- Email: [email protected]
Comb spectra and coherent optical pulse propagation in a size-imbalanced coupled ring resonator
Optics Communications 396 (2017) 44–49
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Comb spectra and coherent optical pulse propagation in a size-imbalanced coupled ring resonator
MARK
⁎
Ryuta Suzuki, Makoto Tomita
Department of Physics, Faculty of Science, Shizuoka University, 836, Ohya, Suruga, Shizuoka 422-8529, Japan
A R T I C L E I N F O
A BS T RAC T
Keywords: Coherent optical effects 0π pulse Resonator Pulse propagation
Transmission spectra and coherent optical pulse propagation though a size-imbalanced coupled ring resonator are investigated, where the size of the first ring is extremely large and has a narrow free-spectral-range with an extremely high Q-value, and the second ring is small with a moderate Q-value. The system shows characteristic comb spectra due to interference effects between the two resonators. When an arbitrary-shaped coherent pulse propagates through this system, a series of oscillating output pulses appears. It is shown that this pulse train develops into coherent 0π optical pulses.
1. Introduction Optical resonators have been highly significant in optics and photonics. The high spectral selectivity provided by resonators has been used in spectroscopic applications, and their ability to enhance light–matter interactions demonstrates their potential in nonlinear optics, as well as quantum manipulation of photons [1]. Besides traditional applications, there have also been a number of new developments; optical-mechanical phenomena in micro cavities [2,3], interaction-free measurements in optical resonators [4], and perfect absorption in critically coupled resonators [5] are examples where the unique properties of resonators have been used. When two resonators are coupled, interference effects provide further interesting and important characteristics. For example, coupled-resonator-induced transparency (CRIT) has attracted extensive interest. A coupled resonator can be described as analogous to a Λtype three-level atomic system [6–8] and exhibits similar effects to electromagnetically induced transparency (EIT) in atoms [9]. Slowlight propagation [10] has been investigated using CRIT. When more resonators are directly chained, the system is referred to as a coupled resonator optical waveguide (CROW) [11]. This architecture has been realised in many different material platforms and various types of CROW have been proposed. They were conceived as a new technology for integrated devices, demonstrating potential in applications such as Sagnac effects [12], slow or stopped light, optical switching [13]. Here, we discuss an unusual coupled resonator, where the size of the first ring is extremely large, providing a narrow free-spectral-range (FRS) with an extremely high Q-value [1010], and the second ring is small with a moderate Q-value [108]. Fig. 1(a) shows schematic
⁎
Corresponding author.
http://dx.doi.org/10.1016/j.optcom.2017.03.024 Received 3 February 2017; Received in revised form 8 March 2017; Accepted 8 March 2017 0030-4018/ © 2017 Elsevier B.V. All rights reserved.
illustration of the size-imbalanced coupled resonator, where R1 and R2 are the large and small rings, respectively. Our motivation is illustrated schematically in Fig. 1(b). Recognizing the first ring as a feedback loop, we reconfigured the system (Fig. 1(a)) as a serial array of identical ring resonators (Fig. 1(b)). In contrast to CROW systems, the resonance frequency and Q-value of the resonators in the equivalent system of the serial array of resonators can be set to be identical. Using this system, we can investigate the propagation of pulses through a serial array of identical ring resonators. We investigated transmission spectra and coherent optical pulse propagation through the system. The spectra showed a characteristic comb structures due to the interference effect between the first and second resonators. When coherent optical pulses propagate through this system, a series of oscillating pulse trains appeared, related to the spectral comb structures. It is shown that this pulse train develops into coherent 0π optical pulses. That is, the pulse area, defined by the time integral of the slowly varying envelope of the electric fields, decayed exactly exponentially to 0π during propagation, independently of the input pulse duration or the input pulse shape, thus strictly obeying the McCall and Hahn [14,15] pulse area theorem in a linear regime [16–19]. The unique feature of the weak coherent 0π pulse is that the decay of the pulse area does not necessarily imply that the pulses lose their energy. The slowly varying electric field envelope oscillates between positive and negative values in such a way that the area theorem is satisfied. Recently, the development of coherent 0π pulses has been demonstrated successfully in a ring resonator with a dynamic recurrent loop [20]. In this dynamic system, a fast optical switch was employed and optical pulses were injected into the dynamic recurrent loop. Pulse states were examined after passing through resonators an arbitrary number of times. The
Optics Communications 396 (2017) 44–49
R. Suzuki, M. Tomita
R2
(a)
R1
C2 Ein (t )
C1 RA
RB
RC
CB
CC
Output
are couplers. R1 (blue line) and R2 (red line) are the large and small rings, respectively. (b) Illustration of a conceptual serial array of ring resonators where RA ,RB ,RC ,…are ring resonators with the same resonance frequency and CA ,CB ,CC ,.., are couplers.
present coupled resonator could be reorganised as a static version of the above dynamic system. The differences between the present static system and the dynamic system are briefly discussed.
