Phase coherence length in silicon photonic platform

Yisu Yang; Yangjin Ma; Hang Guan; Yang Liu; Steven Danziger; Stewart Ocheltree; Keren Bergman; Tom Baehr-Jones; Michael Hochberg

doi:10.1364/OE.23.016890

1. Introduction

Silicon photonics makes use of CMOS compatible fabrication technologies to realize large-scale, energy-efficient photonic integrated circuits (PICs) with high yield and low cost [1,2]. Applications have been identified in on-chip optical interconnections [3–6], sensors [7,8] and all optical processing [9–12]. Reliable fabrication processes and process design kits (PDKs) are required to provide the infrastructure for the manufacture of practical, competitive PICs in silicon [13]. One of the significant considerations in system level integration is the fabrication non-uniformity that results in fluctuations in waveguide's effective index. The issue creates large phase uncertainty that makes the peak wavelength distribution in resonators hard to predict [14–18]. Thus, extra power as well as development effort for control systems will be spent on the tuning of these devices' resonance positions. This is undesirable in power budget limited scenarios [3,15,19].

The phase coherence length is a key parameter to characterize these fabrication non-uniformities. Traditionally, coherence length was used to characterize the phase noise of a laser in the interferometric measurement. This concept was then applied to study the phase noise of low confinement non-silicon waveguides. For instance, typical fiber Bragg grating's coherence length is about 10-100 cm [20] and the silica channel waveguide's coherence length is about 27 m [21]. However, former coherence length reports in integrated optics didn't show strong statistical significance as they were extracted from small numbers of samples, due to fabrication and test limitations [21, 22].

To the best of our knowledge, coherence length has not yet been reported in silicon photonic waveguides. This is because analyzing coherence length requires a large amount of data from special designed interferometer structures across the entire wafer. Unlike many researchers [20–23] in silicon photonics, the authors had access to a complete multi-project-wafer (MPW) Si photonics wafer from a commercial foundry. This gave us a unique opportunity to take the extensive cross-wafer data set needed to accurately measure the coherence length. Developing reliable statistical methods to measure random phase is another hard problem. Researchers have made important progress by stitching a hundred scanning electron microscope (SEM) images of one waveguide to quantify phase noise in integrated silicon Bragg gratings [23], but the cost would be high if hundreds of devices are characterized using SEM.

In this paper, we report the typical coherence length of silicon waveguides fabricated in a foundry's silicon photonics MPW run. We propose a new analytic method to verify the phase coherence length model. 800 silicon Mach-Zehnder interferometers (MZIs) in clusters were measured across the wafer. MZIs' spectra clearly show the fabrication non-uniformity similar to previous work [14–16]. We cite [16] as a validation but our work is not the extension of it [16]. is a significant study of the non-uniformity of chip-scale silicon photonics by ring resonators in one die. However [16], didn't consider the coherence length value because there was only one type of ring. But our work analyzed a large number of MZI structures with different optical path differences across the wafer. The coherence length extracted from MZIs shows the relationship between waveguide's random phase variance as a function of the waveguide length. For rings, this relationship is not clear, because of the small device footprint. We find the coherence lengths are 4.17 ± 0.42 mm and 1.61 ± 0.12 mm for silicon strip and rib waveguides respectively. We hope to use the coherence length as a standard figure of merit to evaluate the fabrication non-uniformity to support system level integration efforts in silicon photonics.

2. Theory

2.1 Effective index and geometry

In this work, the principal devices were the strip waveguide and the rib waveguide as shown in Fig. 1. The input optical wavelength is 1550 nm. The effective index of the waveguides will change due to variations in waveguide thickness (t), width (w), sidewall angle (α) and slab layer thickness (s). The relationships between the effective index and geometry can be considered to be linear if the standard deviations satisfy that δt<10nm, δw<10nm, δs<10nm and δα<10° as shown in Fig. 2. The proportionality constants that link geometries to the effective index are defined as C_w, C_t, C_α and C_s. Their values are simulated by a finite element mode solver [24] and summarized in Table 1. From simulations, we find that the effective index is very sensitive to the silicon layer thickness and sidewall angle. The effective index of the rib waveguide is less sensitive to sidewall angle as compared to the strip waveguide.

