavior also cannot be explained by changes in waveguide transmission alone, because according to Eq. (12) variations in Τavg and/or η would produce a symmetric response with respect to ϕSV = 0. Instead, the anti-symmetric response in Fig. 4(b) indicates that each after-pulse was primarily the result of a change in ϕSV. Specifically, each after-pulse corresponded to a decrease in ϕ0. That the amplitude of the after-pulses approached zero near ϕSV≅0, where ΤMZ(SV) varies most weakly with changes in ϕSV, but most strongly with changes in Τavg or η, further indicates that the variations in waveguide transmission during each after-pulse were negligible. Repeating this experiment with a negative voltage ramp led to the same conclusions.

A second feature of Fig. 4(b) is that modulator transmission during the application of each RF pulse remained consistently close to half its maximum value, Τmax/2. Comparing this result to the predicted value of Τavg/2 in Eq. (11) indicates that any changes in optical waveguide transmission during the high-voltage pulse, if any, were very slight. The weak variation in modulator output observed from pulse to pulse could easily have been the result of fluctuations in RF pulse power, through the non-zero J0 Bessel function in Eq. (10), rather than induced waveguide loss.

Given the clear dominance of relative phase in modulator after-pulse response, and the lack of any clear evidence for induced waveguide loss during the RF pulse, it was concluded that modulator transient response could be confidently treated as a variation in the modulator relative phase ϕ.

3.3 Single pulse response

In order to study the relationship between pulse properties and observed transient response, as well as to avoid possible cumulative effects due to repetitive pulsing, a series of single-pulse RF tests were next performed. These involved applying a solitary RF pulse to the modulator and measuring link response at zero bias voltage. The resulting MZ transmission for various pulse powers and a pulse width of 1 μs is shown in Fig. 5(a) . It is clear that a single RF pulse was sufficient to induce a strong, sub-microsecond after-pulse in the modulator transmission. It is also clear that the 1 μs pulse was sufficiently narrow to produce a near-impulse response.

 

Fig. 5 Modulator transmission following the application of a single RF pulse (a), and the corresponding change in MZ relative phase, normalized to its maximum value (b). The curves overlap in (b), which indicates that the shape of each after-pulse was independent of the RF pulse power. The change in relative phase Δϕ0 was not plotted during the RF pulse (0≤t≤1μs) due to measurement uncertainty.

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The transmission data were converted to relative phase by first noting that the interferometric visibility was better than 0.99, and the bias voltage zero, so that

TMZ(SV)Tavg12(1+cosφ0)
before and after the pulse. The change in relative phase was then calculated using definitions
Δφ0(t)φ0(t)φ0(0)
|Δφ0|maxmax(|Δφ0(t)|),
where t = 0 corresponds to the time at the beginning of the pulse. Figure 5(b) plots the resulting relative phase change Δϕ0, normalized to its peak value |Δϕ0|max. Note that data could not be plotted during the RF pulse (due to uncertainty in Δϕ0), so that the resulting curves are discontinuous. From the figure it is clear that the shape of each after-pulse was independent of amplitude, at least for the range of powers tested. Figure 6(a) plots the amplitude of each after-pulse as a function of peak pulse power. This in turn demonstrates that the change in relative phase following each RF pulse was directly proportional to pulse power. Together, the two results demonstrate a linear relationship between Δϕ0(t) and RF pulse power.

 

Fig. 6 Relationship between after-pulse amplitude and (a) peak pulse power, and (b) pulse width, following the application of a single RF pulse. Solid lines correspond to linear fits to the data, and demonstrate that 0|max was linearly proportional to both pulse power and pulse width in the ranges shown.

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In a similar fashion, single-pulse tests were performed for a variety of pulse widths and a constant pulse power of 8 W. The resulting after-pulse amplitude is plotted in Fig. 6(b), which demonstrates that the relative phase change was also directly proportional to pulse width. Together, the linear dependences on pulse power and pulse width suggest a more fundamental dependence on pulse energy. For this reason, single-pulse tests were repeated using a constant RF pulse energy of 64 μJ, but pulse widths and powers in the ranges 1-64 μs and 1-64 W, respectively. In each case, there was no appreciable variation in the after-pulse amplitude.

In summary, the single-pulse tests demonstrate that the lingering relative phase change induced by a short RF impulse had the following properties: 1) the peak value was directly proportional to RF pulse energy, and 2) the temporal evolution was independent of pulse power. From the linear fits in Fig. 6, the ratio of peak relative phase change to input RF energy was estimated to be -π(9.5x101 J−1), (i.e. −3.0x102 J−1).

3.4 Two-pulse response

In order to study the cumulative nature of RF-induced transient response, a series two-pulse tests were performed. Each consisted of applying a pair of 4 μs width, 36 W peak power pulses to the modulator, while varying the inter-pulse spacing. The resulting change in MZ relative phase is depicted in Fig. 7(a) , with time t = 0 corresponding to the beginning of the second pulse. As shown, reducing the pulse spacing caused the pulse amplitude following the second RF pulse, |Δϕ0(2)|max, to progressively increase.

 

Fig. 7 Change in MZ relative phase induced by a single pair of pulses (a), and the resulting increase in after-pulse amplitude following the second pulse (b). The different curves in (a) correspond different pulse spacing, incremented from 100 to 500 μs. Time t = 0 coincides with the start of the second pulse. In (a) data is not plotted during each RF pulse. The solid line in (b) corresponds to an exponential fit, with time constant 270 μs.

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Figure 7(b) plots this value, |Δϕ0(2)|max versus the inter-pulse spacing on a semi-logarithmic scale. Also plotted is an exponential fit to the data. That the change in relative phase following the second RF pulse approached a value roughly twice that of the first pulse is significant. It indicates that the phase change induced by individual RF pulses was not only cumulative, but linear in RF power. That the time constant of the exponential fit, 270 μs, closely matched the 280 μs 1/e decay constant for a single after-pulse further supports this conclusion.

3.5 Repetitive pulse response

In order to fully establish the cumulative nature of the induced phase delay, a burst of twenty RF pulses was applied to the modulator, each with 4 μs width, 36 W peak power pulses equally spaced by 50 μs. The resulting response is shown in Fig. 8 . During the initial burst, the relative phase decreased exponentially with time, demonstrating that the modulator approached a steady-state with repeated pulsing. The time constant associated with this steady-state buildup was 270 μs, closely matching that of the subsequent relaxation. This similarity provides further confirmation that the physical mechanism underlying the RF-induced phase change had a 1/e time constant of 270 μs.

 

Fig. 8 Change in MZ relative phase induced by a pulse train of 20 identical RF pulses. The exponential decay of the relative phase envelope during pulsing indicates that the modulator approached a steady state. The time constant of the steady-state buildup, 270 μs, closely matched that of the subsequent relaxation. Data is not plotted during each pulse, leading to discontinuity in the curve.

