Abstract

In this work, we propose and demonstrate the performance of silicon-on-insulator (SOI) off-axis microring resonator (MRR) as electro-optic modulator (EOM). Adding an extra off-axis inner-ring in conventional microring structure provides control to compensate thermal effects on EOM. It is shown that dynamically controlled bias-voltage applied to the outer ring has the potency to quell the thermal effects over a wide range of temperature. Thus, besides the appositely biased conventional microring, off-axis inner microring with pre-emphasized electrical input message signal enables our proposed structure suitable for high data-rate dense wavelength division multiplexing scheme of optical communication within a very compact device size.

© 2014 Optical Society of America

1. Introduction

The reduction in feature size due to the advancement of semiconductor industry and integration of millions of transistors within the same chip area as prognosticated by Moore’s law have steered on-chip communications to the paradigm of multi-core processors which are fast enough to execute huge calculations within billionth of a second. However, drift velocity of electron limits the speed of electronic circuits. The delay at metallic interconnects remains the same which forces asynchronous data transfer between the processor and the other parts of a computational system and consequently creates an important bottleneck problem [1–3]. Also, with the increasing demand, communication bandwidth of a system is becoming a big issue day-by-day which has to be addressed very soon.

Photonic circuits have nascent potentiality to overcome almost all the drawbacks of present-day electronic technology [4, 5]. Low power consumption, low latencies, less interference, ultra compact size and wide bandwidth are the key advantages of optical circuits [6]. Such key benefits have engendered interest among scientists to find out an efficient design of electro-optic modulator (EOM), which is an indispensible active optical component of photonic circuits [7]. Recent advancements in silicon-on-insulator (SOI) technology [8, 9] has opened up a new window for the fabrication of micro optical devices with microelectronics on chip [10]. Nonetheless, to design a compact, high-speed EOM is still a big challenge due to the absence of Pockels effect, very low Franz-Keldysh and Kerr effect of pure crystalline Si [7]. Carrier plasma dispersion effect of Si is the only effective way to achieve a considerable amount of refractive index (RI) variation by applying moderate external electromagnetic field avoiding the breakdown of Si-device. Soref et al. have predicted the change in RI, Δn and absorption coefficient, Δα of Si due to carrier plasma dispersion effect [11]. Carrier dispersion and absorption [7] at λ = 1.55μm can be approximated by Eqs. (1) and (2),

Δn=Δne+Δnh=[8.8×1022ΔN+8.5×1018(ΔP)0.8]
Δα=Δαe+Δαh=[8.5×1018ΔN+6×1018ΔP]
Δne and Δαe(cm−1) are the changes in RI and absorption coefficient variation due to the electron concentration change ΔN (cm3), Δnh and Δαh (cm1) are the RI change and absorption coefficient variation due to the change in hole concentration ΔP (cm3), respectively.

According to the previous results [7], a carrier depletion/injection of ~1018 cm−3 can yield a RI change Δn ~2 × 10−3. A modulating device typically of the order of few millimetres is capable to provide a high modulation with extinction ratio (ER) as high as 20dB with sufficient variation of RI [7] until the development of resonator based SOI EOM. MRR confines light within the ring cavity and can provide relatively longer path length without increasing device size. A small change in RI can detune a MRR and yields a very high modulation [7].One of the critical issues of Network on Chip (NoC) systems is to keep power budgets manageable while increasing performance. DWDM is one such solution which offers massive parallelism and is considered very important in order to realize high performance optical interconnects which can be achieved through MRRs.

But, Si has high thermo-optic coefficient [12] (TOC = ΔnT), ~1.86 × 10˗4/K. Small amount of change in RI results drastic change in transmission characteristics of SOI based MRR due to its large Q-factor and narrow bandwidth [13]. MRRs are highly susceptible to the thermal fluctuations. Heat generated from the surroundings, off-chip continuous-wave laser sources, local hot-spots etc. can cause shift in resonance wavelength. This instability can disturb precise channel allocations for DWDM systems. The shift in resonant wavelength can be represented as [13],

dλdT=(neffαsub+neffT)λ0ng
where λo is the resonant wavelength at the room temperature, neff denotes the effective RI and ng is the group index. The substrate expansion coefficient, αsub, of SiO2 can be neglected in Eq. (3) due to its small value [13]. The Eq. (3) can be modified and written as,

dλdT=neffTλ0ng

In recent years, a few attempts were made to build athermal EOMs. A detailed comparative study on these approaches has been discussed in [13]. Methods to compensate the thermal effects can be broadly classified in two major categories (i) athermal solutions and (ii) control-based solutions. One of the most well-known control-based solution procedures to combat thermal degradation is the use of heat sink and integrated heater simultaneously [14–16] which later on has been proved highly power-inefficient [17]. Padmaraju et al. proposed an adaptive, innovative and dynamic athermal design exploiting the heating effect of current [17], which is more effective to deliver/extract heat using an external bias, albeit, this bias voltage can affect the message signal. This approach also fails to compensate wide range of temperature change. Moreover, it is not suitable for high data-rate due to its large response delay and inability to use pre-emphasized [18] input signal. Parallel efforts (athermal solution) to neutralize the thermal effect using negative temperature coefficient materials e.g. polymers [19–21] as the key component of the device cladding also exist. Mathematical representation of the effective TOC for a waveguide structure with Si-core and polymer-clad can be given by,

neffT=Γcore(Si)ncoreT+Γclad(polymer)ncladT+ΓsubnsubT

Where Γis the modal confinement factor. To overcome the thermal effects, one must produce a zero effective TOC. Since the TOC of Si is very high, according to Eq. (5) either we need a polymer with very high negative TOC or the mode confinement factor of the core must be deliberately reduced. Mode confinement can be reduced using narrow-ridged or slotted waveguides but these structures associate with high losses, greater radiation, lower Q-factor, small and corrupted FSR. The weak confinement of the mode also leads to higher device footprint. Furthermore, this method is not a successful commercial solution till date because of the CMOS-incompatibility of these polymers. CMOS compatible titanium oxide (TiO2) with negative TOC [22] may be a possible future solution which allows moderate optical confinement in Si-core. Still the performance of these devices is not satisfactory due to low Q-factor and low FSR. Temperature independent operation over a small range has been shown employing MZI-coupled MRRs [23] and multiple cavity coupled devices [24]. However, relatively large footprints make them ineffectual for high-density on-chip applications. Adiabatic microdisk/microring modulators and cascaded MRRs [25,26] are capable of operating at as high data rate as 40Gbps and provide uncorrupted FSR but these devices are also very susceptible to the change in ambient temperature.