Fig. 1 Cross-sections of two kinds of waveguides used to build MZIs: (a) the rib waveguide, (b) the strip waveguide.

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Fig. 2 The relationship between the effective index and the waveguide geometry. The geometry parameters are (a) waveguide thickness, (b) waveguide width, (c) sidewall angle and (d) slab layer thickness. The inset figures show the definition of the geometry parameters. In (c), the inset figure also shows the mode profile when the sidewall angle is 45°. Default geometry parameters are the same as shown in Fig. 1. Input wavelength is 1550nm.

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Table 1. The proportionality constants that link the waveguide width, thickness, sidewall angle and slab height variations to an effective index variation

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2.2 Phase noise model

The vertical geometry variations come from the wafer top silicon layer thickness non-uniformity and etching process. The thickness can vary up to ± 20 nm between wafers [25] even before the wafer processing begins. The lateral geometry variations come from sidewall roughness created during the etching process. Although the standard 248 nm lithography CMOS processes were used in the fabrication of the samples, there were uncertainties in the photoresist thickness/roughness, mask alignment positions, the developing rate, the silicon dry etching rate and the thermal oxidation growth rate. Since a lot of random factors contribute to the change of effective index, based on the central limit theorem [21], the accumulated output phase of the waveguide is assumed to have a zero-centered component that has a random walk under Gaussian distribution. The randomly introduced phase noises are assumed to be independent. As we use the standard and fixed CMOS-compatible processes [26], the relationship between the fabrication parameters and the coherence length is not studied in this paper that focuses on the measurements of coherence length in silicon photonics waveguides. But it is an area for future study.

2.3 Coherence length model

The coherence length characterizes the physical phenomena that when the waveguide's length (L) is equal to the coherence length (L_coh), the random phase (Δϕ(L) is a Gaussian random variable with zero average) inside the waveguide as a result of fabrication non-uniformity will make the signal phase vary in a certain range. The average range (deviation) is expressed as <e^{iΔϕ∙Lcoh)}> = e⁻¹ in order to follow the laser community's tradition. In other words, the random phase noise's standard deviation is (2L/L_coh)^1/2 for the waveguide of length L. If we use the waveguide to build interferometers like rings and MZIs, we can expect the average difference in resonant wavelengths between two as fabricated interferometers will be ± FSR × Δϕ(L_coh)/2π ≈ ± 0.11 × FSR when the path-difference length equals L_coh. FSR is the free spectral range (unit: nm). The theory predicts that the random phase noise variances increase linearly with the waveguide length if the coherence length is constant. In order to calculate the coherence length, we propose a new analyzing method to extract optical phase noise from unbalanced MZIs. Although each MZI's waveguide has a different overall effective index after real fabrication processes, we can assume the average effective index (<n_eff>) to be same for the same type of MZIs. Then for the i-th MZI of the same type, the effective index can be expressed as a function of position (x):

n_{e f f, i} (x) = < n_{e f f} > + Δ n_{e f f, i} (x),

where Δn_eff_,_i(x) is a random deviation of effective index from its average value at the position x with Gaussian distribution centered at zero. The destructive interference condition of a MZI can be written as below:

\frac{2 π}{λ} \int_{0}^{d L} n_{e f f} (x) d x = (2 m + 1) π,

where dL is the length difference between two arms of the MZI; m is an integer standing for the MZI azimuthal mode index; n_eff (x) is the effective index of the fundamental mode at the position x; λ is the wavelength of the input light. So the destructive interference condition can be re-written as below:

\frac{2 π}{λ} < n_{e f f} > + Δ ϕ_{i} (d L) = (2 m + 1) π,

Δ ϕ_{i} (d L) = \frac{2 π}{λ} \int_{0}^{d L} n_{e f f} (x) d x .

Although ideally Eq. (3-b) should be written:

Δ ϕ_{i} (d L) = \frac{2 π}{λ} \int_{0}^{L_{1} + L_{2}} n_{e f f} (x) d x .

