A simple model system to study coupled photonic crystal microcavities
aa r X i v : . [ phy s i c s . e d - ph ] A ug A simple model system to study coupled photonic crystal microcavities
A. Perrier, Y. Guilloit, E. Le Cren, and Y. Dumeige
1, 2, a) Université de Rennes 1, IUT de Lannion, rue Edouard Branly, 22300 Lannion, France Université de Rennes 1, Institut FOTON, 6 rue de Kerampont, 22300 Lannion, France (Dated: 13 August 2020)
In this paper, we designed and experimentally studied several systems of standard coaxial cables with differentimpedances which mimic the operation of so called photonic structures like coupled photonic crystal microcavities.Using elementary cells of half-meter long coaxial cables we got resonances around 100 MHz, a range of frequenciesthat can be easily studied with a standard teaching laboratory apparatus. Resonant mode frequency splitting has beenobtained in the case of double and triple coupled cavities. A good agreement between experimental results and transfermatrix model has been observed. The aim here is to demonstrate that standard coaxial cable system is a very cheap wayand an easy to implement structure to explain to undergraduate students complex phenomena that usually occur in theoptical domain.
I. INTRODUCTION
Optical micro-resonators are of great interest for fundamen-tal studies in optics and for applications in photonics such asintegration of optical sources, optical filtering or bio an chem-ical sensing . One very popular method to integrate opticalmicro-resonators is to create a defect in a periodic photoniccrystals . As it is the case of atomic crystals, the periodicitybreaking creates resonant modes localized in the defect witha resonance frequency lying within the photonic bandgap .The coupling of optical micro-resonators gives additional de-grees of freedom and enables the design of complex photonicstructures, with optimized optical characteristics . As thestates of photons confined in an optical microcavity are similarto confined electron states in atoms, optical micro-resonatorsare often referred to photonic atoms. In this physical pic-ture, coupled resonators will support hybridized states andcan be compared to photonic molecules . From a funda-mental point of view these objects are still the subject of in-tense research efforts in quantum photonics, nonlinear opticsor laser physics . The interaction of the resonant modes oftwo cavities leads to interesting phenomena such as frequencysplitting or induced transparency and can be used for disper-sion management or optical storage . The symmetricfrequency splitting obtained in a photonic molecule composedof three coupled resonators made of whispering gallery moderesonators or photonic crystals microcavities has been used toreach the phase matching condition in the four wave mixingprocess . By increasing the number of coupled resonatorsit is even possible to obtain resonant waveguides or delaylines with applications in optical signal buffering .From another point of view, coaxial cable structures havebeen used to emulate one-dimensional photonic crystals,Bragg mirrors or quasi-periodic photonic structures in theradio-frequency domain . Doing a periodic system con-sisted of two sets of meter long coaxial cables, it is possibleto observe a Bragg diffraction due to reflections at impedancetransitions . Adding an extra cable in the middle of the sys-tem, a Fabry-Perot type resonance appears in the center of the a) Electronic mail: [email protected] stop-band; this effect has already been demonstrated in therange 10 −
50 MHz using coaxial cables . From an edu-cational point of view, the great asset of this approach is thatstudents can build their own mirrors or cavities without theneed of complex technological facilities. This is not possiblein the optical domain since in this case the involved wave-lengths are very short. In this paper we propose to extendand generalize this radio-frequency (RF) analogy to coupledresonator structures made of one-dimensional photonic crys-tal defect cavities and show that usual laboratory equipmentcan be used to teach the basics of nanophotonic circuitry atthe undergraduate level.The paper is organized as follows: in section II we intro-duce the transfer matrix method (TMM) which will be usedall along the paper to model our structures. Then, we detailthe physics underlying the coupling of optical cavities focus-ing on the case of two and three defects. Section III is devotedto the experimental demonstration of the coupling of modelphotonic crystal cavities consisted of coaxial cables with twodifferent impedances of 50 Ω and 75 Ω . II. THEORYA. Periodic structure