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The magnitude of the steady-state phase change in Fig. 8 is also noteworthy. With repetitive pulsing at 36W peak power and 8% duty cycle, the relative phase change approached an asymptotic value of -π(7.7x10−2). Given that the pulse width and spacing were both small compared to the relaxation time constant, this steady-state response should have approximated a continuous-wave RF input with identical average power. Thus, the ratio of the steady-state relative phase change to time-averaged RF input power was estimated to be -π(2.6x10−2 W−1) for this modulator.

4. Potential mechanisms

Having investigated the primary characteristics of modulator pulse response, we now turn to deducing its underlying physical mechanisms. In order to do so, it is useful to consider the EO modulator cross-section in Fig. 9 . This depicts the location of the two optical waveguides, with normalized mode intensity distributions 1(x,z)|2 and 2(x,z)|2, in relation to the overlaid coplanar waveguide (CPW) RF electrode. The axes coincide with the LN crystal principal axes. Note that any changes in the modulator relative phase Δϕ(t) are directly related to changes in the local refractive index Δn(x,y,z,t) and longitudinal strain ΔSy(x,y,z), according to the relations (derived from [14,15]).

Δφ(t)=Δφ(n)(t)+Δφ(S)(t)
Δφ(n)(t)=2πλ0y0Loy0[Δneff,1(y,t)Δneff,2(y,t)]dy
Δφ(S)(t)=2πnλ0y0Loy0[ΔSy,eff,1(y,t)ΔSy,eff,2(y,t)]dy
where
Δneff,i(y,t)=|ψi(x,z)|2Δn(x,y,z,t)dxdz,
ΔSy,eff,i(y,t)=|ψi(x,z)|2ΔSy(x,y,z,t)dxdz,
λ0 is the free-space optical wavelength, c is the speed of light in vacuum, Lo is the interferometer length (depicted in Fig. 2), y0 is the position at the end of the waveguides, t’ ≡ t + n(y-y0)/c ≅ t, and

 

Fig. 9 Schematic diagram of the modulator cross-section. The axes coincide with the principal axes of the LN crystal.

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|ψi(x,z)|2dxdz=1.

Thus, light in each waveguide experiences an optical phase shift that is proportional to the weighted average of Δn(x,y,z,t) + nΔSy(x,y,z,t), across its intensity profile. In turn, the relative phase shift, Δϕ(t), is the difference in phase shifts experienced by the two optical guides.

The transient changes in relative phase observed in the previous section could result from a number of possible mechanisms. It is clear from the linear relationship between Δϕ0 and RF power that the first step in the process was linear RF absorption in the modulator. Under normal conditions, RF absorption occurs primarily in the metal conductor of the electrode [16]. However, due to the unusually strong electromagnetic fields involved in this study, it is necessary to consider that significant RF absorption may also have occurred within the dielectric. In this regard it is useful to revisit Fig. 5(b), which shows that the induced differential phase shift between the two waveguides was negligible immediately following a short pulse. This result is important, because it implies that the absorbed RF energy across the waveguides was either 1) negligible, 2) not immediately manifested in changes in refractive index, or 3) perfectly balanced between the two waveguides.

The third situation is rather unlikely, because perfectly balanced RF absorption would require a highly symmetric modulator (with respect to z = 0), but then the subsequent change in relative phase would also require that the energy dissipation be highly asymmetric. The second case could include mechanisms such as free carrier excitation in the LN crystal or ferroelectric domain inversion. Carrier excitation in the LN could lead to delayed EO modulation, since positive and negative charges excited in the crystal drift in opposite directions along the z-axis due to the spontaneous z-polarization of the crystal [17]. However, the time required for space charge to dissipate in LN is typically on the order of seconds to days [18], much longer than the observed 270 μs decay. Ferroelectric domain inversion could also lead to a transient phase shift, due to the finite time required for domain reversal [13]. However, the fact that the relative phase remained unchanged immediately following a short pulse implies a lack of induced stress in the crystal, which would be necessary to induce domain inversion.

Based on these observations, it is far more likely that the relevant RF absorption occurred away from the optical waveguides. The most probable mechanism would then be resistive loss in the electrode. In such a case, the short RF pulse would primarily heat the metal conductor, not the LN substrate, and thus avoid immediate changes to the waveguide refractive index. Subsequent thermal transfer to the waveguides would then produce a delayed refractive index shift via thermo-optic and strain-optic effects. Such behavior closely matches that observed experimentally. It was therefore hypothesized that resistive heating in the metal conductor was the driving force behind the observed RF-induced modulator transients.

5. Thermal simulation

In order to support this hypothesis, heat transfer simulations were performed using commercial finite-element software. The thermal transfer was modeled by solving the heat equation [19]

ρCpTt(κT)=Q
in 2D across the modulator cross-section, where ρ, Cp and κ represent the density, specific heat capacity and thermal conductivity of the constituent materials, respectively, and T is the absolute temperature. Although the 2D model neglected heat flux in the y-direction, it is shown in the Appendix that the impact of this approximation was minimal.

The thermal power generated per unit volume, Q, was estimated from the RF power loss along the electrode, which from the exponential decay [16]

PRF(y,t)=Pin,RF(t)e2αy
was calculated to be
dPRF(y,t)dy=2αPin,RF(t)e2αy,
where α is the field attenuation coefficient and t”≡ t’-ny/c ≅ t. By neglecting RF loss in the dielectric, and noting that the thermal conductivity of the gold electrode would allow thermal energy generated at the metal surface (due to the skin effect) to rapidly conduct to the center, Q was approximated as
Qc(y,t)=2αFcPin,RF(t)e2αywche
Qo(y,t)=α(1Fc)Pin,RF(t)e2αywohe
within the center and outer conductors, respectively. Outside the metal, Q was zero. Note that Eqs. (25) and (26) only apply along the length of the electrode, 0≤y≤Le. The fraction of total power loss occurring in the center CPW conductor was denoted as Fc.

Since the heat flux in the y-direction was negligible, and Q varied along each conductor according to

Q(y,t)Q(0,t)e2αy,
it follows that the induced temperature change also had an exp(−2αy) dependence. Precise analytical expressions relating this temperature change to changes in refractive index and longitudinal strain were not readily available in the literature, since the crystal was heated non-uniformly and partially-clamped. However, structural mechanics simulations, a description of which is beyond the scope of this paper, indicated that longitudinal strain would be too weak, and ultimately too slow to develop, to explain the experimental relative phase change Δϕ(t). This allowed the longitudinal strain term Δϕ(S) to be neglected in Eq. (16), yielding Δϕ(Δϕ(n). The change in refractive index with temperature was approximated to first-order from the expression for an unclamped crystal [20]
Δn(x,y,z,t)AΔT(x,y,z,t),
with the understanding that the value of A could differ somewhat from the known, unclamped value of 3.3x10−5 K−1 [20] due to localized transverse strain. From the exponential dependence of the temperature change along the electrode length, it follows from Eq. (27) that the refractive index change Δn would also vary as an exp(−2αy). Completing the integration of Eq. (17) along the interaction region 0≤y≤Le then leads to the expression
Δφ(t)παλ0(1e2αLe)[Δneff,1(0,t)Δneff,2(0,t)],
which relates the relative phase change to the refractive index distribution at y = 0. This simplifies analysis by reducing the problem to that of finding the refractive index distribution in a 2D plane at y = 0.

