Selective linewidth control in a micro-resonator with a resonant interferometric coupler
Paula L. Pagano Massimo Borghi Federica Moroni Alice Viola Francesco Malaspina Marco Liscidini Daniele Bajoni Matteo Galli


APS/123-QED





Centro de Investigaciones Ópticas, CONICET-CICBA-UNLP, Camino Centenario y 505 s/n , 1897 Gonnet, Argentina.
Dipartimento di Ingegneria Industriale e dell'Informazione, Università di Pavia, Via Ferrata 5, 27100 Pavia, Italy.

corresponding author: massimo.borghi@unipv.it 



Dipartimento di Fisica, Università di Pavia, Via Bassi 6, 27100 Pavia, Italy.






Dipartimento di Fisica, Università di Pavia, Via Bassi 6, 27100 Pavia, Italy.


Dipartimento di Ingegneria Industriale e dell'Informazione, Università di Pavia, Via Ferrata 5, 27100 Pavia, Italy.


Dipartimento di Fisica, Università di Pavia, Via Bassi 6, 27100 Pavia, Italy.












             


Optical microresonators are characterized by a comb of resonances that preserve similar characteristics over a broad spectral interval. However, for many applications it is beneficial to selectively control of the quality factor (Q) of one or only some resonances. 

In this work we propose and experimentally validate the use of a resonant interferometric coupler to selectively change the Q-factor of a target resonance in an integrated silicon nitride microresonator. We show that its Q-factor can be continuously tuned from 6.5×10^4 to 3×10^6, leaving the untargeted resonances uperturbed. Our design can be scaled to independently control several resonances.




                              
Selective linewidth control in a micro-resonator
 with a resonant interferometric coupler
    Matteo Galli
    April 29, 2024
=========================================================================================







§  INTRODUCTION

Integrated optical microresonators are versatile devices which find use in several applications, such as spectral filters, switches <cit.>, delay lines <cit.>, sensors <cit.>, modulators <cit.> and light sources <cit.>. Microresonators are typically characterized by a comb of resonances that preserve similar characteristics, such as the quality factor or the extinction rate, over a broad spectral interval. This feature is harnessed to generate broadband frequency combs <cit.>, soliton pulses <cit.>, or photon pairs in high dimensional entangled states <cit.>. However, some applications may require the selective and dynamic control of the quality factor (Q) of only some of the resonances. Examples include Q-switched lasers <cit.>, single frequency lasers <cit.>, tunable modulators <cit.>, tunable delay lines <cit.>, optical memories <cit.>, pseudo-random binary sequence generators <cit.> and photon pair sources with tailored spectral correlations <cit.> or high heralding efficiency <cit.>. Several approaches have been investigated to control the Q-factor <cit.>, and some of them are wavelength selective <cit.>. For example, a strong control pulse has been used to induce Raman scattering in silicon on a target resonance <cit.>. However, this operation can be done efficiently only at the maximum of the Raman gain of the material, and demands strict fabrication tolerances <cit.>. 
The Q-factor can also be tuned by varying the phase shift in an unbalanced interferometer forming the coupling region of a microresonator <cit.>. Alternatively, the Q-factor of a single resonance can be changed due to the strong coupling with another resonator <cit.>. 
or by a combination of the previous strategies <cit.>.


In this work, we propose and experimentally demonstrate the selective control of the Q-factor of a single resonance in a microresonator by using a  resonant interferometric coupler. A Main resonator is coupled twice to a bus waveguide to form an unbalanced Mach-Zehnder interferometer (MZI). Then, we introduce an auxiliary resonator (Aux) in one of the arms to impart a wavelength-selective phase. This changes the interference within the MZI and thus the effective coupling coefficient with the input waveguide. The article is structured as follows: in Sec.<ref> we describe the working principle of the device and explore different configurations which allows us to selectively change the Q-factor of a target resonance. The device is then fabricated on a silicon nitride chip, and its experimental characterization is reported in Sec.<ref>. Here we show that the Q-factor of the target resonance can be continuously changed from 6.5×10^4 to 3×10^6 with minimal perturbation of the adjacent ones. 