Recently, resonant mode-splitting method to negate the temperature effect has been reported [27]. Though, this method has potential to eliminate most of the previously mentioned drawbacks, it possesses a few serious disadvantages. Negative TOC CMOS-incompatible polymers have been used again to achieve satisfactory performance. Device size is large and it is impossible to implement this device in DWDM system due to its inability of using broad range of adjacent channels during the operation. It is also cumbersome to define and retain the material properties intact throughout the entire operational range.

The inefficacy of all the aforementioned mechanisms motivate us to propose a novel design of EOM using off-axis MRR. We have recently revealed many striking features of MRR with off-axis inner ring [28], where tunability aspects of conventional and extra resonant notches usher us the way to obtain a highly efficient and compact EOM. Two rings within the same structure provide us an extra degree of freedom of designing; we use one ring to deliver message and another to minimize the thermal effects. According to [13], our method falls in control-based solutions to avoid thermal degradation. Pre-emphasized input message signal can be applied to the conventional outer MRR to obtain high data-rate DWDM communication systems.

In this work, device parameters are extracted from commercial TCAD electro-optic device simulator based on finite difference time domain (FDTD)-Gummel solver. RI variations due to carrier injection or depletion are calculated from mode solver. Other necessary design parameters are calculated from transfer matrix method (TMM) and verified by FDTD method.

2. Design of off-axis MRR as an EOM

In Fig. 1(a), it is depicted how a MRR with off-axis inner ring can be used as an EOM. One can apply a modulating signal, Vm as an input to the off-axis ring and another voltage source VB to the outer MRR to nullify the thermal effect. Note that, Vm and VB are interchangeable. Cross-sectional view of the waveguide is shown in Fig. 1(b). A rib structure has been grown on buried oxide layer. Waveguide with dimension of 450nm × 350nm is placed on a 50nm thick plateau which provides electrical connections to the optical device. The junction volume is 3.48 × 10˗12 cm3.

 

Fig. 1 (a) Schematic of MRR with off-axis inner ring structure used as a modulator, rsh, rse and RT represent the shunt, series and temperature dependent equivalent resistances of the device, Vm and VB are voltage sources. (b) Cross-section of p+n-n+ SOI waveguide structure.

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We have created a p+n˗n+ type structure. As the conductivity of pure Si is very low, n˗ epitaxial layer with nominal carrier concentration ~1015 cm˗3 has been sandwiched between p+ and n+ regions to keep the carrier flow uninterrupted. Moreover, this denuded layer reduces the number of recombination centres which further prevents the alteration of diode current (R-G current [29]) with the change in temperature. One must keep in mind that high doping concentration in n˗ region will obtain a better RI sensitivity (ΔnV) for the waveguide with respect to applied bias but it will increase the device loss due to higher absorption coefficient [7]. We optimize our structure to limit the absorption loss due to extra carrier injection within 20 dB/cm. Doping concentration of p+ and n+ layers are ~1019 cm˗3; metallic contacts are made through n++ and p++ with doping concentration of ~1020 cm˗3. The doping concentration of n˗ layer is much lesser than the concentration of n+ or p+ layer. Basically, this layer acts as an intrinsic layer.

3. Theory, results and discussions

We have applied an 8-bit, 3Gbps pre-emphasized NRZ signal, Vm = ± 1V (Vp-p = 2V) to the inner off-axis ring with an off-set dc bias of 1V. The minority charge carriers diffuse from the junction to the terminal when a voltage is applied to the p+n˗n+ device (forward biasing). If np0and pn0are the minority carrier concentrations at p and n sides at room temperature in thermal equilibrium, we can write the extra charge carrier density as the function of distance x [29,30] from the depletion layer as,

ΔN(x)=np0(eeVkT1)exp(xp+xLp)
ΔP(x)=pn0(eeVkT1)exp(xn+xLn)
where, xn and xp denote the depletion region widths at the n and p sides, Ln and Lp represent the diffusion lengths of electrons and holes, respectively. The Lp (~22μm) and Ln (~35μm) are greater than the depletion layer width (~640nm) which assures moderate current flow through the device.

Figure 2(a) shows the schematic cross-section of an off-axis micro ring resonator with three coupling regions dictated by dashed rectangular boxes where R1 and R2 are the radii of outer and inner rings, respectively. In order to understand the behavior of off-axis micro ring resonator, we have to first compute the effective index of the guided mode which can be either obtained by effective index method [31], frequently used in integrated optics by combining Eqs. (6), (7) and (1), (2) or through full-vectorial finite element method (FEM) solver. One can yield effective index of the guided mode neff for a particular mode and at a specific wavelength. Effective RI of a particular quasi-TE mode for outer ring (R1 = 3μm), neff1 and off-axis ring (R2 = 2μm), neff2 have been determined by changing applied bias and plotted in Fig. 2(b), whereas the spectral variation of the effective indices of the quasi-TE mode for different ring radii and different biasing voltages is depicted in Fig. 2(c). At wavelength 1.55μm, the dispersion coefficient D(=δneff/δλ)for outer and inner rings are obtained as ˗0.98965μm−1 and ˗1.0135μm−1, respectively, when no voltage is applied and ˗0.9897μm−1 and ˗1.01295μm−1, respectively when an external bias of 1V is applied across the p+n-n+ junction of outer and inner rings, respectively. Since, we intend to implement the structure for several DWDM channels within a broad operating wavelength range, neff1 and neff2 will decrease with increasing wavelength due to dispersion which must be taken into account in TMM method. We use the dispersion coefficients to calculate accurate neff values over this broad range of operating wavelengths. Both the effective RI and dispersion coefficient of off-axis MRR are greater than the effective RI and dispersion coefficient of outer ring since the bending radius of the inner MRR is smaller than the outer one. Both the values of effective RIs and dispersion coefficients depend upon external voltage bias.