Dimensions used in the thermal simulations were based on microscope measurements of the modulator (with the modulator package opened after completion of the tests), and are listed in Table 1 . Also listed are material properties from references [2126], with those of the polymer buffer layer estimated from a range of typical values. The RF attenuation coefficient was taken from reference [27], and Fc from quasi-static EM simulations. The mode intensity profiles of the diffused waveguides were modeled as [28]

|ψi|2=(642x2πwmhm3)exp[8(zziwm)24(xhm)2]
for x≥0, where wm is the mode width, hm/2 is the distance from the substrate-buffer interface (x = 0) to the peak of the intensity profile, and z1 and z2 are the center coordinates of the two waveguide modes. The mode spacing s denotes the center-to-center spacing, s = z2 – z1.

Tables Icon

Table 1. Physical Properties Used in Thermal Model

Due to the symmetry of the electrode with respect to z = 0, the simulated temperature distribution was also symmetric in z. Thus, for waveguides positioned perfectly on-center with the CPW (i.e. symmetrically in z) the temperature distribution across the two waveguides would be identical. In order to induce a relative change in refractive index, under the thermo-optic model in Eq. (28), it would therefore necessary for the modulator to have some asymmetry. Since the CPW dimensions were clearly observed to be symmetric, a reasonable explanation would be that the CPW electrode was slightly misaligned with respect to the optical waveguides.

Figure 10 plots the simulated relative phase change resulting from a 1 μs RF pulse, for the case in which both waveguides were shifted off-center by 0.24 μm in the –z direction. As shown, the temporal variation in relative phase closely matched that measured experimentally. Furthermore, using the thermo-optic coefficient A = 3.3x10−5 K−1 [20], the peak relative phase change from simulation was -π(9.9x101 J−1), closely matching the experimental value of -π(9.5x101 J−1). Simulations also predicted a steady-state relative phase change of -π(2.4x10−2 W−1), which closely matched the experimental value -π(2.6x10−2 W−1). Trying different values of A (due to its uncertainty) produced similar results, after slight adjustment of the waveguide offset. This excellent agreement between simulation and experiment clearly demonstrates that the observed RF-induced transients were a thermo-optic response to resistive heating within the CPW conductors.

 

Fig. 10 Measured and simulated temporal evolution of the RF-induced relative phase change. Simulated results correspond to the case in which both optical waveguides were shifted off center with respect to the CPW electrodes by 0.24 μm in the –z direction.

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Figure 11 plots the sensitivity of the thermally-induced phase change to the electrode-waveguide center-to-center offset, Δzoffset≡(z2–z1)/2, determined from simulations with the properties in Table 1, and the RF pulse energy denoted as Upulse. The figure shows that the thermo-optic phase change scales linearly with Δzoffset for small offsets, which in turn implies that some reduction in short-term RF disruption may be possible through careful alignment of the RF and optical waveguides.

 

Fig. 11 Simulated dependence of the RF-induced steady-state (left axis) and transient (right axis) relative phase change on the waveguide-electrode center-to-center offset, Δzoffset.

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Figure 11 also indicates that the thermo-optic relative phase change peaks when |Δzoffset| ≅ 14μm ~s/2, that is, when one waveguide is placed below the CPW center conductor. In such a case the thermo-optic phase shift would be more than an order of magnitude greater for a given RF input than for the modulator tested in this study. This is significant because z-cut LN modulators, in contrast to x-cut, typically position one waveguide directly below the CPW center conductor [29]. Thus, these results imply that the RF-induced, thermo-optic relative phase shift should be considerably greater in z-cut LN modulators than in x-cut.

6. Discussion

The results presented in this paper have significant implications for the development of fast-response, high-power EO limiters. Most importantly, high-field effects (such as avalanche breakdown or ferroelectric domain reversal) were not observed in the EO modulator for the entire range of RF inputs tested. The observed short-term disruption was instead a simple matter of resistive heating within the conductor, which implies that mitigating RF disruption, and thereby achieving fast-response, may be as straightforward as improving device thermal management. Potential approaches include reducing RF attenuation, increasing conductor cross-sectional area, and/or the addition of a dielectric heat sink.

Secondly, the finding that short-term disruption was due to optical phase modulation strongly suggests that there may be advantages to replacing a MZ intensity modulator with an otherwise identical EO phase modulator, as shown in Fig. 12 . In contrast to intensity-modulated links, phase detection (e.g. via an asymmetric MZ interferometer) is irresponsive to slow-variations in optical phase [30], which implies that short-term disruption could be greatly reduced, if not eliminated, with such a setup.

 

Fig. 12 Phase modulated version of the photonic link in Fig. 1, which utilizes a LN EO phase modulator, asymmetric MZ phase demodulator, and balanced photodiode detection. This link is irresponsive to slow variations in optical phase.

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Lastly, these results offer insight into the physical mechanisms underlying long-term disruption and permanent modulator damage reported previously at higher RF power [7]. Given that resistive heating was significant in this study, but high-field effects were not, it is reasonable to examine the impact of RF heating at elevated power. A rough estimate of the temperature increase within the center conductor, immediately following a short RF pulse, is given by

ΔTc(y)UpulseQc(y,0)ρCpPin,RF(0)=2αFce2αywcheρCp.

Using the properties in Table 1, and for simplicity neglecting their temperature dependence, this yields the value 2.5x104 K/J at the RF input, y = 0. In the previous study [7], long-term disruption occurred at the pulse energy 1 mJ, which translates to an initial temperature change of 25 K. Permanent damage was later observed at the pulse energy 8 mJ, which corresponds to an initial temperature change of 200 K. Such substantial increases in temperature, and their corresponding temperature gradients, would almost certainly have been a factor in the long-term disruption and damage reported previously, if not the driving factor. Thus, a clear path towards a higher-power, high-speed photonic limiter is to address RF heating within the modulator electrode.

Appendix

Heat flux in the y-direction should result from uneven heating along the RF electrode, as well as the heat sinking property of the flared electrode ends. In order to estimate the importance of this longitudinal flux, the conduction term in Eq. (22) was first separated into transverse and longitudinal components, yielding

(κT)=xz(κxzT)+κ2Ty2
where xz is the 2D differential operator in x and z. A set of thermal simulations were next performed in 2D in (y,z) along the modulator center conductor, neglecting all heat flux out of the conductor. By neglecting the outward flux, the heat equation in the center conductor could be approximated from Eq. (22) as
ρeCp,eTyztκe2Tyzy2=Qc
along the interaction length, 0≤y≤Le. By comparison, the 2D heat equation in (x,z) solved in Section 5 was given by
ρeCp,eTxztxz(κxzTxz)=Qc
within the center conductor (the subscripts xz and yz are used here to denote results from the two different simulations). The relative magnitude of the longitudinal conduction was estimated by combining Eqs. (A2) and (A3), which reduces to the relation
κe2Tyz/y2(κT)=(1+Txz/tTyz/t)1for(Qc = 0),
following the completion of a pulse.