2. Size imbalanced coupled resonator and transmission spectra Before the analysis of coherent optical pulse propagation, we first describe the steady state transmission spectra. Fig. 1(a) shows a schematic illustration of the coupled ring resonator, where the first ring R1 is large with an extremely high Q-value and the second ring R2 is small with a moderate Q-value. The steady-state output of the electric field, Eout (ω), normalised by the incident light electric field Ein(ω), is described in a similar manner to conventional coupled resonators [21],
T (ω) exp[iθ (ω)],
where
⎡ y − x R (ω)exp[iφ (ω)] ⎤ 1 2 1 ⎥ Res(ω) = (1 − γ1)1/2 ⎢ 1 ⎢⎣ 1 − x1y1R2(ω)exp[iφ1(ω)] ⎥⎦ ⎡ y − x exp[iφ (ω)] ⎤ 2 2 ⎥ R2(ω) = (1 − γ2 )1/2 ⎢ 2 ⎢⎣ 1 − x 2y2 exp[iφ2(ω)] ⎥⎦
(x1 + y1)2 (1 + x1y1)2
(2)
(3)
The renormalized loss includes the loss and phase shift from the resonance R2 . In Case [I], the coupling conditions for both R1 and R2 are under-coupling conditions, i.e. x1 < y1 and x 2 < y2 . Fig. 2(a) shows the transmission spectrum for this case. The inherent loss of R1 is strong compared with the relevant coupling. As the frequency of the incident light approaches the resonance of R2 , x1(ω) decreases. The coupling of R1 further departs from critical coupling, i.e. x1(ω) < < y1, which results in an increase in Tbottom . This means that the depth of the resonance comb becomes shallow around the resonance frequency of R2 . As a result, the bottom of the envelope function of the comb exhibits a “Λ”shaped structure (Fig. 2(a1)). The comb is on resonance at ω =0; that is, one of the dips in the comb is centred at δω =0 (Fig. 2(a2)). Similarly, the top of the envelope function shows a shallow dip structure around the resonance of R2 because Ttop in Eq. (2) decreases as the frequency of the incident light approaches the resonance of R2 . Fig. 2(b) shows the transmission spectra in case [II], where the coupling conditions for R1 and R2 correspond to under-coupling and over-coupling conditions, respectively; x1 < y1 and x 2 > y2 . A “Λ”-shaped transmission spectrum similar to that in Fig. 2(a1) is obtained. In this case, however, the comb is off resonance at ω =0, i.e. ω =0 is located at the middle of the neighbouring two dips of the comb (Fig. 2(b2)). This occurs because the phase is π rad-shifted when the electric field transmits through the over-coupled R2 . Next, we consider cases [1II] and [IV], where R1 is prepared in the over-coupling condition, i.e. x1 > y1. In contrast to the previous cases of [I] and [II], when the incident light frequency approaches the resonance of R2 , the coupling of R1 approaches critical coupling. The depth of the resonance comb increases around the resonance of R2 . There are two cases. First, when x1(ω = 0) > y1, the bottom of the envelope function of the comb exhibits a “V”-shaped structure (Fig. 2(c1)). For the second case, x1(δω = 0) < y1, the bottom of the envelope function of the comb exhibits a “W”-shaped structure (Fig. 2(d1)). This “W”-shaped structure appears because R1 passes the critical coupling condition, i.e. x1(ω) = y1, twice when the frequency is increased across the resonance of R2 . The frequency corresponding to the critical coupling conditions are denoted as w0 in Fig. 2(d1). In case [III] when R2 is in the under-coupling condition, the comb is off resonance at ω =0. In case [IV] when R2 is in the over-coupling condition, the comb is on resonance at ω =0. Therefore, regarding the coupling condition of R2 , a reversed relationship compared to cases [I] and [II] is observed. Fig. 2(c) and (d) show examples of the transmission spectra for cases [III] and [IV], where “V” and “W”-shaped structures appear.
Fig. 1. (a) Schematic illustration of the size-imbalanced coupled resonator. C1, and C2
Eout (ω) = Res(ω) = Ein(ω)
(1 − x1y1)2
x1(ω) = x1 + R2(ω).
Eout (t )
CA
(x1 − y1)2
When the ring R2 is coupled, the transmission spectra are modulated by the resonance of the second ring, and exhibit a characteristic comb structure. We categorize the transmission spectra in four cases, case [1], [II], [III], and [IV], depending on the coupling conditions in R1 and R2 . For convenience, we introduce a renormalized loss parameter for R1:
Output
Ein (t )
Input
Ttop = (1 − γ1)
Eout (t )
Input
(b)
Tbottom = (1 − γ1)
(1)
ω is detuning frequency from the resonance of R1 and R2 , xi = (1 − γi )1/2exp( − ρi Li /2) and yi = cos(κi ) are the loss and coupling parameters, respectively, ρi is the roundtrip loss, κi is the coupling strength, and γi is the excess loss. φi(ω) = ωnLi / c is the phase shift in the circulation orbit, Li is the length of the ring resonator and n is the effective refractive index, i =1, 2 indicates the first and second resonators, respectively [21]. Note that smaller values of x and y indicate stronger attenuation and stronger coupling, respectively. In the present coupled resonator, we consider a situation where the first ring is large and the free spectral range of the first ring FSR1 satisfies the relationship FSR1 < δν2 , where δν2 is the resonance bandwidth of the second ring R2 . When the ring R2 is decoupled, i.e. y2 = 1, the bottom Tbottom and top Ttop of the comb appear when R1 is on- resonance and offresonance, respectively and obtained from Eq. (1) as
3. Coherent pulse propagation and 0π pulse We analyse the coherent optical pulse propagation through the sizeimbalanced coupled ring resonator. Arbitrary-shaped coherent pulses that propagate through this system transform in a series of oscillating output pulses in a train caused by the comb structures, and the pulse train develops into a weak coherent 0π optical pulse. We denote the slowly varying envelope of the electric field of the pulse of input and output light as Ein(t ) and Eout (t ), respectively. The Fourier transforms 45
Optics Communications 396 (2017) 44–49
R. Suzuki, M. Tomita
1
-1
Intensity (Normalized)
(a2)
(a1)
0 0
0
1 -0.1
0.1
1
FSR 2
(b1)
0 -1
FSR 1
0
(b2) 0.1
0
1 -0.1
1
(c1)
0 -1
0
(c2) 1 -0.1
0.1
0
1
w0 -1
(d2)
(d1)
0
1 -0.1
0
0.1
0
Frequency (Normalized) Fig. 2. Typical examples of the calculated transmission spectra through the imbalanced coupled ring resonator. The coupling conditions of R1 and R2 are (a1) under–under coupling, (b1) under–over coupling, (c1) over–under coupling, and (d1) over–over coupling, respectively. The parameters,x1, y1, x 2 , and y2 are (a) 0.7, 0.9, 0.9, and 0.95; (b) 0.7, 0.9, 0.9, and 0.7; (c) 0.9, 0.7, 0.7, and 0.96; and (d) 0.9, 0.7, 0.9, and 0.7, respectively. In the right column, (a2), (b2), (c2), and (d2) are the expansions of (a1), (b1), (c1), and (d1) around the resonance region of R2 . The red curves in the right column represent the resonance profiles of R2 without R1. FSR1 and FSR2 indicate the free spectral range of ring R1 (large ring) and ring R2 (small ring), respectively. The symbol w0 in (d1) indicates the frequency at the critical coupling condition.