§   WORKING PRINCIPLE OF THE DEVICE


The proposed structure is sketched in Fig.<ref>. A main ring resonator (Main) of radius R is coupled twice to a bus-waveguide at points A and B, with coupling coefficients κ_A and κ_B respectively, to form an interferometric coupler <cit.>.
An auxiliary resonator of radius R_Aux is also coupled to the bus waveguide 
at point C. 



The highlighted area in Fig.<ref> can be seen as a Mach-Zehnder interferometer (MZI), with a first arm of length π R along the Main ring, and a second arm of length l along the bus-waveguide.

The effective coupling of the Main ring resonator to the bus waveguide depends on the phase difference between the two MZI arms <cit.>. 
Such a difference is given by two distinct contributions: the imbalance in the optical length of the two interferometer arms, and the transmission of the Aux ring.
The former contribution acts at all frequencies, the latter is relevant only when its frequency is within the spectral width of an Aux ring resonance.
Thus, one has two sets of resonances that can be controlled independently: those that belong only to the Main ring and those that are shared with the Aux.
For instance, by choosing incommensurate values for R and R_Aux, it is possible to selectively control one single resonance. Using a microheater, one can adjust the refractive index of the bus waveguide to set the desired coupling for all the other Main resonances.  An additional microheater can act on the Aux to align one of its resonances to the target Main ring resonance.












With reference to Fig.<ref>, the device transmission T at the output port can be written as 

    T = |e^iϕ̃_Mainσ_Aσ_B  δ_MZI - κ_Aκ_B - δ_MZIe^i2ϕ̃_Main/1 - [σ_Aσ_B - κ_Aκ_B  δ_MZI] e^i2ϕ̃_Main|^2,

where σ_j and κ_j are the self- and cross-coupling coefficients at point j ∈{A, B, C}, ϕ̃_Main = β̃π R  and β̃= β + iα/2 is the complex wavevector, which accounts for both propagation (β) and linear attenuation (α). Finally,

    δ_MZI = t_Auxe^iβ̃(l -π R)

describes the attenuation 
and phase difference acquired between the two arms of the MZI, and

    t_Aux = σ_C - e^iϕ̃_Aux/1 - σ_C e^iϕ̃_Aux,

is the transmission amplitude of the Aux resonator, with ϕ̃_Aux = β̃2π R_Aux.







Unsurprisingly, Eq.(<ref>) is equivalent to the transmission of a resonator in the all-pass filter configuration in which the effective coupling coefficient κ_^2 with the input waveguide is given by 

    |κ_|^2= |κ_Aσ_B +κ_Bσ_Aδ_|^2,

which is the expression for the cross port transmission of the MZI, where Δϕ_MZI = β(l -π R) + [t_Aux]. 

We now consider the special case in which l = 3π R. Thus, for all of the resonances that are spectrally far from those of the Aux (for which |δ_|∼ 1 and arg[t_]∼ 0) we have that the phase difference in the MZI is a multiple of 2π, and the interference is constructive at the cross port of the MZI. In this situation, when κ_A = κ_B, the effective coupling is maximum and equal to |κ_|^2 = 4κ_A^2σ_A^2. Changing Δϕ_MZI through a micro-heater placed in the long arm of the MZI modifies κ_^2 for all the resonances. However, if the Aux resonator is used to change the interference condition at the MZI, the effective coupling is modified only in the neighborhood of the Aux resonances. It is then possible to vary the effective coupling of the Main ring from a completely uncoupled condition, when interference is destructive, to an over-coupled condition, when the interference is constructive. 


We further set R_Aux=3/4R so that only one resonance out of four in the Main ring will overlap with one of the Aux, leaving the other resonances largely detuned from any Aux resonance. We consider the case in which the Aux is massively over-coupled, so that most of the light is efficiently coupled into it and back to the MZI before it is lost by scattering. 