Figure 13 plots this ratio, calculated from simulated values of ∂Txz/∂t and ∂Tyz/∂t, for an identical 1 μs input RF pulse. Within the first millisecond following the pulse, the time frame of interest to this study, the mean value of this ratio was less than 2% along the length of the electrode. Thus, the impact of longitudinal heat flux was minimal, and the 2D approximation in Section 5 therefore justified.

 

Fig. 13 Simulated fraction of thermal conduction that occurs along the y-direction, within the center CPW conductor, as a function of distance along the electrode. Times indicated refer to the time elapsed from the start of the 1 μs pulse.

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Acknowledgment

This work was sponsored by DARPA. The authors would like to thank Marc Currie and Nicholas J. Condon for helpful discussions on nonlinear optic and thermal phenomena.

References and links

1. J. Benford, J. Swegle, and E. Schamiloglu, High Power Microwaves, 2nd Ed. (Taylor and Francis, 2007).

2. “High-Altitude Electromagnetic Pulse Protection for Ground-Based Facilities Performing Critical, Time-Urgent Missions,” MIL-STD-188–125–1, July 17, 1998.

3. Micronetics - Enon Microwave Inc, “New High Power Limiters” (Enon Microwave 2009). http://www.micronetics.com/products/enon/limiters.html.

4. Plasma Sciences Corp, “Products” (Plasma Sciences Corp 2009) http://www.plasmasciences.net/Products/products.html

5. Accurate Automation Corporation, “AAC-TVS Plasma Limiters” (AAC 2006). http://www.accurate-automation.com/w_content/images/f/f6/Plasma_Limiter_Brochure_May_2006.pdf.

6. R. C. J. Hsu, et al., “All-Dielectric Wireless Receiver,” IEEE/MTT-S International Microwave Symposium, 221–224 (June 2007).

7. F. Bucholtz, C. A. Villarruel, P. F. Knapp, J. Shue, T. D. Andreadis, R. T. Schermer, J. Gil Gil, and K. J. Williams, “Susceptibility of a Lithium Niobate Modulator to High-Power Microwave Pulses,” Electron. Lett. 45(5), 272–273 ( 2009). [CrossRef]  

8. G. F. Lipscomb, et al., “Electro-Optic Effect,” in Characterization Techniques and Tabulations for Organic Nonlinear Optical Materials, M. G. Kuzyk and C. W. Dirk, eds., (Marcel Dekker 1998).

9. C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the Performance of RF-Over-Fiber Links and Their Impact on Device Design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 ( 2006). [CrossRef]  

10. G. L. Li and P. K. L. Yu, “Optical Intensity Modulators for Digital and Analog Applications,” J. Lightwave Technol. 21(9), 2010–2030 ( 2003). [CrossRef]  

11. C. J. G. Kirkby, and C. Florea, “Electro-optic coefficients of LiNbO3,” in Properties of Lithium Niobate, K. K. Wong, ed., (IEE 2002).

12. G. E. Betts, “LiNbO3 External Modulators and Their Use in High Performance Analog Links,” in RF Photonic Technology in Optical Fiber Links, W. S. C. Chang, ed., (Cambridge 2002).

13. V. Gopalan, V. Dierolf, and D. A. Scrymgeour, “Defect-Domain Wall Interactions in Trigonal Ferroelectrics,” Annu. Rev. Mater. Res. 37(1), 449–489 ( 2007). [CrossRef]  

14. R. C. Alferness, “Waveguide Electrooptic Modulators,” IEEE Trans. Microw. Theory Tech. 30(8), 1121–1137 ( 1982). [CrossRef]  

15. G. Cocorullo and I. Rendina, “Thermo-Optical Modulation at 1.5 μm in Silicon Etalon,” Electron. Lett. 28(1), 83–84 ( 1992). [CrossRef]  

16. D. M. Pozar, Microwave Engineering, 2nd Ed. (Wiley 1998).

17. A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51(2), 1520–1531 ( 1995). [CrossRef]   [PubMed]  

18. F. Zhao et al.., “Temperature Dependence of Light-Induced Scattering in a Ce:Fe:LiNbO3 Photorefractive Crystal,” Opt. Eng. 35(7), 1985–1992 ( 1996). [CrossRef]  

19. J. P. Holman, Heat Transfer, 9th Ed. (McGraw Hill, 2002).

20. L. Moretti, M. Iodice, F. G. Della Corte, and I. Rendina, “Temperature Dependence of the Thermo-Optic Coefficient of Lithium Niobate, from 300 to 515 K in the Visible and Infrared Regions,” J. Appl. Phys. 98(3), 036101 ( 2005). [CrossRef]  

21. J. Wen, “Heat Capacities of Polymers,” in Physical Properties of Polymers Handbook, 2nd Ed., J. E. Mark, ed. (Springer 2007).

22. M. J. Weber, Handbook of Optical Materials (CRC 2003).

23. T. H. Lin, S. H. Lee, and D. Craig, “Thermal Conductivity Properties of LiNbO3,” in Properties of Lithium Niobate, K. K. Wong, ed., (IEE 2002).

24. Y. Yang, “Thermal Conductivity,” in Physical Properties of Polymers Handbook, 2nd Ed., J. E. Mark, ed. (Springer 2007).

25. R. A. Orwoll, “Densities, Thermal Expansions and Compressibilities,” in Physical Properties of Polymers Handbook, 2nd Ed., J. E. Mark, ed. (Springer 2007).

26. D. H. Jundt, and G. Foulon, “Crystal Structure of LiNbO3 ” in Properties of Lithium Niobate, K. K. Wong, ed. (IEE 2002).

27. G. Ghione, et al., “Characterization and Optimization of CPW Electro-Optic Modulators for Microwave and MM-Wave Applications,” Proceedings of the Gallium Arsenide Applications Symposium, 375–380 (1999).