Since every slice (dx) of waveguide in both MZI arms (L₁ and L₂ are arm lengths and L₁ < L₂) contributes to the phase error, Eq. (3-b) is a good approximation when dL >> L₁ as dL~L₁ + L₂ or L₁ <<L_coh (see Appendix). Another reason to use Eq. (3-b) is that, when two waveguides are placed close (as shown in the GDS layout of the inset of Fig. 3(c), distance (x)< 700 μm), they become statistically correlated in terms of fabrication. (see Appendix). We then calculate the effective index dispersion slope (k) by a mode solver [24] in order to find the mean of effective index at the wavelength of interest.

Fig. 3 Automatic wafer-scale test setup and measured of MZI spectra: (a) the photograph of wafer-scale auto-test setup with a 6” silicon photonics wafer loaded and (b) zoom in on-chip devices and fiber array. (c) Transmission spectra of six nominally identical MZIs in the same die. Device layout is shown in the inset. dL = 144μm. (d) Extracted group index versus resonant wavelength for MZIs (dL = 144μm) in one die (about 100 samples). The dots close to the dashed line have same azimuthal mode index m.

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k = \frac{d < n_{e f f} >}{d λ} .

We simulated that k = −1.13 × 10⁻³ nm⁻¹ and −9.81 × 10⁻⁴ nm⁻¹ for strip and rib waveguides respectively. From the MZI spectrum, we measured the FSR and the corresponding resonance wavelengths at the spectrum’s null points. We can calculate the average effective index as shown by <n_eff (λ₁)> by the following formula:

\frac{2 π}{λ_{1}} < n_{e f f} (λ_{1}) > d L - \frac{2 π}{λ_{2}} < n_{e f f} (λ_{2}) > d L = 2 π,

\frac{2 π}{λ_{1}} < n_{e f f} (λ_{1}) > d L - \frac{2 π}{λ_{2}} [< n_{e f f} (λ_{1}) > + k \cdot F S R] d L = 2 π,

where λ₁ and λ₂ are adjacent resonant wavelengths (λ₁ < λ₂). The interferometer's azimuthal mode m is estimated by the method in [16]. That is, using measured resonant wavelength (λ_i), FSR and dL to extract group index (

n_{g}

) by:

F S R = \frac{λ_{1} λ_{2}}{n_{g} L} .

Then we can determine the m from the closely gathered cluster in the scatter plotting of (λ_i, $n_{g}$ ) as shown in Fig. 3(d). Substituting MZI_i's <n_eff (λ₁)> and m into Eq. (3-a), we get the random phase shift Δϕ_i (dL). Finally, we can calculate the coherence length (L_coh) from the linear regression of the variance (<Δϕ_i(dL)²>) and dL. The relationship is shown as:

< Δ ϕ {(d L)}^{2} > = \frac{2 L}{L_{c o h}} .

In our method, we replace L by dL in Eq. (7).

3. Experiment and discussion

In this work, the MZIs were fabricated by BAE Systems [26] through an MPW passive run with 248 nm lithography. The devices were built on a 6” SOI wafer from SOITEC with 220 nm top silicon, 3 μm buried oxide layer with 10 Ω-substrate [25]. The MZIs were made by the waveguides as shown in Fig. 1. In the measurement, a linearly transverse electric (TE) polarized light beam from a tunable laser (Agilent 81980A) centered at a wavelength around of 1550 nm was coupled into the MZI through a fiber array and an on-chip grating coupler (GC). The MZI's output light was coupled out through another GC to the fiber array and measured by a lightwave multimeter (Agilent 8163B) as shown in Fig. 3(a, b). The measured spectra were normalized against the transmission of a reference GC loop connected by the same waveguide. We tracked each resonance wavelength in the spectra of a set of MZIs by sine square function curve fitting [27]. Spectra of strip waveguide based MZIs with dL = 144 μm are shown in Fig. 3(c). The azimuthal mode m was determined by the scatter plot of group index and resonant wavelength as shown in Fig. 3(d).