1 2π 1 = 2π
are defined as
Ein(t ) = Ein(ω) =
∞ 1 E (ω)e−iωt dω 2π −∞ in ∞ 1 E (t )eiωt dt . 2π −∞ in
Eout (t ) =
∫
∫
(4)
= The output field from the coupled resonator is
1 2π
∫ Eout(ω)e−iωtdω ∫ Res(ω)Ein(ω)e−iωtdω ⎡ y − x R (ω)exp(iφ ) ⎤
∫ (1 − γ1)1/2⎢⎢⎣ 1 1− x y1 R2 (ω)exp(iφ1 ) ⎥⎥⎦Ein(ω)e−iωtdω 11 2
1/2
=(1 − γ1)
1 2π
∫
⎡ Ein(ω)⎢y1 + ⎢⎣
⎤ {x1R2(ω)exp(iφ1)}N ⎥e−iωt dω ⎥⎦ ∞ ⎛ NnL1 ⎞ = ∑ ηN E ⎜N , t − ⎟ ⎝ c ⎠ N =1
46
1
∞
∑ y1
N −1
(y12 − 1)
N =1
(5)
Optics Communications 396 (2017) 44–49
R. Suzuki, M. Tomita
1.0
1.0
where ∞
θ (0) = μ
∫−∞ E (0, t )dt,
(9)
Transmission Intensity (Normalized)
and
(b)
(a)
0 -4.0
0
0 4.0 -4.0
1.0
1
R2(0) = (1 − γ2 )2
4.0
0
1.0
(y2 − x 2 ) (1 − x 2y2 )
.
(10)
Given the under-coupling condition (x 2 < y2 ), the function R2(ω) satisfies 1 > R2(ω = 0) > 0 , and Eq. (8) is given by
θ (N ) = θ (0)Exp{ − N | ln [R2(0)]|}.
(c)
0 -4.0
0
(d)
0 4.0 -4.0
1.0
Eq. (11) states that the pulse area decays exponentially to 0 π as a function of N , with a decay rate of | ln [Res(ω = 0)]|. Note that this decay constant is independent of details about the pulse, such as the pulse shape or duration. In contrast to the atomic system, the over-coupling condition occurs in ring resonators. In this case, the phase shift in the circulated light appears directly in the transmitted light, and the function R2(ω) satisfies −1 < R2(ω = 0) < 0 ; hence,
4.0
0
1.0
(f)
(e)
0 -4.0
0
0 4.0 -4.0
0
(11)
4.0
θ (N ) = ( − 1)N θ (0)Exp{ − N | ln [{|R2(0)|]|}.
Frequency(MHz)
(12)
In the over-coupling condition, the absolute value of the pulse area |θ (N )| also decays exponentially to 0 π as a function of N . It is important that the fact that the pulse area decays exponentially does not necessarily imply that the pulses lose their energy. The electric-field envelope oscillates between positive and negative values in such a way that the area theorem is satisfied.
Fig. 3. (a)–(d) Experimental observation of the transmission spectra as a function of the detuning frequency. (a), (c) On-resonance frequency region of R2 . (b), (d) Off-resonance frequency region of R2 . (a), (b) Transmission spectra observed without R1. (c), (d) Transmission spectra observed wit R1, where the resonance comb appears. (e), (f) Calculated curves of the transmission spectra corresponding to the experiments in (c) and (d), respectively.
4. Experiments where
1 2π
E (N , t ) ≡
The theoretical results of Eqs. (11) and (12) predict that the pulse area in the output pulse train decays exactly exponentially to 0 π as a function of N , obeying the McCall and Hahn pulse area theorem in the linear regime. We used the experimental setup schematically shown in Fig. 1(a) and performed the 0 π pulse experiment in the typical case (II) discussed in Section 2. The imbalanced ring resonator was created using a single-mode polarisation-maintaining (PM) fibre. The physical lengths of the large (R1) and small (R2 ) rings were L1=230 m and L 2 =6.2 m, respectively. An 80:20 coupler was used for the small ring R2 . The proper value of coupler for R1 depends on N to be observed, as the output pulse intensity is proportional to ηN , given by Eq. (6). In the present study, a 80:20 coupler was used for the large ring R1. The resonance condition of the large ring was extremely sensitive to the laboratory environment. The fibre ring system was temperature-controlled within an accuracy of < 1 mK for stabilisation of the large ring. An Er-fibre laser at 1556 nm was used as the incident light source. The spectral width was 1 kHz, and the laser frequency was tuned by piezoelectric control of the cavity length. For observation of the steady state transmission spectra, a LiNbO3 (LN) modulator was operated in open mode. For the pulse transmission experiments, single-sided exponential pulses with a decay time of τ =80 ns and a repetition rate of 100 kHz were generated using a LN modulator. The average power of the pulses was 0.1 mW. The transmitted profiles of the system were observed using a 1-GHz InGaAs photodetector and recorded using a 600-MHz digital oscilloscope. Note that in the present system, the output profiles of different N values appeared serially in time at intervals of nL1/ c . Fig. 3 shows the transmission spectra as a function of detuning frequency. Fig. 3(a) and (c) show the experimental observation for the on-resonance frequency region of R2 , and Fig. 3(b) and (d) display the experimental observation for the off-resonance frequency region of R2 . Fig. 3(a) and (b) show the transmission spectra observed without R1 (i.e. single resonator of R2 ), and Fig. 3(c) and (d) show the transmission spectra observed with R1 (coupled resonator). Fig. 3(c) and (d) show the comb spectral structure with an FSR1 of 0.9 MHz. The full width at half
∫ {R2(ω)}N Ein(ω)e−iωtdω
1/2 N N −1 2 ⎧ x1 y1 (y1 − 1) N ≠ 0 ⎪ η ≡ (1 − γ ) 1 ⎨ N 1/2 ⎪ N=0 ⎩ η0 ≡ (1 − γ1) y1
(6)
In the derivation of Eq. (5), the denominator of the response function is expanded. The total output electric field Eout (t ) is represented as a sum of sub-electric fields E (N , t ), defined by Eq. (6). When the pulse duration tp is shorter than the circulation time in the ring R1, nL
NnL