In Fig.<ref> we show the spectral transmission T, the cosine of Δϕ_MZI, and the intensity enhancement (IE) in the Main ring for different MZI and Aux configurations. The spectral position of the Aux resonances can be clearly seen as sharp variations in the otherwise smooth sinusoidal oscillations of cos(Δϕ_MZI). In Fig.<ref>(a), β (2π R) = 2mπ, so κ_^2 is maximized and the Main ring is maximally over-coupled. The Aux is detuned from all the Main resonances, which are then unperturbed and have the same value of the IE.  
In Fig.<ref>(b) we show the response of the structure with an Aux resonance matching a target Main resonance at the wavelength λ_0, which is highlighted in yellow. The Main ring becomes completely uncoupled at λ_0 due to the additional phase imparted by the Aux resonator, which causes Δϕ_MZI(ω_0)=π and sets the MZI to a completely destructive interference at its cross port. 

Figure <ref>(c) shows the effect of a slight detuning of the MZI from the configuration shown in Fig.<ref>(b). Here Δϕ_MZI>0 for all the resonances, and the π phase shift given by the Aux does not uncouple the target resonance anymore because it causes Δϕ>π. 
We choose the MZI detuning such that the target resonance is brought to a critical coupling condition when it is overlapped by the Aux (κ_=κ_cc, where κ_cc is the coupling coefficient for which the intrinsic loss perfectly balances the coupling loss with the input waveguide), while the other ones are still over-coupled. This condition is also clearly witnessed from the large IE at the target resonance.

In the last configuration, reported in Fig.<ref>(d), we show that reverse behavior is possible as well: a target resonance in a comb of critically coupled ones can be brought to overcoupling due to the use of the Aux resonator. The IE at the target resonance is now greatly reduced compared to the neighboring ones.
In Sec.<ref> we will show that the Aux resonator can be swept continuously across the Main resonance, thus realizing all the coupling conditions described above. 











§  EXPERIMENTAL RESULTS

The device described in Sec.<ref> is fabricated on a silicon nitride photonic chip. An optical microscope image is shown in Fig.<ref>(a). Three metallic heaters allow us to respectively change the resonance wavelength of the Main ring, of the Aux resonator, and the phase of the bus-waveguide arm ϕ_=e^iβ d through the thermo-optic effect.
The sample is mounted on a holder that is thermally stabilized, and the metallic micro-heaters are controlled by a voltage driver module. 
The waveguide has width of  and  thickness of , while the gap between the Main resonator and the bus-waveguide is .

The setup used to probe the device transmission T is shown in Fig. <ref>(a). A tunable laser is coupled to a fiber polarization controller (FPC) and the polarization is controlled to match that of the TE mode of the silicon nitride waveguide.

Light is coupled into and out of the chip through a fiber array of UHNA4 fibers with a coupling loss of approximately 1.7 dB/facet. Then, the output signal is detected by a powermeter and recorded in real-time by an oscilloscope. 



Initially, ϕ_ is set to ensure that all the resonances of the Main resonator which are not perturbed by the Aux are critically coupled. This configuration mimics that shown in Fig.<ref>(d).

Figure <ref>(b) shows the device transmission when the Aux resonances are all detuned from those of the Main resonator. We can distinguish the Main from the Aux resonances from their different free spectral range (FSR),  linewidth and extinction. In particular, the resonances of low extinction are those of the Aux resonator because it is heavily overcoupled. 
The Main resonator has an FSR of 1.56 nm. From the resonance linewidth in the critical coupling configuration we estimated an intrisic Q-factor of ∼ 3.2× 10^6 and κ_A^2∼ 0.03. The FSR of the auxiliary resonator is 2.1 nm and κ_C^2 ∼ 0.21, yielding a Q of ∼ 23000.