Figure. 1.a) shows a finite one-dimensional photonic crys-tal or Bragg mirror made of N identical cells constituted bytwo dielectric material layers of refractive indices n and n and thicknesses ℓ and ℓ . E in , E r and E t are respectivelythe input, reflected and transmitted optical fields. To obtain amaximal reflection at frequency ν the phase accumulated bythe wave after propagation within a unit cell have to be equalto π and thus the following condition must be verified : n ℓ + n ℓ = λ , (1)where λ is the Bragg wavelength. The radio-frequency (RF)analog of this Bragg mirror can be obtained by a periodic sys-tem whose elementary cell consists of two coaxial cables ofdifferent length and impedance. In the RF domain it is moreconvenient to use voltages and we define here V in , V r and V t a)b) FIG. 1. a) Finite one-dimensional Bragg mirror or photonic crystalof period Λ . b) Analog of the photonic crystal made of two coaxialcables of impedance Z and Z and length ℓ and ℓ . z is the spacecoordinate, z = as the input, reflected and transmitted voltages respectively.By introducing the phase velocity v φ and v φ of the cable ofimpedance Z and Z , it is possible to write the Bragg condi-tion as ℓ v φ + ℓ v φ = ν , (2)where ν is the resonant frequency. The propagation along thestructure can be modeled using the TMM . The reflec-tion and transmission at each interface between two media ofimpedance Z i and Z j is obtained via the matrix M i , j definedby M i , j = t j , i (cid:18) r j , i r j , i (cid:19) (3)where r j , i = Z i − Z j Z j + Z i and t j , i = Z i Z j + Z i . At angular frequency ω = πν , the propagation in a layer of thickness ℓ i and phasevelocity v φ i is taken into account thanks to matrix Q i whichreads Q i = (cid:18) exp (cid:0) j ω ℓ i / v φ i − κ i ℓ i (cid:1)
00 exp (cid:0) − j ω ℓ i / v φ i + κ i ℓ i (cid:1)(cid:19) (4)where κ i is the attenuation coefficient of the medium ofimpedance Z i . We define the period of the photonic crystal Λ = ℓ + ℓ ; V + ( z ) and V − ( z ) are the signals propagating re-spectively in the forward and backward directions (see Fig.1). Assuming that m ∈ N , the matrix M associated to the unitcell defined by (cid:18) V + ([ m + ] Λ ) V − ([ m + ] Λ ) (cid:19) = M (cid:18) V + ( m Λ ) V − ( m Λ ) (cid:19) , (5)is thus given by M = M , Q M , Q . (6)
75 80 85 90 95 100 105 110 115 120 1250.00.20.40.60.81.0 P o w e r t r a n s f e r f un c t i o n s n (MHz) R T FIG. 2. Power transmission ( T ) and reflection ( R ) for a periodicstructure made of N =
20 cells with Z = Ω , Z = Ω , v φ = v φ = c and ℓ = ℓ =
50 cm.
We consider now that input and output media have both animpedance Z , thus the matrix M Bragg of a periodic structuremade of N elementary cells is given by M Bragg = M N . (7)In the linear regime, the voltage amplitude transmission t andreflection r can be deduced by using the following relation (cid:18) t (cid:19) = M Bragg (cid:18) r (cid:19) . (8)The power transmission T and reflection R coefficients arethus obtained by T = | t | and R = | r | . Figure 2 gives anexample of the transmission and reflection spectra of a 20-cells periodic structure made of coaxial cables with the fol-lowing parameters: Z = Ω , Z = Ω , v φ = v φ = c and ℓ = ℓ =
50 cm. We consider in this section a loss-less mate-rial ( κ i =
0) for a sake of clarity. The propagation is forbiddenwithin a spectral range of 27 MHz centered at ν =
100 MHz,see Eq. (2), this effects manifests itself by a high reflectionand a low transmission.
B. Defect modes in one-dimensional photonic crystalstructures
By inserting defects or impurities in the periodic structureas shown in Fig. 3, it is possible to create localized modeswhose resonant frequencies appear within the photonic band-gap . The defects have a length ℓ D and an impedance Z D .In this work we focus on structures with one, two or threeidentical defects.