tp < < c 1 , each sub electric field E (N , t − c 1 ) appears separately in time and generates a pulse train. The sub electric field E (N , t ) represents the N th pulse in the train and corresponds to the light component that circulated in the ring resonator N times. The pulse area is defined as the integral of the slowly varying electric field with respect to time [14,15]. In coherent optics, this quantity represents the total angle of rotation of the atomic state vector around the electric field. To show that the output pulse train develops into a coherent 0π optical pulse, we defined the pulse area θ (N ) and pulse energy W (N ) for the N th sub-pulse as
θ (N ) = μ ∫
∞
E (N , t )dt
−∞ ∞
W (N ) = κ ∫
−∞
E (N , t ) 2 dt
(7)
respectively, where μ and κ are constants. The pulse area of the th subpulse is calculated as
θ (N ) = μ ∫
∞
−∞
=
1 2π
E (N , t )dt ∞
∞
∫−∞ ∫−∞ μEin(ω){R2(ω)}N e−iωt dtdω ∞
= 2π ∫ δ (ω)μE (0, ω){R2(ω)}N dω −∞ =θ (0){R2(0)}N ,
(8) 47
Optics Communications 396 (2017) 44–49
R. Suzuki, M. Tomita
1
1
Fig. 4. Left column: red curves show the experimental observation of the output pulses through the imbalanced coupled resonator. The blue curves in the left column are the theoretically fitted curves for the output intensity. (a1) the original input pulses N =0. (b1) The sub-pulse of N =1 in output pulse train, (c1) N =2, and (d1) N =3. The right column represents the slowly varying electric field amplitude calculated using the same parameters as the left column. The parameters x 2 = 0.98 , y2 = 0.89 were used in the
of the input pulse transformed into oscillatory structures as N increased. The temporal duration of the earlier oscillation peak became shorter as N increased (Fig. 4(d1)), which is one of the characteristic features of the coherent 0 π pulse [17]. The blue curves in the left column represent the fitted curves to the experimental data. We used Fourier space analyses based on Eqs. (4) and (6). The input pulse was Fourier-transformed once, and the response function was multiplied. The result was then inverseFourier-transformed, and the transmitted pulse profiles were obtained. The green curves in the right column correspond to the electric field amplitude calculated using the same parameters as in the left column. In Fig. 5, the pulse area |θ (N )| and pulse energy W (N ) are plotted as a function of N , as defined by Eq. (7), in which the pre-factor ηN in Eq. (6) was compensated. The pulse area decayed exponentially to 0π, obeying the small-area pulse theorem represented by Eq. (12). The observed decay rates were Γ =−0.37/stage, which is in good agreement with the theoretically calculated value of Γ =−0.37/stage based on Eq. (12) using parameters of x 2 =0.98 and y2 =0.89. In contrast to the pulse area, the pulse energy did not exhibit exponential decay, as shown in Fig. 5. In an ideal system in which there is no energy dissipation, the pulse energy does not decay, while the pulse area decays to zero. In an atomic system, this occurs when the pulse duration is much shorter than the phase relaxation time T2 and the energy relaxation time T1. In cavity systems, this occurs when the pulse duration is much shorter than the cavity decay time Q / ω . In real systems, however, there are inherent dissipations leading to greater or less energy decay (which is not critical to the pulse area decay). This energy decay rate differs from that of simple absorption described by the Lambert–Beer law. The pulse energy did not exhibit exponential decay in Fig. 5.
calculations. All intensities were normalised with respect to the maximum of the input pulse.
5. Discussion
(a1)
0
(b1) N =1
0 1
(c1) N =2
0 1
(d1) N =3
Electric field (Normalized)
Intensity (Normalized)
N =0 0 1
(a2)
-0.5 1
(b2)
0 -0.5 1
(c2)
0 -0.5 1
(d2)
0 -0.5
0
400
800
0
1000
2000
Time(ns)
A single-stage resonator corresponds to a single atom, or at best, a thin atomic gas sheet, in which electromagnetic waves interact only once with the atom rather than propagating over a distance. To emulate the propagation effect in atomic gases, we need to prepare a serial array of resonators. In a previous experiments with the dynamic recurrent system [20], the development of 0 π optical pulses was successfully demonstrated over a large number of recurrence times as the excess loss due to the coupler, i.e. C1 in Fig. 1(a) was absent. Dependences on the pulse shape, pulse duration, and the parameters of the resonator were systematically investigated. In the present system, owing to the inevitable coupling loss, which is relevant to the pre-factor ηN in Eq. (6), it is not easy to observe the output pulses in the large N regions. However, in contrast to the dynamic recurrent system in Ref. [20]., the present system can be recognised as a coupled resonator. This system exhibits the characteristic comb spectral structures relevant to the under- and over-coupling conditions. In traditional coupled resonator systems, the first resonator has a moderate Q-value and the second resonator has a ultra-high Q-value [6–8,10]. In the present sizeimbalanced coupled resonator, therefore, the combination of the Qvalues was inverted with respect to that of the traditional ones. So far, various types of coupled resonators have been proposed in many different material platforms. They were conceived as a new technology in integrated devices. The size-imbalanced coupled resonator may also have unique features for demonstrating phenomena, including optical procedures [22], cavity coherent transient effects, and fast and slow light.
Log[Abs[Pulse area]] Log[Pulse energy]
0
-0.5
-1 -1.5 0
3 1 2 N* (Number of resonator)
Fig. 5. Pulse area |θ (N )| (solid green squares) and the pulse energy W (N ) (solid blue circles) as a function of N , obtained from the experimental data in Fig. 4.