To demonstrate the change of the linewidth of a single resonance at λ_0=1542.97 (indicated with an arrow in Fig.<ref>(b)) in the Main resonator, we recorded the transmission spectra at different detunings Δλ_=λ_-λ_0 between the target resonance and the closest auxiliary resonance at λ_. We then define the normalized distance between the two resonances as Δλ_/γ_, where γ_ = 0.067 nm is the linewidth of the Aux resonance. The wavelength of the Aux resonance is swept from Δλ_/γ_ = 5 to Δλ_/γ_∼ -2 in small steps.  
The different spectra are shown stacked in Fig.<ref>(a), while Fig.<ref>(b) reports T calculated using Eq.(<ref>), which shows a good agreement with the experimental results. 
From the stacked spectra in Fig.<ref>(a) we extracted the extinction and the linewidth Δλ of the target resonance. In  Fig.<ref>(c) it is shown the quantity λ/Δλ, which coincides with the Q-factor when the Aux is highly detuned from the target Main resonance, as the latter restores its natural Lorentzian shape. 

The simultaneous analysis of the extinction and of the linewidth allows us to assess how the effective coupling κ_^2 between the Main resonator and the bus-waveguide changes due to the phase imparted by the Aux resonator. When Δλ_/γ_∼ 2.5, the extinction is almost zero, and λ/Δλ approaches the value of the intrinsic Q. In this configuration, the resonator is totally under-coupled at λ_0. This is also confirmed from the simulation of κ_^2 as a function of  Δλ_/γ_ (inset in Fig.<ref>(b)), where κ_ reaches its minimum value at Δλ_/γ_∼ 2.5. At Δλ_/γ_∼ 1.4, the extinction is maximized (∼-10 dB) and λ/Δλ is half the intrinsic Q. This implies that the resonance is critically coupled, meaning that κ_eff=κ_cc at λ_0, as shown in the inset of Fig.<ref>(b). When Δλ_/γ_→ 0, we have that λ/Δλ reaches the minimum value of 6.5×10^4, with an extinction of -0.25 dB, indicating an highly over-coupled resonance which is consistent with the maximum in κ^2_eff shown in the inset of Fig.<ref>(b).

To summarize, by sweeping the Aux resonance across the target resonance of the Main, we can continuously control λ/Δλ between 6.5×10^4 and ∼ 3× 10^6, realizing all the coupling regimes described in Sec.<ref>.
To confirm that the control of the Q-factor is wavelength-selective, in Fig.<ref>(c) we also report the linewidth of a resonance at λ_n∼ 1544.6 nm, which is largely detuned from all Aux resonances. As  Δλ_ changes, the Q-factor reMains mostly unperturbed. 
The stacked spectra around the resonance at λ_n are shown in Fig.<ref>(d). The slight red-shift of the resonance wavelength is caused by the thermal cross-talk between the micro-heater on top of the Aux resonator and the Main resonator.


















































§ CONCLUSIONS
 
We proposed and experimentally validated an integrated photonic device consisting of a microresonator where the Q-factor of a single resonance can be selectively changed, leaving that of the other resonances unperturbed. This is accomplished by a resonant interferometric coupler, in which an auxiliary resonator allows us to change the interference condition in a very narrow-band spectral range. We theoretically investigated several configurations, in which one  resonance of the Main resonator is tuned from being maximally over-coupled to be totally under-coupled, leaving the other resonances unchanged.   
We experimentally validated the working principle of the device on the silicon nitride platform. The fabricated device enables the selective switching of the quality factor of a target resonance from 6.5× 10^4 to 3× 10^6. 
By adding more than one Aux resonator in the long arm of the MZI we could target different resonances and independently control them. Thus, our design is highly modular and scalable.
Many applications that require the resonant control of the light matter interaction in nonlinear and quantum optics could benefit from our structure. Examples includes Q-switched lasers, wavelength-selective integrated parametric oscillators, entangled photon pair generation and photon pair sources with simultaneously high brightness heralding efficiency.






§ ACKNOWLEDGEMENTS

P.L.P. acknowledges funding from the HyperSpace project (project ID 101070168). D.B acknowledges the support of Italian MUR and the European Union - Next Generation EU through the PRIN project number F53D23000550006 - SIGNED. M.B., M.G and M.L. acknowledge the PNRR MUR project PE0000023-NQSTI. All the authors acknowledge the support of Xanadu Quantum Technology for providing the samples and the useful discussions.






















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