28. S. K. Korotky, W. Minford, L. Buhl, M. Divino, and R. Alferness, “Mode Size and Method for Estimating the Propagation Constant of a Single-Mode Ti:LiNbO3 Strip Waveguides,” IEEE J. Quantum Electron. 18(10), 1796–1801 ( 1982). [CrossRef]  

29. E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 ( 2000). [CrossRef]  

30. M. J. LaGasse and S. Thaniyavarn, “Bias-Free High-Dynamic-Range Phase-Modulated Fiber-Optic Link,” IEEE Photon. Technol. Lett. 9(5), 681–683 ( 1997). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. J. Benford, J. Swegle, and E. Schamiloglu, High Power Microwaves, 2nd Ed. (Taylor and Francis, 2007).
  2. “High-Altitude Electromagnetic Pulse Protection for Ground-Based Facilities Performing Critical, Time-Urgent Missions,” MIL-STD-188–125–1, July 17, 1998.
  3. Micronetics - Enon Microwave Inc, “New High Power Limiters” (Enon Microwave 2009). http://www.micronetics.com/products/enon/limiters.html .
  4. Plasma Sciences Corp, “Products” (Plasma Sciences Corp 2009) http://www.plasmasciences.net/Products/products.html
  5. Accurate Automation Corporation, “AAC-TVS Plasma Limiters” (AAC 2006). http://www.accurate-automation.com/w_content/images/f/f6/Plasma_Limiter_Brochure_May_2006.pdf .
  6. R. C. J. Hsu, et al., “All-Dielectric Wireless Receiver,” IEEE/MTT-S International Microwave Symposium, 221–224 (June 2007).
  7. F. Bucholtz, C. A. Villarruel, P. F. Knapp, J. Shue, T. D. Andreadis, R. T. Schermer, J. Gil Gil, and K. J. Williams, “Susceptibility of a Lithium Niobate Modulator to High-Power Microwave Pulses,” Electron. Lett. 45(5), 272–273 (2009).
    [CrossRef]
  8. G. F. Lipscomb, et al., “Electro-Optic Effect,” in Characterization Techniques and Tabulations for Organic Nonlinear Optical Materials, M. G. Kuzyk and C. W. Dirk, eds., (Marcel Dekker 1998).
  9. C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the Performance of RF-Over-Fiber Links and Their Impact on Device Design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
    [CrossRef]
  10. G. L. Li and P. K. L. Yu, “Optical Intensity Modulators for Digital and Analog Applications,” J. Lightwave Technol. 21(9), 2010–2030 (2003).
    [CrossRef]
  11. C. J. G. Kirkby, and C. Florea, “Electro-optic coefficients of LiNbO3,” in Properties of Lithium Niobate, K. K. Wong, ed., (IEE 2002).
  12. G. E. Betts, “LiNbO3 External Modulators and Their Use in High Performance Analog Links,” in RF Photonic Technology in Optical Fiber Links, W. S. C. Chang, ed., (Cambridge 2002).
  13. V. Gopalan, V. Dierolf, and D. A. Scrymgeour, “Defect-Domain Wall Interactions in Trigonal Ferroelectrics,” Annu. Rev. Mater. Res. 37(1), 449–489 (2007).
    [CrossRef]
  14. R. C. Alferness, “Waveguide Electrooptic Modulators,” IEEE Trans. Microw. Theory Tech. 30(8), 1121–1137 (1982).
    [CrossRef]
  15. G. Cocorullo and I. Rendina, “Thermo-Optical Modulation at 1.5 μm in Silicon Etalon,” Electron. Lett. 28(1), 83–84 (1992).
    [CrossRef]
  16. D. M. Pozar, Microwave Engineering, 2nd Ed. (Wiley 1998).
  17. A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51(2), 1520–1531 (1995).
    [CrossRef] [PubMed]
  18. F. Zhao and ., “Temperature Dependence of Light-Induced Scattering in a Ce:Fe:LiNbO3 Photorefractive Crystal,” Opt. Eng. 35(7), 1985–1992 (1996).
    [CrossRef]
  19. J. P. Holman, Heat Transfer, 9th Ed. (McGraw Hill, 2002).
  20. L. Moretti, M. Iodice, F. G. Della Corte, and I. Rendina, “Temperature Dependence of the Thermo-Optic Coefficient of Lithium Niobate, from 300 to 515 K in the Visible and Infrared Regions,” J. Appl. Phys. 98(3), 036101 (2005).
    [CrossRef]
  21. J. Wen, “Heat Capacities of Polymers,” in Physical Properties of Polymers Handbook, 2nd Ed., J. E. Mark, ed. (Springer 2007).
  22. M. J. Weber, Handbook of Optical Materials (CRC 2003).
  23. T. H. Lin, S. H. Lee, and D. Craig, “Thermal Conductivity Properties of LiNbO3,” in Properties of Lithium Niobate, K. K. Wong, ed., (IEE 2002).
  24. Y. Yang, “Thermal Conductivity,” in Physical Properties of Polymers Handbook, 2nd Ed., J. E. Mark, ed. (Springer 2007).
  25. R. A. Orwoll, “Densities, Thermal Expansions and Compressibilities,” in Physical Properties of Polymers Handbook, 2nd Ed., J. E. Mark, ed. (Springer 2007).
  26. D. H. Jundt, and G. Foulon, “Crystal Structure of LiNbO3 ” in Properties of Lithium Niobate, K. K. Wong, ed. (IEE 2002).
  27. G. Ghione, et al., “Characterization and Optimization of CPW Electro-Optic Modulators for Microwave and MM-Wave Applications,” Proceedings of the Gallium Arsenide Applications Symposium, 375–380 (1999).
  28. S. K. Korotky, W. Minford, L. Buhl, M. Divino, and R. Alferness, “Mode Size and Method for Estimating the Propagation Constant of a Single-Mode Ti:LiNbO3 Strip Waveguides,” IEEE J. Quantum Electron. 18(10), 1796–1801 (1982).
    [CrossRef]
  29. E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
    [CrossRef]
  30. M. J. LaGasse and S. Thaniyavarn, “Bias-Free High-Dynamic-Range Phase-Modulated Fiber-Optic Link,” IEEE Photon. Technol. Lett. 9(5), 681–683 (1997).
    [CrossRef]

2009 (1)

F. Bucholtz, C. A. Villarruel, P. F. Knapp, J. Shue, T. D. Andreadis, R. T. Schermer, J. Gil Gil, and K. J. Williams, “Susceptibility of a Lithium Niobate Modulator to High-Power Microwave Pulses,” Electron. Lett. 45(5), 272–273 (2009).
[CrossRef]

2007 (1)

V. Gopalan, V. Dierolf, and D. A. Scrymgeour, “Defect-Domain Wall Interactions in Trigonal Ferroelectrics,” Annu. Rev. Mater. Res. 37(1), 449–489 (2007).
[CrossRef]

2006 (1)

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the Performance of RF-Over-Fiber Links and Their Impact on Device Design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

2005 (1)

L. Moretti, M. Iodice, F. G. Della Corte, and I. Rendina, “Temperature Dependence of the Thermo-Optic Coefficient of Lithium Niobate, from 300 to 515 K in the Visible and Infrared Regions,” J. Appl. Phys. 98(3), 036101 (2005).
[CrossRef]

2003 (1)

2000 (1)

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

1997 (1)

M. J. LaGasse and S. Thaniyavarn, “Bias-Free High-Dynamic-Range Phase-Modulated Fiber-Optic Link,” IEEE Photon. Technol. Lett. 9(5), 681–683 (1997).
[CrossRef]

1996 (1)

F. Zhao and ., “Temperature Dependence of Light-Induced Scattering in a Ce:Fe:LiNbO3 Photorefractive Crystal,” Opt. Eng. 35(7), 1985–1992 (1996).
[CrossRef]

1995 (1)

A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51(2), 1520–1531 (1995).
[CrossRef] [PubMed]

1992 (1)

G. Cocorullo and I. Rendina, “Thermo-Optical Modulation at 1.5 μm in Silicon Etalon,” Electron. Lett. 28(1), 83–84 (1992).
[CrossRef]

1982 (2)

S. K. Korotky, W. Minford, L. Buhl, M. Divino, and R. Alferness, “Mode Size and Method for Estimating the Propagation Constant of a Single-Mode Ti:LiNbO3 Strip Waveguides,” IEEE J. Quantum Electron. 18(10), 1796–1801 (1982).
[CrossRef]

R. C. Alferness, “Waveguide Electrooptic Modulators,” IEEE Trans. Microw. Theory Tech. 30(8), 1121–1137 (1982).
[CrossRef]

Ackerman, E. I.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the Performance of RF-Over-Fiber Links and Their Impact on Device Design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

Alferness, R.