 

Fig. 2 (a) Schematic of an off-axis MRR with necessary design parameters. (b) Effective refractive index variation of outer and inner rings with respect to bias voltage, VB at 1.55μm, (c) spectral variation of guided mode index for different ring radii at different biasing voltages, and (d) loss as a function of applied voltage Vm, at wavelength 1.55μm.

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In Fig. 2(d), we have plotted total losses (absorption, bending, radiation etc.) as a function of bias voltage Vm. Note that, Vm and VB are interchangeable. As discussed earlier, absorption losses have been restricted within 20dB/cm. It is noticed that both absorption loss and effective RI change rapidly after Vm>0.7V (built-in potential of Si p-n junction). In this work, we have achieved Δn = −1.163 × 10−3 and Δn = −1.179 × 10−3 for outer and inner rings with an external bias voltage of 1V when applied individually on outer and inner rings, respectively at λ = 1.55μm.

Off-axis inner ring as shown in Fig. 2(a) gives rise to the extra resonant notches in the transmission spectrum [28]. Off-axis MRR follows extra resonant conditions which results in the appearance of extra resonant notches in the transmission characteristics of off-axis MRR. These extra resonant notches are impeccably discernible beside the conventional notches due to their high Q-factor. The shift in resonant notches (conventional and extra) with respect to applied bias, Vm and VB can be calculated through propagation matrix P. For the off-axis MRR, propagation matrix is calculated [28] as,

P=[ejϕ100ejϕ1χ]
where,
χ=τ2+(jκ2)2(ejϕ2τ2)1
Half trip phase delay, ϕ1at outer ring and round trip phase delay, ϕ2 at off-axis inner ring can be written as,

ϕ1=2π2neff1R1λ
ϕ2=4π2neff2R2λ

whereτ2 and κ2 are the transmission and coupling coefficients between off-axis inner and outer rings, respectively. The change in ambient temperature of MRR affects the effective RI (neff1 and neff2) of outer and inner rings, modifying the transmission characteristics. Voltages applied at the individual rings change the RI of respective rings only. Propagation matrix P is a function of both ϕ1(neff1) and ϕ2(neff2). A positive voltage to the outer ring modifies Pand blue shifts the conventional resonant notches as well as extra resonant notches. It is found that, in that case the shift in conventional resonant notches is larger in comparison to the extra notches. If we apply a positive voltage at the inner ring, blue-shift in extra notches will be greater than the blue-shift in conventional notches. It has been observed that the shift of extra resonant notches due to the change in neff1 is more than the shift of conventional resonant notches due to the change in neff2. These facts have been theoretically established and validated by FDTD method [28,32]. Mathematically, the phase matching conditions (which cause the conventional notches) for the outer ring can be given by [28],

ϕ1=2π2neff1R1fc=mπ
whereas, phase matching conditions [28] for the off-axis inner ring are,
ϕ1+κ22sin(ϕ2)τ2(1+τ2)cos(ϕ2)(1+τ22)=mπ
where m = 1, 2, 3…

Both ϕ1 and ϕ2 depend on neff1 and neff2, respectively. From Eq. (13) it is viable that change in neff1 (due to the voltage applied to outer MRR) is capable for a slight shift in the extra resonant notches. On contrary, from Eq. (12), it is conspicuous that change in neff2 (due to the voltage applied to the inner MRR) has ideally no effect on the conventional resonant notches. The details are tabulated in Table 1.

Tables Icon

Table 1. Applied voltage at outer and inner MRRs and corresponding resonant notch shifts

This makes us confident to negate the thermal effects on MRR using an extra biasing voltage VB at outer ring. The position of Vm and VB can be swapped at the cost of overall performances. For example, if we apply a message signal at the outer ring and an extra bias voltage to the inner ring, it will take higher voltage and consume more power to compensate the thermal effect as the effect of external bias in the inner microring (or, change in neff2) is negligible on conventional notches. Moreover, due to high Q-factor and narrow bandwidth of the extra resonant notches, only a small message signal applied to inner ring would be sufficient to achieve acceptable ER. Thereof, it is always preferable to apply the message signal at the inner ring. The normalized power transmission of off-axis MRR within optical communication wavelength range is depicted in Fig. 3(a).

 

Fig. 3 Normalized power transmission (in dB) characteristics of an off-axis MRR with change in (a) Vm, (b) VB and (c) transmission characteristics of thermally-compensated off-axis MRR.

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It is conspicuous from inset of Fig. 3(a) that Vm = + 1V results into ~380pm blue-shift of extra resonant notches (negligible blue-shift in conventional notches) which is sufficient to achieve almost 100% (>20 dB ER) modulation due to an ultra-high Q-factor of extra resonant notches of off-axis MRR [28]. Figure 3(b) shows the effect of VB on conventional and extra resonant notches. Applying ~1V at conventional ring (i.e. outer ring) causes a blue shift of ~250-300pm in the conventional resonant notches while ~50-100pm blue shift of extra resonant notches. It is found that per Kelvin temperature rise can incur ~40-50pm and ~50-70pm red shifts for conventional and extra notches, respectively. Even 5K change in temperature may result into 20-30% change in modulation as Q-factor is very high. Temperature effect, a nonlinear phenomenon in silicon waveguides, can be compensated through proper biasing voltage VB at the outer ring. In Fig. 3(c), it is shown that VB = 3.4V has nullified the effect of 20K rise in temperature. The solid blue curve shows an extra resonant notch at T = 300K with no voltage bias applied at outer ring. When 1V modulating signal is applied to the inner ring the notch is blue-shifted towards the left-hand side as shown by the solid violet curve. At 1V, the off-axis MRR renders an extinction ratio of ~38dB. The solid red curve shows the transmission of the MRR with 20K rise in temperature without modulating signal while the solid green curve depicts the transmission of the MRR with 20K rise in temperature with an applied message signal of 1V to the inner ring. The solid black curve represents the transmission characteristics of MRR with a temperature compensating extra bias VB = 3.4V applied to the outer ring. This bias voltage can counterbalance the thermal effect, which is clearly visible from the inset where the solid black curve overlaps the solid violet curve. Therefore, for the Fig. 3(c) one can write,