1. Photonic crystal cavities
The first structure is obtained by inserting a single defect .The structure is described in Fig. 3.a): it consists of a one- a) b)c) FIG. 3. Defect resonant structures obtained in a one-dimensionalphotonic crystal: a) one-dimensional photonic crystal cavity, b) twocoupled cavities and c) three coupled cavities. p is the numberof elementary cells of the confinment barriers, the defects have animpedance Z D = Z and a length ℓ D = ℓ , q is the number of elemen-tary cells of the barrier between cavities. Note that in each case inputand output media have an impedance Z . In all the examples givenin this figure, p = q =
80 90 100 110 1200.000.250.500.751.00 T r a n s m i ss i o n T n (MHz) a) b) FIG. 4. Transmission of a one-dimensional photonic crystal cavitymade with parameters given in the caption of Fig. 2 and p = Z ( z ) and voltage V ( z ) spatial distributions givenfor ν = ν . Inset b) zoom of the transmission spectrum close to theresonance ν . dimensional photonic crystal with N = p cells where an ex-tra layer of length ℓ D = ℓ and impedance Z D = Z is addedafter p periods. This structure can also be seen as a singlemode Fabry-Perot resonator made of two Bragg mirrors and acentral layer of optical length λ . The matrix M D associatedto this structure is given by: M D = M p Q M p , (9)and can be used to determine the spectral response of thedefect-structure. Figure 4 shows the transmission spectrumof such a structure with p =
10 corresponding to the peri-odic structure studied in Fig. 2 with a single defect. More-over, the inset a) of Fig. 4 gives the voltage (or field) V ( z ) = V + ( z ) + V − ( z ) and impedance Z ( z ) distributions in-side the structure at ν = ν . An inspection of these two plotsdemonstrates clearly that a mode localized in the defect ap-pears in the middle of the photonic bandgap at the frequency ν =
100 MHz. The inset b) of Fig. 4 is a zoom of thetransmission peak which displays a Lorentzian profile. Thisresonator can also be described in the coupled mode theory(CMT) framework by writing the evolution equation ofthe localized mode amplitude a ( t ) shown in Fig. 3.a): dadt = (cid:18) j ω − τ (cid:19) a ( t ) + r τ e V in ( t ) , (10)where ω = πν , τ e is the coupling rate of the mode to theexternal media through the Bragg mirrors and τ is the modeamplitude lifetime. Since we have neglected the losses, wehave τ = τ e . The output signal can thus be written as V t ( t ) = q τ e a ( t ) . In the stationary regime at angular frequency ω , a ( t ) = ˜ ae j ω t , V in ( t ) = ˜ V in e j ω t and V t ( t ) = ˜ V t ( ω ) e j ω t . It isstraightforward to solve Eq. (10) and the transmission of thesystem then reads T ( ω ) = (cid:12)(cid:12)(cid:12)(cid:12) ˜ V t ( ω ) ˜ V in (cid:12)(cid:12)(cid:12)(cid:12) = τ ( ω − ω ) + τ . (11)The transmission resonance has thus a Lorentzian shape asshown in the inset b) of Fig. 4, its width ∆ν is related to themode amplitude lifetime and the quality factor Q of the cavityby Q = ω τ = ν ∆ν . (12)
2. Two coupled photonic crystal cavities
Figure 2.b) shows a system composed of two coupled cav-ities. It consists of two identical defect separated by a barrierwith q cells . The input and output coupling is obtained bytunneling through two barriers with p periods. The matrix ofthe whole system reads M D = M p Q M q Q M p (13)Calculations of the transmission are shown Fig. 5 for q = q =