maximum (FWHM) of the resonance dip of R2 was δν =1.7 MHz, in good agreement with the theoretical value of 1.8 MHz calculated by Eq. (1). The free spectral range of R1 (FSR1) was much narrower than that of R2 (FSR2 ). In Fig. 3(c), the envelope function of the comb structure showed a Λ-shaped structure. The spectral width of the Λ-shaped envelope was the same as that of the naked resonance width of R2 shown in Fig. 3(a). The comb was off resonance at ω =0. These experimental observations show good accordance with the theoretically calculated curve in Fig. 3(e) and (f), which indicates that R1 and R2 were prepared in the under-coupling and over-coupling conditions, respectively. Next we examined coherent pulse propagation. The red curves in the left column of Fig. 4 show the experimental observation. Fig. 4(a1) shows the pulse of N =0, i.e. the input pulse. Fig. 4(b1), (c1), and (d1) show the pulse of N =1, 2, and 3, respectively. The long exponential tail
6. Summary We investigated the propagation of optical pulses a though a sizeimbalanced coupled ring resonator, where the first ring was large, and had a narrow free spectral range with an extremely high Q value. Theoretical analyses showed that the output light consisted of an oscillating pulse train. The pulse energy of the sub-pulses in the train 48
Optics Communications 396 (2017) 44–49
R. Suzuki, M. Tomita
[6] D.D. Smith, H. Chang, K.A. Fuller, A.T. Rosenberger, R.W. Boyd, Coupledresonator-induced transparency, Phys. Rev. A 69 (2004) 063804. [7] A. Naweed, G. Farca, S.I. Shopova, A.T. Rosenberger, Induced transparency and absorption in coupled whispering-gallery microresonators, Phys. Rev. A 71 (2005) 043804. [8] D.D. Smith, N.N. Lepeshkin, A. Schweinsberg, G. Gehring, R.W. Boyd, Q.Han Park, H. Chang, D.J. Jackson, Coupled-resonator-induced transparency in a fiber system, Opt. Commun. 264 (2006) 163–168. [9] S.E. Harris, Electromagnetically Induced Transparency, Phys. Today July 36 (1997). [10] K. Totsuka, N. Kobayashi, M. Tomita, Slow Light in Coupled-Resonator-Induced Transparency, Phys. Rev. Lett. 98 (2007) 213904. [11] F. Morichetti, C. Ferrari, A. Canciamilla, A. Melloni, The first decade of coupled resonator optical waveguides: bringing slow light to applications, Laser Photonics Rev. 6 (2012) 74–96. [12] J. Scheuer, A. Yariv, Sagnac effect in coupled-resonator slow-light waveguide structures, Phys. Rev. Lett. 96 (2006) 053901. [13] K. Liu, X.D. Yuan, W.M. Ye, Z.H. Zhu, C. Zeng, Frequency, clockwise to counter clockwise and even to odd lasing mode switching based on optical feedback mechanism, Opt. Commun. 285 (2012) 2316–2319. [14] S.L. McCall, E.L. Hahn, Self-induced transparency by pulsed coherent light, Phys. Rev. Lett. 18 (1967) 908. [15] S.L. McCall, E.L. Hahn, Self-Induced Transparency, Phys. Rev. 183 (1969) 457. [16] M.D. Crisp, Propagation of small-area pulses of coherent light through a Resonant medium, Phys. Rev. A 1 (1970) 1604. [17] J.E. Rothenberg, D. Grischkowsky, A.C. Balant, Observation of the formation of the 0π pulse, Phys. Rev. Lett. 53 (1984) 552. [18] H. Jeong, U. Österberg, Coherent transients: optical precursors and 0π pulses, J. Opt. Sco. Am. B 25 (2008) B1–B5. [19] J.F. Chen, S. Wang, D. Wei, M.M.T. Loy, G.K.L. Wong, S. Du, Optical coherent transients in cold atoms: from free-induction decay to optical precursors, Phys. Rev. A 81 (2010) 033844. [20] Y. Suzuki, M. Tomita, Development of weak coherent 0π optical pulses in a ring resonator with a dynamic recurrent loop, J. Opt. Soc. Am. B 34 (2017) 489–496. [21] K. Totsuka, M. Tomita, Slow and fast light in a microsphere-optical fibre system, J. Opt. Soc. Am. B23 (2006) 2194. [22] T. Oishi, R. Suzuki, P. Sultana, M. Tomita, Optical precursors in coupled-resonatorinduced transparency, Opt. Lett. 37 (2012) 2964–2966.
did not decay exponentially, but the pulse area decayed exponentially to 0π, independently of the pulse duration or the pulse shape, thus strictly obeying the pulse area theorem by McCall and Hahn in the linear regime. We experimentally observed the development of a coherent 0π optical pulse using the size-imbalanced coupled ring resonators. There have been continuous developments in designs in coupled resonators, demonstrating various physical effects. The 0π pulses were analyzed to describe the same physical phenomenon as optical precursors and many results from coherent transient pulse propagation in atoms [18,19]. The present coupled resonator may have unique potential for demonstration and application of phenomena including optical procedures, cavity coherent transient effects, and fast and slow light. Funding Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant number 26287091 ). References [1] K.J. Vahala, Optical microcavities, Nature 424 (2003) 839–846. [2] S. Weis, R. Rivière, Sl. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, TJ. Kippenberg, Optomechanically induced transparency, Science 330 (2010) 1520. [3] Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, KJ. Vahala, O. Painter, Coherent mixing of mechanical excitations in nano-optomechanical structures, Nat. Photonics 4 (2010) 236. [4] T. Tsegaye, E. Goobar, A. Karlsson, G. Björk, M.Y. Loh, K.H. Lim, Efficient interaction-free measurements in a high-finesse interferometer, Phys. Rev. A 57 (1998) 3987. [5] Y.D. Chong, Li Ge, Hui Cao, A.D. Stone, Coherent perfect absorbers: time-reversed lasers, Phys. Rev. Lett. 105 (2010) 053901.
49
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Comb spectra and coherent optical pulse propagation in a size-imbalanced coupled ring resonator
MARK
⁎
Ryuta Suzuki, Makoto Tomita
Department of Physics, Faculty of Science, Shizuoka University, 836, Ohya, Suruga, Shizuoka 422-8529, Japan
A R T I C L E I N F O
A BS T RAC T
Keywords: Coherent optical effects 0π pulse Resonator Pulse propagation
Transmission spectra and coherent optical pulse propagation though a size-imbalanced coupled ring resonator are investigated, where the size of the first ring is extremely large and has a narrow free-spectral-range with an extremely high Q-value, and the second ring is small with a moderate Q-value. The system shows characteristic comb spectra due to interference effects between the two resonators. When an arbitrary-shaped coherent pulse propagates through this system, a series of oscillating output pulses appears. It is shown that this pulse train develops into coherent 0π optical pulses.