ΔλVm=1V=ΔλΔT=20K+ΔλVB=3.4V+ΔλVm=1V
where ΔλVm=1V,ΔλΔT=20KandΔλVB=3.4Vrepresent the shift in resonant wavelength due to Vm = 1V, ΔT = 20K and VB = 3.4V, respectively.

The athermal behavior has been further validated and corroborated through eye diagrams. Figure 4(a) shows the waveform of user-defined pre-emphasized message signal. NRZ signal is passed through a low noise amplifier-differentiator block and the output is added with the original NRZ signal itself to provide pre-emphasized NRZ signal. Using pre-emphasized input, rise time, tr and fall-time, tf can be reduced significantly (as small as 100ps and 40ps respectively) [18]. Figure 4(b) shows the eye-diagram for the optical transmission in room temperature.

 

Fig. 4 (a) Pre-emphasized modulating signal waveform. (b) Eye-diagram of transmission at room temperature (c) Eye-diagram at ΔT = 5K and VB = 0V. (d) Voltage compensated eye-diagram of the transmission at ΔT = 5K. (e) Eye-diagram at ΔT = 10K. (f) Voltage compensated eye-diagram at ΔT = 10K.

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The parameters μ1 and μ2 refer to the eye amplitude levels, the standard deviations (SDs) σ1 and σ2 measure the noises at the levels μ1 and μ2, respectively. The eye-opening is defined as μ2μ1 which varies from 0 to 1 without thermal effects. The eye signal-to-noise ratio (SNR) can be defined as:

SNR=μ2μ1σ1+σ2

From TMM, it has been found numerically that the modulator amplitude varies from ~0.42 to 1 (3.82dB) at 5K rise in temperature. Consequently, the eye squints, which is demonstrated in Fig. 4(c). Figure 4(d) establishes that an extra bias of 2.2V at the outer ring can nullify the thermal effect. It is clearly observed that(μ2μ1)>(μ25μ15), σ25>σ2 and σ15>σ1 due to the thermal effects, but at the same time(μ2μ1)(μ25μ15), where the superscript 5 has been used to explicate the amplitude levels at ΔT = 5K and a prime is used in superscript to indicate the thermally compensated eye-levels. One should also note carefully that even after the recompense, σ25σ2 and σ15σ1 as the thermal noise in Si-waveguide remains intact and cannot be eliminated through this procedure. If temperature is raised to 10K the eye opening further decreases and the transmission amplitude varies from 0.65 to 1 (swing 1.86 dB). The closed eye can be reopened by counteracting the thermal effect using a bias voltage VB of ~2.82V. These two cases have been delineated in Figs. 4(e) and 4(f). Further, an additional voltage of only 0.58V can negate the temperature effect of 20K.

Padamraju et al., has clearly mentioned that thermal compensation with control-based solution consumes more power than athermal based solutions, which is the main disadvantage of the control-based solutions. Our process can be conceived as control-based solution. Besides the fabrication complexity, low fabrication tolerance and the requirement of an extra power source are the challenges in the off-axis MRR. While the thermal noise cannot be eliminated, however this does not affect the performance of the proposed device. Moreover, the RI change with respect to the bias voltage is nonlinear due to the exponential I-V characteristics [30] of the PIN diode. Until the device breaks down, this characteristic enables the device worthwhile to control the temperature effects more easily when the bias voltage is already very high. Note that, the extra bias voltage (ΔVB) applied to the outer ring in order to compensate the thermal effect will be lesser for the same amount of temperature rise (ΔT) while the temperature is already high in comparison to the required feedback voltage for low initial temperature value, for example, if ΔVB1 and ΔVB2 are the bias voltages to compensate the effect of same amount of temperature rise ΔT at two different room temperatures T1 and T2 (T1T2), respectively, then within the break-down limit of the SOI EOM one can write mathematically,

ΔVB1>ΔVB2
The power consumption in the proposed device is therefore not high in comparison with the other control-based solutions, which is one of the important features of the proposed device. Apart from previously reported techniques [13,17], dynamic temperature control can also be achieved using proper thermal coefficient of the resistance RT (i.e. ΔRTT) (Fig. 1a) by choosing suitable material and doping concentration. An ultra-sensitive on-chip optical MRR sensor, temperature monitoring integrated photo-detector or fibre Bragg grating sensor can be used to detect the ambient temperature. One can also use pre-calibrated external feedback mechanism to control and maintain the required value of VB dynamically to compensate thermal effect. Other possibilities are still under investigations.

4. Conclusion

To conclude, off-axis MRR EOM can obviate the thermal effects for wide range of temperatures very effectively through an extra ring within the same MRR structure which not only creates highly sensitive and sharp resonant notches (with high Q-factor) but also offers another degree of freedom of designing. This renders us an opportunity to build up an electro-optic modulator untarnished by thermal effects within the same compact dimension of a single MRR. The eye diagrams of the proposed off-axis MRR show the potential to overcome the thermal detuning for DWDM applications.

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30. D. Neamen, Semiconductor Physics And Devices: Basic Principles (McGraw-Hill, 2003).