S. K. Korotky, W. Minford, L. Buhl, M. Divino, and R. Alferness, “Mode Size and Method for Estimating the Propagation Constant of a Single-Mode Ti:LiNbO3 Strip Waveguides,” IEEE J. Quantum Electron. 18(10), 1796–1801 (1982).
[CrossRef]

Alferness, R. C.

R. C. Alferness, “Waveguide Electrooptic Modulators,” IEEE Trans. Microw. Theory Tech. 30(8), 1121–1137 (1982).
[CrossRef]

Anderson, D. Z.

A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51(2), 1520–1531 (1995).
[CrossRef] [PubMed]

Andreadis, T. D.

F. Bucholtz, C. A. Villarruel, P. F. Knapp, J. Shue, T. D. Andreadis, R. T. Schermer, J. Gil Gil, and K. J. Williams, “Susceptibility of a Lithium Niobate Modulator to High-Power Microwave Pulses,” Electron. Lett. 45(5), 272–273 (2009).
[CrossRef]

Attanasio, D. V.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Betts, G. E.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the Performance of RF-Over-Fiber Links and Their Impact on Device Design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

Bossi, D. E.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Bucholtz, F.

F. Bucholtz, C. A. Villarruel, P. F. Knapp, J. Shue, T. D. Andreadis, R. T. Schermer, J. Gil Gil, and K. J. Williams, “Susceptibility of a Lithium Niobate Modulator to High-Power Microwave Pulses,” Electron. Lett. 45(5), 272–273 (2009).
[CrossRef]

Buhl, L.

S. K. Korotky, W. Minford, L. Buhl, M. Divino, and R. Alferness, “Mode Size and Method for Estimating the Propagation Constant of a Single-Mode Ti:LiNbO3 Strip Waveguides,” IEEE J. Quantum Electron. 18(10), 1796–1801 (1982).
[CrossRef]

Cocorullo, G.

G. Cocorullo and I. Rendina, “Thermo-Optical Modulation at 1.5 μm in Silicon Etalon,” Electron. Lett. 28(1), 83–84 (1992).
[CrossRef]

Cox, C. H.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the Performance of RF-Over-Fiber Links and Their Impact on Device Design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

Della Corte, F. G.

L. Moretti, M. Iodice, F. G. Della Corte, and I. Rendina, “Temperature Dependence of the Thermo-Optic Coefficient of Lithium Niobate, from 300 to 515 K in the Visible and Infrared Regions,” J. Appl. Phys. 98(3), 036101 (2005).
[CrossRef]

Dierolf, V.

V. Gopalan, V. Dierolf, and D. A. Scrymgeour, “Defect-Domain Wall Interactions in Trigonal Ferroelectrics,” Annu. Rev. Mater. Res. 37(1), 449–489 (2007).
[CrossRef]

Divino, M.

S. K. Korotky, W. Minford, L. Buhl, M. Divino, and R. Alferness, “Mode Size and Method for Estimating the Propagation Constant of a Single-Mode Ti:LiNbO3 Strip Waveguides,” IEEE J. Quantum Electron. 18(10), 1796–1801 (1982).
[CrossRef]

Fritz, D. J.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Gil Gil, J.

F. Bucholtz, C. A. Villarruel, P. F. Knapp, J. Shue, T. D. Andreadis, R. T. Schermer, J. Gil Gil, and K. J. Williams, “Susceptibility of a Lithium Niobate Modulator to High-Power Microwave Pulses,” Electron. Lett. 45(5), 272–273 (2009).
[CrossRef]

Gopalan, V.

V. Gopalan, V. Dierolf, and D. A. Scrymgeour, “Defect-Domain Wall Interactions in Trigonal Ferroelectrics,” Annu. Rev. Mater. Res. 37(1), 449–489 (2007).
[CrossRef]

Hallemeier, P. F.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Iodice, M.

L. Moretti, M. Iodice, F. G. Della Corte, and I. Rendina, “Temperature Dependence of the Thermo-Optic Coefficient of Lithium Niobate, from 300 to 515 K in the Visible and Infrared Regions,” J. Appl. Phys. 98(3), 036101 (2005).
[CrossRef]

Kissa, K. M.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Knapp, P. F.

F. Bucholtz, C. A. Villarruel, P. F. Knapp, J. Shue, T. D. Andreadis, R. T. Schermer, J. Gil Gil, and K. J. Williams, “Susceptibility of a Lithium Niobate Modulator to High-Power Microwave Pulses,” Electron. Lett. 45(5), 272–273 (2009).
[CrossRef]

Korotky, S. K.

S. K. Korotky, W. Minford, L. Buhl, M. Divino, and R. Alferness, “Mode Size and Method for Estimating the Propagation Constant of a Single-Mode Ti:LiNbO3 Strip Waveguides,” IEEE J. Quantum Electron. 18(10), 1796–1801 (1982).
[CrossRef]

Lafaw, D. A.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

LaGasse, M. J.

M. J. LaGasse and S. Thaniyavarn, “Bias-Free High-Dynamic-Range Phase-Modulated Fiber-Optic Link,” IEEE Photon. Technol. Lett. 9(5), 681–683 (1997).
[CrossRef]

Li, G. L.

Maack, D.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

McBrien, G. J.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Minford, W.

S. K. Korotky, W. Minford, L. Buhl, M. Divino, and R. Alferness, “Mode Size and Method for Estimating the Propagation Constant of a Single-Mode Ti:LiNbO3 Strip Waveguides,” IEEE J. Quantum Electron. 18(10), 1796–1801 (1982).
[CrossRef]

Moretti, L.

L. Moretti, M. Iodice, F. G. Della Corte, and I. Rendina, “Temperature Dependence of the Thermo-Optic Coefficient of Lithium Niobate, from 300 to 515 K in the Visible and Infrared Regions,” J. Appl. Phys. 98(3), 036101 (2005).
[CrossRef]

Murphy, E. J.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Prince, J. L.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the Performance of RF-Over-Fiber Links and Their Impact on Device Design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

Rendina, I.

L. Moretti, M. Iodice, F. G. Della Corte, and I. Rendina, “Temperature Dependence of the Thermo-Optic Coefficient of Lithium Niobate, from 300 to 515 K in the Visible and Infrared Regions,” J. Appl. Phys. 98(3), 036101 (2005).
[CrossRef]

G. Cocorullo and I. Rendina, “Thermo-Optical Modulation at 1.5 μm in Silicon Etalon,” Electron. Lett. 28(1), 83–84 (1992).
[CrossRef]

Schermer, R. T.