We studied five groups of MZIs in each die (area size: 2.5 × 3.2 cm²). Each group had a different dL and 20 nominally identical MZIs that were spaced 120 μm apart as shown in Fig. 1(b). The dLs of the MZIs are 50 μm, 144 μm, 444 μm, 744 μm and 1044 μm. The transmission spectra of 800 MZIs from 8 dies across the wafer were measured to guarantee statistical significance. Based on the analyzing method mentioned in the first section, we found that the coherence length of the strip and rib waveguides across the wafer were 4.17 ± 0.42 mm and 1.61 ± 0.12 mm, respectively. The linear regressions of <Δϕ(dL ⁾²> and dL are shown in the Fig. 4. We also did statistical T-test to verify the linear relationship as shown in Figs. 4(a) and 4(c). T-test is used because the degree of freedom is small (4) as we characterized 5 types of dLs. Calculated by the experimental data shown in Figs. 4(a) and 4(c), T value proves that the relationship of Eq. (7) has statistical significance. The type-I error's α level is smaller than 0.1% as shown in the Appendix.

Fig. 4 Coherence length measurement results: (a, c) the relationship between strip/rib waveguide length and the variances of random phase shifts; (b, d) the statistics of coherence length of strip/ rib waveguide in 8 dies across a MPW wafer.

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The strip waveguide's coherence length is longer than the rib waveguide's, which indicates that the fabrication process for the strip waveguide has better tolerance to process variations. Both strip and rib waveguides have same waveguide width (500 nm) and the fundamental mode was tightly confined in the center of the waveguide from simulation results. So it can be assumed that the top surface roughness is approximately same. It's clear that the rib waveguide has larger overall non-uniformity due to the additional slab layer. The extra phase error may mainly suffer from the partial etch step that forms the slab layer. Different mechanisms of fabrications uncertainties have been studied by others, including etching, mask aligning, and oxidation step errors. For example, some reports showed the waveguide sidewall roughness as ± 1.8 nm [23] and ± 2.7 nm [28] and the waveguide top surfaces roughness as ± 0.45 nm [28]. The thickness and width variations of slab layer of rib waveguide in one die were about ± 0.1 nm and ± 0.4 nm [14]. Compared with the strip waveguide, we think the rib waveguide has larger non-uniformity in the slab layer fabrication. Although these results came from dedicated runs, they could qualitatively show that slab layer variations degrade the phase noise performance indicated by shorter coherence length. We randomly generated 10000 MZIs with slab thicknesses that is a Gaussian random variable centered at 50 nm. The standard deviation was assumed as 0.25 nm. Each waveguide's effective index at desired wavelength was simulated [24]. By Monte-Carlo method, we can find MZIs resonant wavelengths near 1550 nm under the same azimuthal mode m. The simulated phase noise variance as a function of dL is shown in the Appendix. The simulated coherence length was about 1.8 mm that agreed well with the experimental results (1.6 mm). Thus, our coherence length model is proved. Moreover, 0.25 nm is also a reasonable prediction [14] for the standard deviation of slab thickness. Thus, our method opens a low-cost, high-throughput pathway to characterize fabrication non-uniformity without using SEM test. The measured coherence length was shorter than the simulation results because of other non-uniformities from, for instance, the waveguide thickness, width and sidewall roughness. In this work, we provide a wafer scale variation of coherence length since it is critical to analyze the wafer level fabrication non-uniformities. The small deviation of coherence length across the wafer proves the high uniformity. Depending on the applications, die scale variation may be more important than wafer scale [20, 22]. Therefore, we provide this simple but general analyzing method that can extract the coherence length under different scales.