31. K. Okamoto, Fundamentals of Optical Waveguides (Elsevier, 2006) p. 37.

32. R. Haldar, A. D. Banik, M. S. Sanathanan, and S. K. Varshney, “Compact Athermal Electro-optic Modulator Design Based on SOI Off-axis Microring Resonator,” in Proc. Conference on Lasers and Electro-Optics (CLEO:2014), (Optical Society of America, 2010), San Jose, CA, USA, 8–13 June 2014, paper JW2A.37. [CrossRef]  

References

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  • |

  1. L. C. Kimerling, “Photons to the rescue: Microelectronics becomes microphotonics,” Electrochemical Society Interface 9, 28–31 (2000).
  2. D. A. B. Miller, “Optical Interconnects to Silicon,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1312–1317 (2000).
    [Crossref]
  3. A. Alduino and M. Paniccia, “Interconnects: Wiring electronics with light,” Nat. Photonics 1(3), 153–155 (2007).
    [Crossref]
  4. D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88(6), 728–749 (2000).
    [Crossref]
  5. J. D. Meindl, J. A. Davis, P. Zarkesh-Ha, C. S. Patel, K. P. Martin, and P. A. Kohl, “Interconnect opportunities for gigascale integration,” IBM J. Res. Develop. 46(2.3), 245–265 (2002).
    [Crossref]
  6. A. Shacham, K. Bergman, and L. P. Carloni, “Photonic networks-on-chip for future generations of chip multiprocessors,” IEEE Trans. Comput. 57(9), 1246–1260 (2008).
    [Crossref]
  7. M. Lipson, “Compact Electro-Optic Modulators on a Silicon Chip,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1520–1526 (2006).
    [Crossref]
  8. C. Gunn, “CMOs photonics—SOI learns a new trick,” in Proc. IEEE 2005 Int. Silicon on Insulator (SOI) Conf. Oct. 3–6, 7–13 (2005).
  9. B. Jalali, “Silicon Photonics,” Proc. SPIE 3290, 238–245 (1997).
    [Crossref]
  10. Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
    [Crossref] [PubMed]
  11. R. A. Soref and B. R. Bennett, “Kramers–Kronig analysis of electro-optical switching in silicon,” Proc. SPIE 704, 32–37 (1987).
    [Crossref]
  12. Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica (Amsterdam) 34(1), 149–154 (1967).
    [Crossref]
  13. K. Padmaraju and K. Bergman, “Resolving the thermal challenges for silicon microring resonator devices,” Nanophotonics 13(0), 1–14 (2013).
    [Crossref]
  14. K. Padmaraju, D. F. Logan, X. Zhu, J. J. Ackert, A. P. Knights, and K. Bergman, “Integrated thermal stabilization of a microring modulator,” Opt. Express 21(12), 14342–14350 (2013).
    [Crossref] [PubMed]
  15. C. T. DeRose, M. R. Watts, D. C. Trotter, D. L. Luck, G. N. Nielson, and R. W. Young, “Silicon Microring Modulator with Integrated Heater and Temperature Sensor for Thermal Control,” in Proc. Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS), OSA Technical Digest (CD) (Optical Society of America, May 2010), San Jose, paper CA,CThJ3.
    [Crossref]
  16. S. Manipatruni, R. K. Dokania, B. Schmidt, N. Sherwood-Droz, C. B. Poitras, A. B. Apsel, and M. Lipson, “Wide temperature range operation of micrometer-scale silicon electro-optic modulators,” Opt. Lett. 33(19), 2185–2187 (2008).
    [Crossref] [PubMed]
  17. K. Padmaraju, J. Chan, L. Chen, M. Lipson, and K. Bergman, “Thermal stabilization of a microring modulator using feedback control,” Opt. Express 20(27), 27999–28008 (2012).
    [Crossref] [PubMed]
  18. Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s carrier-injection-based silicon micro-ring silicon modulators,” Opt. Express 15(2), 430–436 (2007).
    [Crossref] [PubMed]
  19. Y. Kokobun, N. Funato, and M. Takizawa, “Athermal waveguides for temperature-independent lightwave devices,” IEEE Photon. Technol. Lett. 5(11), 1297–1300 (1993).
    [Crossref]
  20. J. Teng, P. Dumon, W. Bogaerts, H. Zhang, X. Jian, X. Han, M. Zhao, G. Morthier, and R. Baets, “Athermal Silicon-on-insulator ring resonators by overlaying a polymer cladding on narrowed waveguides,” Opt. Express 17(17), 14627–14633 (2009).
    [Crossref] [PubMed]
  21. J. Lee, D. Kim, H. Ahn, S. Park, and G. Kim, “Temperature dependence of silicon nanophotonic ring resonator with a polymeric overlayer,” J. Lightwave Technol. 25(8), 2236–2243 (2007).
    [Crossref]
  22. B. Guha, J. Cardenas, and M. Lipson, “Athermal silicon microring resonators with titanium oxide cladding,” Opt. Express 21(22), 26557–26563 (2013).
    [Crossref] [PubMed]
  23. B. Guha, B. B. Kyotoku, and M. Lipson, “CMOS-compatible athermal silicon microring resonators,” Opt. Express 18(4), 3487–3493 (2010).
    [Crossref] [PubMed]
  24. Y. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
    [Crossref]
  25. A. Biberman, E. Timurdogan, W. A. Zortman, D. C. Trotter, and M. R. Watts, “Adiabatic microring modulators,” Opt. Express 20(28), 29223–29236 (2012).
    [Crossref] [PubMed]
  26. Y. Hu, X. Xiao, H. Xu, X. Li, K. Xiong, Z. Li, T. Chu, Y. Yu, and J. Yu, “High-speed silicon modulator based on cascaded microring resonators,” Opt. Express 20(14), 15079–15085 (2012).
    [Crossref] [PubMed]
  27. Q. Deng, X. Li, Z. Zhou, and Y. Huaxiang, “Athermal scheme based on resonance splitting for silicon-on-insulator microring resonators,” Photon. Res. 2(2), 71–74 (2014).
    [Crossref]
  28. R. Haldar, S. Das, and S. K. Varshney, “Theory and design of off-axis microring resonator for high-density on-chip photonic applications,” J. Lightwave Technol. 31(24), 3976–3986 (2013).
    [Crossref]
  29. R. E. Pierret, Semiconductor Device Fundamentals (Addison-Wesley, 1996).
  30. D. Neamen, Semiconductor Physics And Devices: Basic Principles (McGraw-Hill, 2003).
  31. K. Okamoto, Fundamentals of Optical Waveguides (Elsevier, 2006) p. 37.
  32. R. Haldar, A. D. Banik, M. S. Sanathanan, and S. K. Varshney, “Compact Athermal Electro-optic Modulator Design Based on SOI Off-axis Microring Resonator,” in Proc. Conference on Lasers and Electro-Optics (CLEO:2014), (Optical Society of America, 2010), San Jose, CA, USA, 8–13 June 2014, paper JW2A.37.
    [Crossref]