1. Introduction Optical resonators have been highly significant in optics and photonics. The high spectral selectivity provided by resonators has been used in spectroscopic applications, and their ability to enhance light–matter interactions demonstrates their potential in nonlinear optics, as well as quantum manipulation of photons [1]. Besides traditional applications, there have also been a number of new developments; optical-mechanical phenomena in micro cavities [2,3], interaction-free measurements in optical resonators [4], and perfect absorption in critically coupled resonators [5] are examples where the unique properties of resonators have been used. When two resonators are coupled, interference effects provide further interesting and important characteristics. For example, coupled-resonator-induced transparency (CRIT) has attracted extensive interest. A coupled resonator can be described as analogous to a Λtype three-level atomic system [6–8] and exhibits similar effects to electromagnetically induced transparency (EIT) in atoms [9]. Slowlight propagation [10] has been investigated using CRIT. When more resonators are directly chained, the system is referred to as a coupled resonator optical waveguide (CROW) [11]. This architecture has been realised in many different material platforms and various types of CROW have been proposed. They were conceived as a new technology for integrated devices, demonstrating potential in applications such as Sagnac effects [12], slow or stopped light, optical switching [13]. Here, we discuss an unusual coupled resonator, where the size of the first ring is extremely large, providing a narrow free-spectral-range (FRS) with an extremely high Q-value [1010], and the second ring is small with a moderate Q-value [108]. Fig. 1(a) shows schematic
⁎
Corresponding author.
http://dx.doi.org/10.1016/j.optcom.2017.03.024 Received 3 February 2017; Received in revised form 8 March 2017; Accepted 8 March 2017 0030-4018/ © 2017 Elsevier B.V. All rights reserved.
illustration of the size-imbalanced coupled resonator, where R1 and R2 are the large and small rings, respectively. Our motivation is illustrated schematically in Fig. 1(b). Recognizing the first ring as a feedback loop, we reconfigured the system (Fig. 1(a)) as a serial array of identical ring resonators (Fig. 1(b)). In contrast to CROW systems, the resonance frequency and Q-value of the resonators in the equivalent system of the serial array of resonators can be set to be identical. Using this system, we can investigate the propagation of pulses through a serial array of identical ring resonators. We investigated transmission spectra and coherent optical pulse propagation through the system. The spectra showed a characteristic comb structures due to the interference effect between the first and second resonators. When coherent optical pulses propagate through this system, a series of oscillating pulse trains appeared, related to the spectral comb structures. It is shown that this pulse train develops into coherent 0π optical pulses. That is, the pulse area, defined by the time integral of the slowly varying envelope of the electric fields, decayed exactly exponentially to 0π during propagation, independently of the input pulse duration or the input pulse shape, thus strictly obeying the McCall and Hahn [14,15] pulse area theorem in a linear regime [16–19]. The unique feature of the weak coherent 0π pulse is that the decay of the pulse area does not necessarily imply that the pulses lose their energy. The slowly varying electric field envelope oscillates between positive and negative values in such a way that the area theorem is satisfied. Recently, the development of coherent 0π pulses has been demonstrated successfully in a ring resonator with a dynamic recurrent loop [20]. In this dynamic system, a fast optical switch was employed and optical pulses were injected into the dynamic recurrent loop. Pulse states were examined after passing through resonators an arbitrary number of times. The
Optics Communications 396 (2017) 44–49
R. Suzuki, M. Tomita
R2
(a)
R1
C2 Ein (t )
C1 RA
RB
RC
CB
CC
Output
are couplers. R1 (blue line) and R2 (red line) are the large and small rings, respectively. (b) Illustration of a conceptual serial array of ring resonators where RA ,RB ,RC ,…are ring resonators with the same resonance frequency and CA ,CB ,CC ,.., are couplers.
present coupled resonator could be reorganised as a static version of the above dynamic system. The differences between the present static system and the dynamic system are briefly discussed.
2. Size imbalanced coupled resonator and transmission spectra Before the analysis of coherent optical pulse propagation, we first describe the steady state transmission spectra. Fig. 1(a) shows a schematic illustration of the coupled ring resonator, where the first ring R1 is large with an extremely high Q-value and the second ring R2 is small with a moderate Q-value. The steady-state output of the electric field, Eout (ω), normalised by the incident light electric field Ein(ω), is described in a similar manner to conventional coupled resonators [21],
T (ω) exp[iθ (ω)],
where
⎡ y − x R (ω)exp[iφ (ω)] ⎤ 1 2 1 ⎥ Res(ω) = (1 − γ1)1/2 ⎢ 1 ⎢⎣ 1 − x1y1R2(ω)exp[iφ1(ω)] ⎥⎦ ⎡ y − x exp[iφ (ω)] ⎤ 2 2 ⎥ R2(ω) = (1 − γ2 )1/2 ⎢ 2 ⎢⎣ 1 − x 2y2 exp[iφ2(ω)] ⎥⎦
(x1 + y1)2 (1 + x1y1)2
(2)
(3)
The renormalized loss includes the loss and phase shift from the resonance R2 . In Case [I], the coupling conditions for both R1 and R2 are under-coupling conditions, i.e. x1 < y1 and x 2 < y2 . Fig. 2(a) shows the transmission spectrum for this case. The inherent loss of R1 is strong compared with the relevant coupling. As the frequency of the incident light approaches the resonance of R2 , x1(ω) decreases. The coupling of R1 further departs from critical coupling, i.e. x1(ω) < < y1, which results in an increase in Tbottom . This means that the depth of the resonance comb becomes shallow around the resonance frequency of R2 . As a result, the bottom of the envelope function of the comb exhibits a “Λ”shaped structure (Fig. 2(a1)). The comb is on resonance at ω =0; that is, one of the dips in the comb is centred at δω =0 (Fig. 2(a2)). Similarly, the top of the envelope function shows a shallow dip structure around the resonance of R2 because Ttop in Eq. (2) decreases as the frequency of the incident light approaches the resonance of R2 . Fig. 2(b) shows the transmission spectra in case [II], where the coupling conditions for R1 and R2 correspond to under-coupling and over-coupling conditions, respectively; x1 < y1 and x 2 > y2 . A “Λ”-shaped transmission spectrum similar to that in Fig. 2(a1) is obtained. In this case, however, the comb is off resonance at ω =0, i.e. ω =0 is located at the middle of the neighbouring two dips of the comb (Fig. 2(b2)). This occurs because the phase is π rad-shifted when the electric field transmits through the over-coupled R2 . Next, we consider cases [1II] and [IV], where R1 is prepared in the over-coupling condition, i.e. x1 > y1. In contrast to the previous cases of [I] and [II], when the incident light frequency approaches the resonance of R2 , the coupling of R1 approaches critical coupling. The depth of the resonance comb increases around the resonance of R2 . There are two cases. First, when x1(ω = 0) > y1, the bottom of the envelope function of the comb exhibits a “V”-shaped structure (Fig. 2(c1)). For the second case, x1(δω = 0) < y1, the bottom of the envelope function of the comb exhibits a “W”-shaped structure (Fig. 2(d1)). This “W”-shaped structure appears because R1 passes the critical coupling condition, i.e. x1(ω) = y1, twice when the frequency is increased across the resonance of R2 . The frequency corresponding to the critical coupling conditions are denoted as w0 in Fig. 2(d1). In case [III] when R2 is in the under-coupling condition, the comb is off resonance at ω =0. In case [IV] when R2 is in the over-coupling condition, the comb is on resonance at ω =0. Therefore, regarding the coupling condition of R2 , a reversed relationship compared to cases [I] and [II] is observed. Fig. 2(c) and (d) show examples of the transmission spectra for cases [III] and [IV], where “V” and “W”-shaped structures appear.