F. Bucholtz, C. A. Villarruel, P. F. Knapp, J. Shue, T. D. Andreadis, R. T. Schermer, J. Gil Gil, and K. J. Williams, “Susceptibility of a Lithium Niobate Modulator to High-Power Microwave Pulses,” Electron. Lett. 45(5), 272–273 (2009).
[CrossRef]

Scrymgeour, D. A.

V. Gopalan, V. Dierolf, and D. A. Scrymgeour, “Defect-Domain Wall Interactions in Trigonal Ferroelectrics,” Annu. Rev. Mater. Res. 37(1), 449–489 (2007).
[CrossRef]

Shue, J.

F. Bucholtz, C. A. Villarruel, P. F. Knapp, J. Shue, T. D. Andreadis, R. T. Schermer, J. Gil Gil, and K. J. Williams, “Susceptibility of a Lithium Niobate Modulator to High-Power Microwave Pulses,” Electron. Lett. 45(5), 272–273 (2009).
[CrossRef]

Thaniyavarn, S.

M. J. LaGasse and S. Thaniyavarn, “Bias-Free High-Dynamic-Range Phase-Modulated Fiber-Optic Link,” IEEE Photon. Technol. Lett. 9(5), 681–683 (1997).
[CrossRef]

Villarruel, C. A.

F. Bucholtz, C. A. Villarruel, P. F. Knapp, J. Shue, T. D. Andreadis, R. T. Schermer, J. Gil Gil, and K. J. Williams, “Susceptibility of a Lithium Niobate Modulator to High-Power Microwave Pulses,” Electron. Lett. 45(5), 272–273 (2009).
[CrossRef]

Williams, K. J.

F. Bucholtz, C. A. Villarruel, P. F. Knapp, J. Shue, T. D. Andreadis, R. T. Schermer, J. Gil Gil, and K. J. Williams, “Susceptibility of a Lithium Niobate Modulator to High-Power Microwave Pulses,” Electron. Lett. 45(5), 272–273 (2009).
[CrossRef]

Wooten, E. L.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Yi-Yan, A.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Yu, P. K. L.

Zhao, F.

F. Zhao and ., “Temperature Dependence of Light-Induced Scattering in a Ce:Fe:LiNbO3 Photorefractive Crystal,” Opt. Eng. 35(7), 1985–1992 (1996).
[CrossRef]

Zozulya, A. A.

A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51(2), 1520–1531 (1995).
[CrossRef] [PubMed]

Annu. Rev. Mater. Res. (1)

V. Gopalan, V. Dierolf, and D. A. Scrymgeour, “Defect-Domain Wall Interactions in Trigonal Ferroelectrics,” Annu. Rev. Mater. Res. 37(1), 449–489 (2007).
[CrossRef]

Electron. Lett. (2)

F. Bucholtz, C. A. Villarruel, P. F. Knapp, J. Shue, T. D. Andreadis, R. T. Schermer, J. Gil Gil, and K. J. Williams, “Susceptibility of a Lithium Niobate Modulator to High-Power Microwave Pulses,” Electron. Lett. 45(5), 272–273 (2009).
[CrossRef]

G. Cocorullo and I. Rendina, “Thermo-Optical Modulation at 1.5 μm in Silicon Etalon,” Electron. Lett. 28(1), 83–84 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. K. Korotky, W. Minford, L. Buhl, M. Divino, and R. Alferness, “Mode Size and Method for Estimating the Propagation Constant of a Single-Mode Ti:LiNbO3 Strip Waveguides,” IEEE J. Quantum Electron. 18(10), 1796–1801 (1982).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A Review of Lithium Niobate Modulators for Fiber-Optic Communications Systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

M. J. LaGasse and S. Thaniyavarn, “Bias-Free High-Dynamic-Range Phase-Modulated Fiber-Optic Link,” IEEE Photon. Technol. Lett. 9(5), 681–683 (1997).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (2)

R. C. Alferness, “Waveguide Electrooptic Modulators,” IEEE Trans. Microw. Theory Tech. 30(8), 1121–1137 (1982).
[CrossRef]

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the Performance of RF-Over-Fiber Links and Their Impact on Device Design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[CrossRef]

J. Appl. Phys. (1)

L. Moretti, M. Iodice, F. G. Della Corte, and I. Rendina, “Temperature Dependence of the Thermo-Optic Coefficient of Lithium Niobate, from 300 to 515 K in the Visible and Infrared Regions,” J. Appl. Phys. 98(3), 036101 (2005).
[CrossRef]

J. Lightwave Technol. (1)

Opt. Eng. (1)

F. Zhao and ., “Temperature Dependence of Light-Induced Scattering in a Ce:Fe:LiNbO3 Photorefractive Crystal,” Opt. Eng. 35(7), 1985–1992 (1996).
[CrossRef]

Phys. Rev. A (1)

A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51(2), 1520–1531 (1995).
[CrossRef] [PubMed]

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Y. Yang, “Thermal Conductivity,” in Physical Properties of Polymers Handbook, 2nd Ed., J. E. Mark, ed. (Springer 2007).

R. A. Orwoll, “Densities, Thermal Expansions and Compressibilities,” in Physical Properties of Polymers Handbook, 2nd Ed., J. E. Mark, ed. (Springer 2007).

D. H. Jundt, and G. Foulon, “Crystal Structure of LiNbO3 ” in Properties of Lithium Niobate, K. K. Wong, ed. (IEE 2002).

G. Ghione, et al., “Characterization and Optimization of CPW Electro-Optic Modulators for Microwave and MM-Wave Applications,” Proceedings of the Gallium Arsenide Applications Symposium, 375–380 (1999).

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D. M. Pozar, Microwave Engineering, 2nd Ed. (Wiley 1998).

G. F. Lipscomb, et al., “Electro-Optic Effect,” in Characterization Techniques and Tabulations for Organic Nonlinear Optical Materials, M. G. Kuzyk and C. W. Dirk, eds., (Marcel Dekker 1998).

C. J. G. Kirkby, and C. Florea, “Electro-optic coefficients of LiNbO3,” in Properties of Lithium Niobate, K. K. Wong, ed., (IEE 2002).

G. E. Betts, “LiNbO3 External Modulators and Their Use in High Performance Analog Links,” in RF Photonic Technology in Optical Fiber Links, W. S. C. Chang, ed., (Cambridge 2002).

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Figures (13)

Fig. 1
Fig. 1

Schematic of an antenna-fed photonic link. Electromagnetic shielding and nonconductive fiber-optic cables isolate the laser and receiver electronics from high-power pickup at the antenna. Only the modulator is exposed to external electromagnetic sources.