Since the fabrication non-uniformity may bring extra loss to waveguide, we measured waveguide insertion loss to study whether it is related to the phase coherence length. We fitted the measured spectra of GC loops (connected by waveguides of different lengths) with parabola in a 40 nm range centered at their peak wavelengths. The insertion loss obtained from linear regression between waveguide length and maximum power at around 1550nm is −4.8 ± 0.03 dB/cm and −5.2 ± 0.06 dB/cm for the strip waveguide and the rib waveguide, respectively, as shown in Fig. 5. Lower insertion loss could be attributed to the better sidewall roughness in strip waveguide, but the difference is only 0.4 dB, which shows that the low insertion loss and the long coherence length are not strongly related. One reason may be that the insertion loss reflects the averaged random phase-shift variance contributed by all fabrication non-uniformities, but the coherence length stands for the total sum of variance as a result of all non-uniformities.

Fig. 5 Typical insertion loss test results in one die (red dots /blue dots) the experiment results of strip/rib waveguide's output power versus its length; (red line/ blue line) The linear regress of strip / rib waveguide's output power versus their lengths.

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Compared with the reported coherence lengths of fiber device and silica waveguide [20–22], the coherence length of silicon waveguide is several orders of magnitude shorter. Note that the index contrast of silicon waveguide is much larger than others. Near 1550nm, silicon and silica refractive index are 3.47 and 1.45 respectively. Moreover, the uncertainty of the silicon thickness is high [25]. Therefore, the shorter coherence length is likely due to the high-index-contrast highly confined waveguide. In other words, the effective index is very sensitive to geometry changes as shown in Fig. 2.

To show the fabrication non-uniformity, we also measured destructive resonance wavelengths and FSRs contours of two types of MZIs (dL = 144 μm, strip waveguide and rib waveguide) across the wafer as shown in Fig. 6. Twenty nominally identical samples were measured in each MZI set. FSRs are 3.92 ± 0.09 nm (strip waveguide) and 4.20 ± 0.08 nm (rib waveguide), respectively. Peak resonance wavelengths are 1548.11 ± 0.71 nm (strip waveguide) and 1548.48 ± 0.96 nm (rib waveguide), respectively. We use experimental results of peak resonance wavelengths (λ) and FSRs (nm) to calculate group index (n_g) by Eq. (6). We find n_g = 4.2 and 4.0 for strip and rib waveguides respectively. The corresponding group indices extracted from effective index simulations using Eq. (4) and Eq. (8) were 4.2 and 3.9 that agree well with experiment results.

Fig. 6 Cross-wafer measurement in MPW for strip/rib waveguide based MZIs (dL = 144μm): (a/b) Contour plots of strip / rib waveguide based MZIs' resonance wavelengths; (c/d) contour plots of strip / rib waveguide based MZIs FSRs

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n_{g} = n_{e f f} (λ) - λ \frac{d n_{e f f}}{d λ} = n_{e f f} (λ) - k λ .

For each MZI, the peak resonance wavelength was picked up under the same azimuthal mode index. The rib waveguide's resonance wavelength random shift is larger than the strip waveguide's, which is consistent with our coherence length analysis. Although the uncertainty of resonance wavelength shift is about ten times larger than the FSR's, which is consistent with the reports in [14,15], long coherence length guarantees that the output phase is still under well control.

4. Coherence length validation and application

We use other reported experimental results to verify the coherence length measurements in this study. In [16], researchers tested 371 racetrack resonators' that have the same designed cross-section of the strip waveguide used in this work. The devices from [16] were also fabricated in a commercial CMOS foundry similar to us. Thus, we can assume our strip waveguide's coherence length can be applied to [16]. In [16], when the device separation distance (this is equivalent to dL in our model) is 1mm, the resonant wavelength average shift (<Δλ>) was about 0.75 nm as shown in [16]'s Fig. 3(b). The racetrack resonator's perimeter was 84.36 μm and FSR was about 6.8 nm at around 1550nm. Therefore, the phase noise's standard deviation is 0.69 rad by 2π × <Δλ>/FSR. By our coherence length model, substituting strip waveguide's L_coh = 4mm and dL = 1mm, we find the phase noise's standard deviation is (2dL/L_coh)^1/2 = 0.71 rad. Therefore [16]'s experiment results agrees quite well with our theory prediction. In general, coherence lengths may vary among different MPW runs.