15 in the case of 10-period input and output mirrors.The transmission spectra show that the initial resonance fre-quency ν is split into two high and low resonant frequencies.This can be understood by writing the evolution equations ofthe two resonant mode amplitudes a and a defined in 3.b)via vector a = ( a , a ) T d a dt = Ka ( t ) + r τ e v in ( t ) (14)where matrix K is given by K = (cid:18) j ω − τ j γ j γ j ω − τ (cid:19) , (15) T r a n s m i ss i o n T n (MHz) q=13 q=15 n > n < FIG. 5. Two coupled photonic crystal cavity transmission spectrafor q =
13 and q =
15. In both cases the calculations have beencarried out with p = ν > and ν < are respectively the high and lowresonant frequencies of the two split modes. -75-50-250255075 -15 -10 -5 0 5 10 15-50-250255075 Z (W) V (arb. u.) a) n=n > b) n=n < z-z (m) FIG. 6. Impedance Z ( z ) and voltage V ( z ) distributions for a doublecavity system obtained for p =
10 and q =
13. a) ν = ν > antisym-metric mode; b) ν = ν < symmetric mode. z is the position of thecenter of the structure. and v in ( t ) = ( V in ( t ) , ) T . In this case τ = τ e and | γ | is thecoupling rate between the two cavities. The eigenvalues of K are j ω S , AS = j ( ω ± γ ) − τ and are associated to the sym-metric and antisymmetric resonant modes of the whole sys-tem. As we consider a loss-less material, γ is real; for an evenvalue of q we have γ > ν > = Re ( ω S ) π and ν < = Re ( ω AS ) π , whereas for an odd value of q , γ < ν > = Re ( ω AS ) π and ν < = Re ( ω S ) π . This control of the sign ofthe coupling coefficient is similar to what has been observedin two dimensional photonic crystal molecules . Figure 6gives the voltage distribution V ( z ) for p =
10 and q =
13 forthe two frequencies ν > an ν < described in Fig. 5. Since q isan odd number, the voltage profile is symmetric with respectto the center of the barrier for ν < and antisymmetric for ν > inboth cases the two mode are localized at the defect locations. T r a n s m i ss i o n T n (MHz) ~ FIG. 7. Transmission spectra for q =
10 and p =
10 in the case of adouble ( k =
2) and a triple ( k =
3) coupled cavity systems. e γ = γ π isthe frequency splitting in Hz. Moreover, by increasing q the coupling between the two cavi-ties is reduced which leads to a smaller frequency splitting asillustrated in Fig. 5.
3. Three coupled photonic crystal cavities
This reasoning can be generalized to k defects, in this casethe transfer matrix M kD is given by M kD = M p Q ( M q Q ) k − M p (16)In Fig. 3.c) we have sketched a coupled cavity system with k = p =
10 and q =
10 is shown in Fig. 7. The initial mode is nowsplit in three modes. A more quantitative study of this splittingcan be carried out using the CMT using Eq. (14) where a = ( a , a , a ) T (see Fig. 3.c) and v in ( t ) = ( V in ( t ) , , ) T . For k =
3, matrix K is now given by K = j ω − τ j γ j γ j ω j γ j γ j ω − τ . (17)If the coupling rate is large enough ( | γ | ≪ τ ) the new resonantfrequencies are the eigenvalues of K which read j ω − τ j ( ω + γ √ ) − τ j ( ω − γ √ ) − τ . (18)Hence, the split resonances have a bandwidth which is halfof that of the central resonance and the frequency splitting isincreased by a √ p =
10 and q = AWG W W OscilloscopeCoaxial cables
FIG. 8. Experimental setup. AWG: arbitrary waveform generatordelivering an harmonic signal with an amplitude of 1 V. Coaxialcables: periodic or defect structures made of coaxial cables RG-58/Uand RG-59/U.