2014 (1)

2013 (4)

2012 (3)

2010 (1)

2009 (1)

2008 (3)

Y. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[Crossref]

S. Manipatruni, R. K. Dokania, B. Schmidt, N. Sherwood-Droz, C. B. Poitras, A. B. Apsel, and M. Lipson, “Wide temperature range operation of micrometer-scale silicon electro-optic modulators,” Opt. Lett. 33(19), 2185–2187 (2008).
[Crossref] [PubMed]

A. Shacham, K. Bergman, and L. P. Carloni, “Photonic networks-on-chip for future generations of chip multiprocessors,” IEEE Trans. Comput. 57(9), 1246–1260 (2008).
[Crossref]

2007 (3)

2006 (1)

M. Lipson, “Compact Electro-Optic Modulators on a Silicon Chip,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1520–1526 (2006).
[Crossref]

2005 (1)

Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

2002 (1)

J. D. Meindl, J. A. Davis, P. Zarkesh-Ha, C. S. Patel, K. P. Martin, and P. A. Kohl, “Interconnect opportunities for gigascale integration,” IBM J. Res. Develop. 46(2.3), 245–265 (2002).
[Crossref]

2000 (3)

D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88(6), 728–749 (2000).
[Crossref]

L. C. Kimerling, “Photons to the rescue: Microelectronics becomes microphotonics,” Electrochemical Society Interface 9, 28–31 (2000).

D. A. B. Miller, “Optical Interconnects to Silicon,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1312–1317 (2000).
[Crossref]

1997 (1)

B. Jalali, “Silicon Photonics,” Proc. SPIE 3290, 238–245 (1997).
[Crossref]

1993 (1)

Y. Kokobun, N. Funato, and M. Takizawa, “Athermal waveguides for temperature-independent lightwave devices,” IEEE Photon. Technol. Lett. 5(11), 1297–1300 (1993).
[Crossref]

1987 (1)

R. A. Soref and B. R. Bennett, “Kramers–Kronig analysis of electro-optical switching in silicon,” Proc. SPIE 704, 32–37 (1987).
[Crossref]

1967 (1)

Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica (Amsterdam) 34(1), 149–154 (1967).
[Crossref]

Ackert, J. J.

Ahn, H.

Alduino, A.

A. Alduino and M. Paniccia, “Interconnects: Wiring electronics with light,” Nat. Photonics 1(3), 153–155 (2007).
[Crossref]

Apsel, A. B.

Baets, R.

Bennett, B. R.

R. A. Soref and B. R. Bennett, “Kramers–Kronig analysis of electro-optical switching in silicon,” Proc. SPIE 704, 32–37 (1987).
[Crossref]

Bergman, K.

K. Padmaraju and K. Bergman, “Resolving the thermal challenges for silicon microring resonator devices,” Nanophotonics 13(0), 1–14 (2013).
[Crossref]

K. Padmaraju, D. F. Logan, X. Zhu, J. J. Ackert, A. P. Knights, and K. Bergman, “Integrated thermal stabilization of a microring modulator,” Opt. Express 21(12), 14342–14350 (2013).
[Crossref] [PubMed]

K. Padmaraju, J. Chan, L. Chen, M. Lipson, and K. Bergman, “Thermal stabilization of a microring modulator using feedback control,” Opt. Express 20(27), 27999–28008 (2012).
[Crossref] [PubMed]

A. Shacham, K. Bergman, and L. P. Carloni, “Photonic networks-on-chip for future generations of chip multiprocessors,” IEEE Trans. Comput. 57(9), 1246–1260 (2008).
[Crossref]

Biberman, A.

Bogaerts, W.

Cardenas, J.

Carloni, L. P.

A. Shacham, K. Bergman, and L. P. Carloni, “Photonic networks-on-chip for future generations of chip multiprocessors,” IEEE Trans. Comput. 57(9), 1246–1260 (2008).
[Crossref]

Chan, J.

Chen, L.

Chu, T.

Das, S.

Davis, J. A.

J. D. Meindl, J. A. Davis, P. Zarkesh-Ha, C. S. Patel, K. P. Martin, and P. A. Kohl, “Interconnect opportunities for gigascale integration,” IBM J. Res. Develop. 46(2.3), 245–265 (2002).
[Crossref]

Deng, Q.

Dokania, R. K.

Dumon, P.

Funato, N.

Y. Kokobun, N. Funato, and M. Takizawa, “Athermal waveguides for temperature-independent lightwave devices,” IEEE Photon. Technol. Lett. 5(11), 1297–1300 (1993).
[Crossref]

Green, W. M. J.

Y. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[Crossref]

Guha, B.

Haldar, R.

Han, X.

Hu, Y.

Huaxiang, Y.

Jalali, B.

B. Jalali, “Silicon Photonics,” Proc. SPIE 3290, 238–245 (1997).
[Crossref]

Jian, X.

Kim, D.

Kim, G.

Kimerling, L. C.

L. C. Kimerling, “Photons to the rescue: Microelectronics becomes microphotonics,” Electrochemical Society Interface 9, 28–31 (2000).

Knights, A. P.

Kohl, P. A.

J. D. Meindl, J. A. Davis, P. Zarkesh-Ha, C. S. Patel, K. P. Martin, and P. A. Kohl, “Interconnect opportunities for gigascale integration,” IBM J. Res. Develop. 46(2.3), 245–265 (2002).
[Crossref]

Kokobun, Y.