The coherence length theory can be applied in PIC system design. For example, considering an on-chip system with 4cm long routing strip waveguide using the waveguide geometry shown in this work, we can find the standard deviation of random phase shift to be 4.47 radians by Eq. (7). The equivalent waveguide length uncertainty's standard deviation is:

Δ L_{e f f} = \frac{Δ ϕ (d L)}{2 π} \frac{λ}{n_{e f f}} .

Then we can calculate the time uncertainty by

Δ t = \frac{Δ L_{e f f}}{c / n_{e f f}} .

Therefore, the uncertainty in time domain due to phase coherence length (assume n_eff = 2.5) is about 3.7 fs. Another example is to apply the coherence length to analyze MZ modulator wavelength tuning range. If the modulator is made by the rib waveguide as shown in Fig. 1(b), the waveguide's coherence length (L_coh) is 1.6mm. Take experimental results from [29]. The V_π is 9 V for a 3 mm (L) length phase shifter. The MZI's dL is 100μm. From a conservative viewpoint, the random phase variance is 3.75 rad² by Eq. (7). Actually, the L in the Eq. (7) could be less than 3 mm due to phase correlation when two arms of the MZI are closely placed (e.g. separation<1mm). The average resonance wavelength difference between two MZ modulators working near 1550nm is about 0.3FSR by:

Δ λ = F S R \frac{s t d (Δ ϕ)}{2 π} .

In order to make one MZ modulator work on the same wavelength of the other, the maximum tuning bias (V_tune) can be expected as 5.4 V by:

V_{t u n e} = V_{π} \frac{Δ λ}{0.5 F S R} .

5. Conclusion

Phase coherence lengths are extracted by a new experimental method and reported in silicon photonics platform for the first time to the best of our knowledge. The measured coherence lengths are 4.17 ± 0.42 mm and 1.61 ± 0.12 mm for strip and rib waveguides respectively. These results show statistical significance and high consistence based on 800 samples of MZIs across a 6” SOI wafer. The phase variation of the strip waveguide has better fabrication tolerance than rib waveguide, which is not strongly correlated to the waveguide insertion loss. Moreover, the coherence length method can be applied to other researchers' study of the non-uniformity on chip-scale silicon photonic integrated circuits. Our prediction of phase noise agrees well with the experimental results of [16]. This work provides theoretical and experimental support of using coherence length as a guideline to evaluate the fabrication non-uniformity of silicon photonics platform.

Appendix

In our experiment, the total arm lengths (L) of MZIs are: 68.8 μm, 306.2 μm, 606.2 μm, 906.2 μm and 1206.2 μm corresponding to arm length differences (dL) of 50 μm, 144 μm, 444 μm, 744 μm and 1044 μm. Although the condition dL >>L₁ is not always satisfied as shown in Table 2. But the condition L_coh>>L₁ always stands for both waveguides. We think that when L_coh>>L₁, the dL can replace the total length's influence to the L_coh. Therefore, replacing Eq. (3-b') by of Eq. (3-b) is a good approximation.

Table 2. MZIs' arm lengths in coherence length experiment. L₁: arm1 length, L₂: arm 2 length, dL = L₂-L₁; total length L = L₁ + L₂.

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We also studied the relationship between the random phase shift's variance and the MZI's total arm length (L = L₁ + L₂) as shown in Fig. 7. The extracted coherence lengths for strip and strip loaded strip waveguides are 4.63 ± 0.35 mm and 1.72 ± 0.11 mm, respectively. The coherence lengths are almost the same as the results extracted from the dL because L_coh>> L₁. However, there is one problem: the random phase shift's variance (<(Δϕ)²>) is not zero when L = 0 by the linear regression as shown in Fig. 7. In contrast, <(Δϕ)²> extract from dL is almost 0 as shown in Fig. 4. So using dL is suitable for our experiment condition.

Fig. 7 The second method of calculating coherence length by the total arm length of MZI: (a, b) The relationship between strip and rib waveguide length and the variances of random phase shifts in 8 dies across a MPW wafer.