Fig. 1. (a) Schematic illustration of the size-imbalanced coupled resonator. C1, and C2
Eout (ω) = Res(ω) = Ein(ω)
(1 − x1y1)2
x1(ω) = x1 + R2(ω).
Eout (t )
CA
(x1 − y1)2
When the ring R2 is coupled, the transmission spectra are modulated by the resonance of the second ring, and exhibit a characteristic comb structure. We categorize the transmission spectra in four cases, case [1], [II], [III], and [IV], depending on the coupling conditions in R1 and R2 . For convenience, we introduce a renormalized loss parameter for R1:
Output
Ein (t )
Input
Ttop = (1 − γ1)
Eout (t )
Input
(b)
Tbottom = (1 − γ1)
(1)
ω is detuning frequency from the resonance of R1 and R2 , xi = (1 − γi )1/2exp( − ρi Li /2) and yi = cos(κi ) are the loss and coupling parameters, respectively, ρi is the roundtrip loss, κi is the coupling strength, and γi is the excess loss. φi(ω) = ωnLi / c is the phase shift in the circulation orbit, Li is the length of the ring resonator and n is the effective refractive index, i =1, 2 indicates the first and second resonators, respectively [21]. Note that smaller values of x and y indicate stronger attenuation and stronger coupling, respectively. In the present coupled resonator, we consider a situation where the first ring is large and the free spectral range of the first ring FSR1 satisfies the relationship FSR1 < δν2 , where δν2 is the resonance bandwidth of the second ring R2 . When the ring R2 is decoupled, i.e. y2 = 1, the bottom Tbottom and top Ttop of the comb appear when R1 is on- resonance and offresonance, respectively and obtained from Eq. (1) as
3. Coherent pulse propagation and 0π pulse We analyse the coherent optical pulse propagation through the sizeimbalanced coupled ring resonator. Arbitrary-shaped coherent pulses that propagate through this system transform in a series of oscillating output pulses in a train caused by the comb structures, and the pulse train develops into a weak coherent 0π optical pulse. We denote the slowly varying envelope of the electric field of the pulse of input and output light as Ein(t ) and Eout (t ), respectively. The Fourier transforms 45
Optics Communications 396 (2017) 44–49
R. Suzuki, M. Tomita
1
-1
Intensity (Normalized)
(a2)
(a1)
0 0
0
1 -0.1
0.1
1
FSR 2
(b1)
0 -1
FSR 1
0
(b2) 0.1
0
1 -0.1
1
(c1)
0 -1
0
(c2) 1 -0.1
0.1
0
1
w0 -1
(d2)
(d1)
0
1 -0.1
0
0.1
0
Frequency (Normalized) Fig. 2. Typical examples of the calculated transmission spectra through the imbalanced coupled ring resonator. The coupling conditions of R1 and R2 are (a1) under–under coupling, (b1) under–over coupling, (c1) over–under coupling, and (d1) over–over coupling, respectively. The parameters,x1, y1, x 2 , and y2 are (a) 0.7, 0.9, 0.9, and 0.95; (b) 0.7, 0.9, 0.9, and 0.7; (c) 0.9, 0.7, 0.7, and 0.96; and (d) 0.9, 0.7, 0.9, and 0.7, respectively. In the right column, (a2), (b2), (c2), and (d2) are the expansions of (a1), (b1), (c1), and (d1) around the resonance region of R2 . The red curves in the right column represent the resonance profiles of R2 without R1. FSR1 and FSR2 indicate the free spectral range of ring R1 (large ring) and ring R2 (small ring), respectively. The symbol w0 in (d1) indicates the frequency at the critical coupling condition.
1 2π 1 = 2π
are defined as
Ein(t ) = Ein(ω) =
∞ 1 E (ω)e−iωt dω 2π −∞ in ∞ 1 E (t )eiωt dt . 2π −∞ in
Eout (t ) =
∫
∫
(4)
= The output field from the coupled resonator is
1 2π
∫ Eout(ω)e−iωtdω ∫ Res(ω)Ein(ω)e−iωtdω ⎡ y − x R (ω)exp(iφ ) ⎤
∫ (1 − γ1)1/2⎢⎢⎣ 1 1− x y1 R2 (ω)exp(iφ1 ) ⎥⎥⎦Ein(ω)e−iωtdω 11 2
1/2
=(1 − γ1)
1 2π
∫
⎡ Ein(ω)⎢y1 + ⎢⎣
⎤ {x1R2(ω)exp(iφ1)}N ⎥e−iωt dω ⎥⎦ ∞ ⎛ NnL1 ⎞ = ∑ ηN E ⎜N , t − ⎟ ⎝ c ⎠ N =1
46
1
∞
∑ y1
N −1
(y12 − 1)
N =1
(5)
Optics Communications 396 (2017) 44–49
R. Suzuki, M. Tomita
1.0
1.0
where ∞
θ (0) = μ
∫−∞ E (0, t )dt,
(9)
Transmission Intensity (Normalized)
and
(b)
(a)
0 -4.0
0
0 4.0 -4.0
1.0
1
R2(0) = (1 − γ2 )2
4.0
0
1.0
(y2 − x 2 ) (1 − x 2y2 )
.