Fig. 2
Fig. 2

Schematic diagram of the integrated-optic, Mach-Zehnder EO modulator used in this study. Light launched into the input optical waveguide of the LN crystal is split equally into two paths at the first y-branch, and then interfered at the second y-branch. Separate RF and bias electrodes apply an electric field to the LN crystal, which in turn modulates the optical phase of the two waveguides via the EO effect.

Fig. 3
Fig. 3

Schematic diagram of the experimental setup. The circuit element after the photodiode represents a low-pass filter.

Fig. 4
Fig. 4

Linear voltage ramp applied to the bias electrode (a), and resulting slowly-varying modulator transmission (b). The red curve in (b) plots the modulator response prior to RF pulsing, while the blue plots the response during an applied RF pulse burst. Each narrow spike on the blue curve coincides with a RF pulse, while the broader after-pulse following each spike is indicative of a lingering, non-instantaneous response.

Fig. 5
Fig. 5

Modulator transmission following the application of a single RF pulse (a), and the corresponding change in MZ relative phase, normalized to its maximum value (b). The curves overlap in (b), which indicates that the shape of each after-pulse was independent of the RF pulse power. The change in relative phase Δϕ0 was not plotted during the RF pulse (0≤t≤1μs) due to measurement uncertainty.

Fig. 6
Fig. 6

Relationship between after-pulse amplitude and (a) peak pulse power, and (b) pulse width, following the application of a single RF pulse. Solid lines correspond to linear fits to the data, and demonstrate that 0|max was linearly proportional to both pulse power and pulse width in the ranges shown.

Fig. 7
Fig. 7

Change in MZ relative phase induced by a single pair of pulses (a), and the resulting increase in after-pulse amplitude following the second pulse (b). The different curves in (a) correspond different pulse spacing, incremented from 100 to 500 μs. Time t = 0 coincides with the start of the second pulse. In (a) data is not plotted during each RF pulse. The solid line in (b) corresponds to an exponential fit, with time constant 270 μs.

Fig. 8
Fig. 8

Change in MZ relative phase induced by a pulse train of 20 identical RF pulses. The exponential decay of the relative phase envelope during pulsing indicates that the modulator approached a steady state. The time constant of the steady-state buildup, 270 μs, closely matched that of the subsequent relaxation. Data is not plotted during each pulse, leading to discontinuity in the curve.

Fig. 9
Fig. 9

Schematic diagram of the modulator cross-section. The axes coincide with the principal axes of the LN crystal.

Fig. 10
Fig. 10

Measured and simulated temporal evolution of the RF-induced relative phase change. Simulated results correspond to the case in which both optical waveguides were shifted off center with respect to the CPW electrodes by 0.24 μm in the –z direction.

Fig. 11
Fig. 11

Simulated dependence of the RF-induced steady-state (left axis) and transient (right axis) relative phase change on the waveguide-electrode center-to-center offset, Δzoffset .

Fig. 12
Fig. 12

Phase modulated version of the photonic link in Fig. 1, which utilizes a LN EO phase modulator, asymmetric MZ phase demodulator, and balanced photodiode detection. This link is irresponsive to slow variations in optical phase.

Fig. 13
Fig. 13

Simulated fraction of thermal conduction that occurs along the y-direction, within the center CPW conductor, as a function of distance along the electrode. Times indicated refer to the time elapsed from the start of the 1 μs pulse.

Tables (1)

Tables Icon

Table 1 Physical Properties Used in Thermal Model

Equations (35)

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T M Z = T a v g 2 ( 1 + η cos φ ) ,
φ = φ R F + φ S V
φ R F = π V i n V π , R F
φ S V = π V b i a s V π , b i a s + φ 0
V o u t ( t ) = ( P l a s e r T f i b e r Z o u t 1 + Z T / Z o u t ) T M Z ( t ) ,
T M Z = T a v g 2 [ 1 + η ( cos φ R F cos φ S V sin φ R F sin φ S V ) ] .
V i n ( t ) = V 0 ( t ) sin ω R F t ,
cos φ R F = J 0 ( π V 0 V π , R F ) + 2 m = 1 J 2 m ( π V 0 V π , R F ) cos 2 m ω R F t
sin φ R F = 2 m = 0 J 2 m + 1 ( π V 0 V π , R F ) sin ( 2 m + 1 ) ω R F t ,
T M Z ( S V ) = T a v g 2 [ 1 + η cos φ S V J 0 ( π V 0 V π ) ] .
T M Z ( S V ) T a v g 2 ( V 0 > > V π )
T M Z ( S V ) = T a v g 2 ( 1 + η cos φ S V ) ( V 0 = 0 ) .
T M Z ( S V ) T a v g 1 2 ( 1 + cos φ 0 )
Δ φ 0 ( t ) φ 0 ( t ) φ 0 ( 0 )
| Δ φ 0 | max max ( | Δ φ 0 ( t ) | ) ,
Δ φ ( t ) = Δ φ ( n ) ( t ) + Δ φ ( S ) ( t )
Δ φ ( n ) ( t ) = 2 π λ 0 y 0 L o y 0 [ Δ n e f f , 1 ( y , t ) Δ n e f f , 2 ( y , t ) ] d y
Δ φ ( S ) ( t ) = 2 π n λ 0 y 0 L o y 0 [ Δ S y , e f f , 1 ( y , t ) Δ S y , e f f , 2 ( y , t ) ] d y
Δ n e f f , i ( y , t ) = | ψ i ( x , z ) | 2 Δ n ( x , y , z , t ) d x d z ,
Δ S y , e f f , i ( y , t ) = | ψ i ( x , z ) | 2 Δ S y ( x , y , z , t ) d x d z ,
| ψ i ( x , z ) | 2 d x d z = 1 .
ρ C p T t ( κ T ) = Q
P R F ( y , t ) = P i n , R F ( t ) e 2 α y
d P R F ( y , t ) d y = 2 α P i n , R F ( t ) e 2 α y ,
Q c ( y , t ) = 2 α F c P i n , R F ( t ) e 2 α y w c h e
Q o ( y , t ) = α ( 1 F c ) P i n , R F ( t ) e 2 α y w o h e
Q ( y , t ) Q ( 0 , t ) e 2 α y ,
Δ n ( x , y , z , t ) A Δ T ( x , y , z , t ) ,
Δ φ ( t ) π α λ 0 ( 1 e 2 α L e ) [ Δ n e f f , 1 ( 0 , t ) Δ n e f f , 2 ( 0 , t ) ] ,
| ψ i | 2 = ( 64 2 x 2 π w m h m 3 ) exp [ 8 ( z z i w m ) 2 4 ( x h m ) 2 ]
Δ T c ( y ) U p u l s e Q c ( y , 0 ) ρ C p P i n , R F ( 0 ) = 2 α F c e 2 α y w c h e ρ C p .
( κ T ) = x z ( κ x z T ) + κ 2 T y 2
ρ e C p , e T y z t κ e 2 T y z y 2 = Q c
ρ e C p , e T x z t x z ( κ x z T x z ) = Q c
κ e 2 T y z / y 2 ( κ T ) = ( 1 + T x z / t T y z / t ) 1 for (Q c = 0),

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