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To prove our method's feasibility, we also designed and measured about 800 MZIs which have the same dL (110 µm) but different total length (L = 1400 µm and 290 µm, respectively). By the same method, the <Δϕ(L)²> are 0.035 rad² when L = 1400 µm and 0.033 rad² when L = 290 µm, which is very close. The small difference in the results proves our analyzing that using dL instead of L to extract coherence length is reasonable. In this experiment, the extra L was added in the x dimension of the layout plane not in the orthogonal dimension as shown in the Fig. 3(c) inset. So we infer that there are correlations in the non-uniformities in the x dimension due to this foundry's fabrication conditions. We think it is possible that other errors in thickness or mask alignment will be correlated to reduce L's influence on the phase errors. The simulation of the random phase noise variance as a function of the rib MZI's optical path difference is shown in Fig. 8 by Monte-Carlo method. We assume that the rib MZI's non-uniformity is only in the slab thickness with a standard deviation of 0.25 nm. We also did t-test to prove the coherence length model's strong statistical significance as shown in Fig. 9.

Fig. 8 The simulated phase noise variance as a function of the dL for the rib MZI by the Monte-Carlo method. The rib waveguide's slab thickness' standard deviation is assumed to be 0.25 nm.

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Fig. 9 The T-test results of the experimental data shown in Fig. 4(a, c). When the degree of freedom (n') is 4, T-value is 8.61 for the Error I's α level of 0.1%, as shown in blue bars. (Green bars): rib waveguide; (Red bars): strip waveguide

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Acknowledgment

The authors thank Mark S. Mirotznik, Lukas Chrostowski, Christophe Galland, Ari Novack, and Nicholas C. Harris for helpful discussions. The authors would like to thank Steven Danziger and Stewart Ocheltree of BAE systems, for their support of device fabrication. All authors gratefully acknowledge support from AFOSR STTR grants, numbers FA9550-12-C-0079 and FA9550-12-C-0038. The authors would like to thank Gernot Pomrenke, of AFOSR, for his support of the OpSIS effort, though both a PECASE award (FA9550-13-1-0027) and ongoing funding for OpSIS (FA9550-10-1-0439), Brett Pokines and AFOSR SOARD office, for their support under grant FA9550-13-1-0176.

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	C_t (nm⁻¹)	C_w (nm⁻¹)	C_α (degree⁻¹)	C_s (nm⁻¹).
Strip waveguide	3.7 × 10⁻³	1.6 × 10⁻³	6.9 × 10⁻³	NA
Rib waveguide	3.2 × 10⁻³	1.2 × 10⁻³	3.5 × 10⁻³	1.4 × 10⁻³

L (μm)	dL (μm)	L₁ (μm)	L₂ (μm)	dL >>L₁?	L_coh >>L₁
68.8	50	9.42	59.42	No	Yes
306.2	144	81.12	225.12	No	Yes
606.2	444	81.12	525.12	No	Yes
906.2	744	81.12	825.12	Yes	Yes
1206.2	1044	81.12	1125.12	Yes	Yes

	C_t (nm⁻¹)	C_w (nm⁻¹)	C_α (degree⁻¹)	C_s (nm⁻¹).
Strip waveguide	3.7 × 10⁻³	1.6 × 10⁻³	6.9 × 10⁻³	NA
Rib waveguide	3.2 × 10⁻³	1.2 × 10⁻³	3.5 × 10⁻³	1.4 × 10⁻³

L (μm)	dL (μm)	L₁ (μm)	L₂ (μm)	dL >>L₁?	L_coh >>L₁
68.8	50	9.42	59.42	No	Yes
306.2	144	81.12	225.12	No	Yes
606.2	444	81.12	525.12	No	Yes
906.2	744	81.12	825.12	Yes	Yes
1206.2	1044	81.12	1125.12	Yes	Yes

Phase coherence length in silicon photonic platform

Abstract

1. Introduction

2. Theory

2.1 Effective index and geometry

2.2 Phase noise model

2.3 Coherence length model

3. Experiment and discussion

4. Coherence length validation and application

5. Conclusion

Appendix

Acknowledgment

References and links

Cited By

Figures (9)

Tables (2)

Equations (15)

Optics Express