III. EXPERIMENTSA. Experimental setup
Experiments have been carried out using two sets of 50 cmlong RG-58/U and RG-59/U coaxial cables whose nominalimpedances are respectively 50 Ω and 75 Ω . Attenuation ofcables are taken into account in the TMM by using the follow-ing values for the attenuation coefficients κ ( ν ) = . × − ν + . × − √ ν (19) κ ( ν ) = . × − ν + . × − √ ν , (20)where ν is given in Hz. For both cable sets the phase ve-locity is v φ = v φ = . c . The experimental setup is de-scribed Fig. 8, the electromagnetic waves with frequencyaround 100 MHz are produced by an arbitrary waveform gen-erator SIGLENT SDG6022X and analyzed using an oscillo-scope Tektronix DPO4104. The transmission spectra are ob-tained by measuring the RMS values of the voltage at the in-put and at the output of the coaxial cable structures at severalfrequencies. B. Bragg mirror and photonic crystal cavity
The first experiment consisted in measuring the transmis-sion of a periodic structure (see Fig. 1.a) made of N = ℓ = ℓ = . ν = . Z and Z . In therest of this work we will use the impedance values Z = Ω and Z = . Ω obtained from the best fit of the periodicstructure transmission experimental data. The photonic crys-tal cavity or single defect resonant structure is obtained byadding an extra cable of impedance Z (see Fig. 3.a). Theassociated experimental transmission spectrum is given Fig.9.b). At the center of the photonic bandgap ( ν ), the transmis-sion spectrum shows an attenuation-limited resonance with aspectral width ∆ν = . Q = .
5. The calculations (full line) which have beencarried out using the length and impedance values obtained
Experiment Fit - TMM T r a n s m i ss i o n T a)b) Experiment TMM T r a n s m i ss i o n T n (MHz) FIG. 9. a) Transmission of a periodic structure with N =
10. b)Single defect structure ( k =
1) with p =
5. Circles: measurements;full lines: calculations obtained by using the TMM. from the previous fit without adjusting any parameter are ingood agreement with the experimental data.
C. Two coupled photonic crystal cavities
In this section we give the experimental results obtained forthe system made of two cavities shown in Fig. 3.b) and an-alyzed from a theoretical point of view at section II B 2. The
60 80 100 120 1400.00.20.40.6 60 80 100 120 1400.00.20.40.660 80 100 120 1400.00.20.40.6 60 80 100 120 1400.00.20.40.6
Experiment TMM T r a n s m i ss i o n T a) q=4 b) q=3c) q=2 d) q=1 T r a n s m i ss i o n T n (MHz) n (MHz) FIG. 10. Transmission spectra of two coupled cavity structures ( p = k =
2) obtained for q values ranging from 4 to 1. c) For q =
2, the measured frequency splitting is ∆ , = . measurements have been done for several values of the num-ber of unit cells separating the two cavities: q ∈ { , } . For q = q , thecavity coupling is increased leading to a stronger frequencysplitting. In particular, for q = ∆ , = .
60 80 100 120 1400.00.20.40.6
Experiment TMM T r a n s m i ss i o n T n (MHz) FIG. 11. Transmission spectrum of a three coupled cavity structure( p = k =
2) obtained with a barrier such as q =
2. The exper-imental frequency splitting is ∆ , = . single defect cavity resonance width. In four cases, the cal-culations plotted in full lines have been carried out using theTMM without adjusting the parameters found at section III B. D. Three coupled photonic crystal cavity
The last experiment consisted in studying a three coupledcavity system ( k =
3) made of two unit cells in the barriers( q = ∆ , = . ∆ , / ∆ , ≈ . √ IV. CONCLUSION
We theoretically and experimentally analyzed the transmis-sion properties of double and triple coupled photonic crys-tal cavities made of coaxial cables with impedance 50 Ω and 75 Ω . We found good agreement between experimen-tal data and simulated spectra obtained using the transfer ma-trix method. This model system is very versatile and easy toimplement which could enable students to design and buildthemselves their own photonic structures during laboratoryclasses. This approach could be extended to chirped pho-tonic structures or composite structures with more than two refractive indices by using cables with higher impedance.Moreover, the instruments required to carry out transmis-sion experiments can be found in any teaching laboratory.Eventually, this approach can illustrate the coupling of res-onators or oscillators which is a fundamental topic in physicsteaching and goes beyond the electrodynamics or photonicsframeworks . ACKNOWLEDGMENTS
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