Y. Kokobun, N. Funato, and M. Takizawa, “Athermal waveguides for temperature-independent lightwave devices,” IEEE Photon. Technol. Lett. 5(11), 1297–1300 (1993).
[Crossref]

Kyotoku, B. B.

Lee, J.

Li, X.

Li, Z.

Lipson, M.

Logan, D. F.

Manipatruni, S.

Martin, K. P.

J. D. Meindl, J. A. Davis, P. Zarkesh-Ha, C. S. Patel, K. P. Martin, and P. A. Kohl, “Interconnect opportunities for gigascale integration,” IBM J. Res. Develop. 46(2.3), 245–265 (2002).
[Crossref]

Meindl, J. D.

J. D. Meindl, J. A. Davis, P. Zarkesh-Ha, C. S. Patel, K. P. Martin, and P. A. Kohl, “Interconnect opportunities for gigascale integration,” IBM J. Res. Develop. 46(2.3), 245–265 (2002).
[Crossref]

Miller, D. A. B.

D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88(6), 728–749 (2000).
[Crossref]

D. A. B. Miller, “Optical Interconnects to Silicon,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1312–1317 (2000).
[Crossref]

Morthier, G.

Padmaraju, K.

Paniccia, M.

A. Alduino and M. Paniccia, “Interconnects: Wiring electronics with light,” Nat. Photonics 1(3), 153–155 (2007).
[Crossref]

Park, S.

Patel, C. S.

J. D. Meindl, J. A. Davis, P. Zarkesh-Ha, C. S. Patel, K. P. Martin, and P. A. Kohl, “Interconnect opportunities for gigascale integration,” IBM J. Res. Develop. 46(2.3), 245–265 (2002).
[Crossref]

Poitras, C. B.

Pradhan, S.

Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

Schmidt, B.

Shacham, A.

A. Shacham, K. Bergman, and L. P. Carloni, “Photonic networks-on-chip for future generations of chip multiprocessors,” IEEE Trans. Comput. 57(9), 1246–1260 (2008).
[Crossref]

Shakya, J.

Sherwood-Droz, N.

Soref, R. A.

R. A. Soref and B. R. Bennett, “Kramers–Kronig analysis of electro-optical switching in silicon,” Proc. SPIE 704, 32–37 (1987).
[Crossref]

Takizawa, M.

Y. Kokobun, N. Funato, and M. Takizawa, “Athermal waveguides for temperature-independent lightwave devices,” IEEE Photon. Technol. Lett. 5(11), 1297–1300 (1993).
[Crossref]

Teng, J.

Timurdogan, E.

Trotter, D. C.

Varshney, S. K.

Varshni, Y. P.

Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica (Amsterdam) 34(1), 149–154 (1967).
[Crossref]

Vlasov, Y.

Y. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[Crossref]

Watts, M. R.

Xia, F.

Y. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[Crossref]

Xiao, X.

Xiong, K.

Xu, H.

Xu, Q.

Xu, Q. F.

Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

Yu, J.

Yu, Y.

Zarkesh-Ha, P.

J. D. Meindl, J. A. Davis, P. Zarkesh-Ha, C. S. Patel, K. P. Martin, and P. A. Kohl, “Interconnect opportunities for gigascale integration,” IBM J. Res. Develop. 46(2.3), 245–265 (2002).
[Crossref]

Zhang, H.

Zhao, M.

Zhou, Z.

Zhu, X.

Zortman, W. A.

Electrochemical Society Interface (1)

L. C. Kimerling, “Photons to the rescue: Microelectronics becomes microphotonics,” Electrochemical Society Interface 9, 28–31 (2000).

IBM J. Res. Develop. (1)

J. D. Meindl, J. A. Davis, P. Zarkesh-Ha, C. S. Patel, K. P. Martin, and P. A. Kohl, “Interconnect opportunities for gigascale integration,” IBM J. Res. Develop. 46(2.3), 245–265 (2002).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

M. Lipson, “Compact Electro-Optic Modulators on a Silicon Chip,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1520–1526 (2006).
[Crossref]

D. A. B. Miller, “Optical Interconnects to Silicon,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1312–1317 (2000).
[Crossref]

IEEE Photon. Technol. Lett. (1)

Y. Kokobun, N. Funato, and M. Takizawa, “Athermal waveguides for temperature-independent lightwave devices,” IEEE Photon. Technol. Lett. 5(11), 1297–1300 (1993).
[Crossref]

IEEE Trans. Comput. (1)

A. Shacham, K. Bergman, and L. P. Carloni, “Photonic networks-on-chip for future generations of chip multiprocessors,” IEEE Trans. Comput. 57(9), 1246–1260 (2008).
[Crossref]

J. Lightwave Technol. (2)

Nanophotonics (1)

K. Padmaraju and K. Bergman, “Resolving the thermal challenges for silicon microring resonator devices,” Nanophotonics 13(0), 1–14 (2013).
[Crossref]

Nat. Photonics (2)

A. Alduino and M. Paniccia, “Interconnects: Wiring electronics with light,” Nat. Photonics 1(3), 153–155 (2007).
[Crossref]

Y. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[Crossref]

Nature (1)

Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

Opt. Express (8)

K. Padmaraju, D. F. Logan, X. Zhu, J. J. Ackert, A. P. Knights, and K. Bergman, “Integrated thermal stabilization of a microring modulator,” Opt. Express 21(12), 14342–14350 (2013).
[Crossref] [PubMed]

K. Padmaraju, J. Chan, L. Chen, M. Lipson, and K. Bergman, “Thermal stabilization of a microring modulator using feedback control,” Opt. Express 20(27), 27999–28008 (2012).
[Crossref] [PubMed]

Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s carrier-injection-based silicon micro-ring silicon modulators,” Opt. Express 15(2), 430–436 (2007).
[Crossref] [PubMed]

A. Biberman, E. Timurdogan, W. A. Zortman, D. C. Trotter, and M. R. Watts, “Adiabatic microring modulators,” Opt. Express 20(28), 29223–29236 (2012).
[Crossref] [PubMed]

Y. Hu, X. Xiao, H. Xu, X. Li, K. Xiong, Z. Li, T. Chu, Y. Yu, and J. Yu, “High-speed silicon modulator based on cascaded microring resonators,” Opt. Express 20(14), 15079–15085 (2012).
[Crossref] [PubMed]

B. Guha, J. Cardenas, and M. Lipson, “Athermal silicon microring resonators with titanium oxide cladding,” Opt. Express 21(22), 26557–26563 (2013).
[Crossref] [PubMed]

B. Guha, B. B. Kyotoku, and M. Lipson, “CMOS-compatible athermal silicon microring resonators,” Opt. Express 18(4), 3487–3493 (2010).
[Crossref] [PubMed]

J. Teng, P. Dumon, W. Bogaerts, H. Zhang, X. Jian, X. Han, M. Zhao, G. Morthier, and R. Baets, “Athermal Silicon-on-insulator ring resonators by overlaying a polymer cladding on narrowed waveguides,” Opt. Express 17(17), 14627–14633 (2009).
[Crossref] [PubMed]

Opt. Lett. (1)

Photon. Res. (1)

Physica (Amsterdam) (1)

Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica (Amsterdam) 34(1), 149–154 (1967).
[Crossref]

Proc. IEEE (1)

D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88(6), 728–749 (2000).
[Crossref]

Proc. SPIE (2)

B. Jalali, “Silicon Photonics,” Proc. SPIE 3290, 238–245 (1997).
[Crossref]

R. A. Soref and B. R. Bennett, “Kramers–Kronig analysis of electro-optical switching in silicon,” Proc. SPIE 704, 32–37 (1987).
[Crossref]

Other (6)

C. T. DeRose, M. R. Watts, D. C. Trotter, D. L. Luck, G. N. Nielson, and R. W. Young, “Silicon Microring Modulator with Integrated Heater and Temperature Sensor for Thermal Control,” in Proc. Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS), OSA Technical Digest (CD) (Optical Society of America, May 2010), San Jose, paper CA,CThJ3.
[Crossref]

C. Gunn, “CMOs photonics—SOI learns a new trick,” in Proc. IEEE 2005 Int. Silicon on Insulator (SOI) Conf. Oct. 3–6, 7–13 (2005).

R. E. Pierret, Semiconductor Device Fundamentals (Addison-Wesley, 1996).

D. Neamen, Semiconductor Physics And Devices: Basic Principles (McGraw-Hill, 2003).

K. Okamoto, Fundamentals of Optical Waveguides (Elsevier, 2006) p. 37.

R. Haldar, A. D. Banik, M. S. Sanathanan, and S. K. Varshney, “Compact Athermal Electro-optic Modulator Design Based on SOI Off-axis Microring Resonator,” in Proc. Conference on Lasers and Electro-Optics (CLEO:2014), (Optical Society of America, 2010), San Jose, CA, USA, 8–13 June 2014, paper JW2A.37.
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 (a) Schematic of MRR with off-axis inner ring structure used as a modulator, rsh, rse and RT represent the shunt, series and temperature dependent equivalent resistances of the device, Vm and VB are voltage sources. (b) Cross-section of p+n-n+ SOI waveguide structure.
Fig. 2
Fig. 2 (a) Schematic of an off-axis MRR with necessary design parameters. (b) Effective refractive index variation of outer and inner rings with respect to bias voltage, VB at 1.55μm, (c) spectral variation of guided mode index for different ring radii at different biasing voltages, and (d) loss as a function of applied voltage Vm, at wavelength 1.55μm.
Fig. 3
Fig. 3 Normalized power transmission (in dB) characteristics of an off-axis MRR with change in (a) Vm, (b) VB and (c) transmission characteristics of thermally-compensated off-axis MRR.
Fig. 4
Fig. 4 (a) Pre-emphasized modulating signal waveform. (b) Eye-diagram of transmission at room temperature (c) Eye-diagram at ΔT = 5K and VB = 0V. (d) Voltage compensated eye-diagram of the transmission at ΔT = 5K. (e) Eye-diagram at ΔT = 10K. (f) Voltage compensated eye-diagram at ΔT = 10K.

Tables (1)

Tables Icon

Table 1 Applied voltage at outer and inner MRRs and corresponding resonant notch shifts

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

Δ n = Δ n e + Δ n h = [ 8.8 × 10 22 Δ N + 8.5 × 10 18 ( Δ P ) 0.8 ]
Δ α = Δ α e + Δ α h = [ 8.5 × 10 18 Δ N + 6 × 10 18 Δ P ]
d λ d T = ( n e f f α s u b + n e f f T ) λ 0 n g
d λ d T = n e f f T λ 0 n g
n e f f T = Γ c o r e ( S i ) n c o r e T + Γ c l a d ( p o l y m e r ) n c l a d T + Γ s u b n s u b T
Δ N ( x ) = n p 0 ( e e V k T 1 ) exp ( x p + x L p )
Δ P ( x ) = p n 0 ( e e V k T 1 ) exp ( x n + x L n )
P = [ e j ϕ 1 0 0 e j ϕ 1 χ ]
χ = τ 2 + ( j κ 2 ) 2 ( e j ϕ 2 τ 2 ) 1
ϕ 1 = 2 π 2 n e f f 1 R 1 λ
ϕ 2 = 4 π 2 n e f f 2 R 2 λ
ϕ 1 = 2 π 2 n e f f 1 R 1 f c = m π
ϕ 1 + κ 2 2 sin ( ϕ 2 ) τ 2 ( 1 + τ 2 ) cos ( ϕ 2 ) ( 1 + τ 2 2 ) = m π
Δ λ V m = 1 V = Δ λ Δ T = 20 K + Δ λ V B = 3.4 V + Δ λ V m = 1 V
S N R = μ 2 μ 1 σ 1 + σ 2
Δ V B 1 > Δ V B 2

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