(10)
Given the under-coupling condition (x 2 < y2 ), the function R2(ω) satisfies 1 > R2(ω = 0) > 0 , and Eq. (8) is given by
θ (N ) = θ (0)Exp{ − N | ln [R2(0)]|}.
(c)
0 -4.0
0
(d)
0 4.0 -4.0
1.0
Eq. (11) states that the pulse area decays exponentially to 0 π as a function of N , with a decay rate of | ln [Res(ω = 0)]|. Note that this decay constant is independent of details about the pulse, such as the pulse shape or duration. In contrast to the atomic system, the over-coupling condition occurs in ring resonators. In this case, the phase shift in the circulated light appears directly in the transmitted light, and the function R2(ω) satisfies −1 < R2(ω = 0) < 0 ; hence,
4.0
0
1.0
(f)
(e)
0 -4.0
0
0 4.0 -4.0
0
(11)
4.0
θ (N ) = ( − 1)N θ (0)Exp{ − N | ln [{|R2(0)|]|}.
Frequency(MHz)
(12)
In the over-coupling condition, the absolute value of the pulse area |θ (N )| also decays exponentially to 0 π as a function of N . It is important that the fact that the pulse area decays exponentially does not necessarily imply that the pulses lose their energy. The electric-field envelope oscillates between positive and negative values in such a way that the area theorem is satisfied.
Fig. 3. (a)–(d) Experimental observation of the transmission spectra as a function of the detuning frequency. (a), (c) On-resonance frequency region of R2 . (b), (d) Off-resonance frequency region of R2 . (a), (b) Transmission spectra observed without R1. (c), (d) Transmission spectra observed wit R1, where the resonance comb appears. (e), (f) Calculated curves of the transmission spectra corresponding to the experiments in (c) and (d), respectively.
4. Experiments where
1 2π
E (N , t ) ≡
The theoretical results of Eqs. (11) and (12) predict that the pulse area in the output pulse train decays exactly exponentially to 0 π as a function of N , obeying the McCall and Hahn pulse area theorem in the linear regime. We used the experimental setup schematically shown in Fig. 1(a) and performed the 0 π pulse experiment in the typical case (II) discussed in Section 2. The imbalanced ring resonator was created using a single-mode polarisation-maintaining (PM) fibre. The physical lengths of the large (R1) and small (R2 ) rings were L1=230 m and L 2 =6.2 m, respectively. An 80:20 coupler was used for the small ring R2 . The proper value of coupler for R1 depends on N to be observed, as the output pulse intensity is proportional to ηN , given by Eq. (6). In the present study, a 80:20 coupler was used for the large ring R1. The resonance condition of the large ring was extremely sensitive to the laboratory environment. The fibre ring system was temperature-controlled within an accuracy of < 1 mK for stabilisation of the large ring. An Er-fibre laser at 1556 nm was used as the incident light source. The spectral width was 1 kHz, and the laser frequency was tuned by piezoelectric control of the cavity length. For observation of the steady state transmission spectra, a LiNbO3 (LN) modulator was operated in open mode. For the pulse transmission experiments, single-sided exponential pulses with a decay time of τ =80 ns and a repetition rate of 100 kHz were generated using a LN modulator. The average power of the pulses was 0.1 mW. The transmitted profiles of the system were observed using a 1-GHz InGaAs photodetector and recorded using a 600-MHz digital oscilloscope. Note that in the present system, the output profiles of different N values appeared serially in time at intervals of nL1/ c . Fig. 3 shows the transmission spectra as a function of detuning frequency. Fig. 3(a) and (c) show the experimental observation for the on-resonance frequency region of R2 , and Fig. 3(b) and (d) display the experimental observation for the off-resonance frequency region of R2 . Fig. 3(a) and (b) show the transmission spectra observed without R1 (i.e. single resonator of R2 ), and Fig. 3(c) and (d) show the transmission spectra observed with R1 (coupled resonator). Fig. 3(c) and (d) show the comb spectral structure with an FSR1 of 0.9 MHz. The full width at half
∫ {R2(ω)}N Ein(ω)e−iωtdω
1/2 N N −1 2 ⎧ x1 y1 (y1 − 1) N ≠ 0 ⎪ η ≡ (1 − γ ) 1 ⎨ N 1/2 ⎪ N=0 ⎩ η0 ≡ (1 − γ1) y1
(6)
In the derivation of Eq. (5), the denominator of the response function is expanded. The total output electric field Eout (t ) is represented as a sum of sub-electric fields E (N , t ), defined by Eq. (6). When the pulse duration tp is shorter than the circulation time in the ring R1, nL
NnL
tp < < c 1 , each sub electric field E (N , t − c 1 ) appears separately in time and generates a pulse train. The sub electric field E (N , t ) represents the N th pulse in the train and corresponds to the light component that circulated in the ring resonator N times. The pulse area is defined as the integral of the slowly varying electric field with respect to time [14,15]. In coherent optics, this quantity represents the total angle of rotation of the atomic state vector around the electric field. To show that the output pulse train develops into a coherent 0π optical pulse, we defined the pulse area θ (N ) and pulse energy W (N ) for the N th sub-pulse as
θ (N ) = μ ∫
∞
E (N , t )dt
−∞ ∞
W (N ) = κ ∫
−∞
E (N , t ) 2 dt
(7)
respectively, where μ and κ are constants. The pulse area of the th subpulse is calculated as
θ (N ) = μ ∫
∞
−∞
=
1 2π
E (N , t )dt ∞
∞
∫−∞ ∫−∞ μEin(ω){R2(ω)}N e−iωt dtdω ∞
= 2π ∫ δ (ω)μE (0, ω){R2(ω)}N dω −∞ =θ (0){R2(0)}N ,
(8) 47
Optics Communications 396 (2017) 44–49
R. Suzuki, M. Tomita
1
1
Fig. 4. Left column: red curves show the experimental observation of the output pulses through the imbalanced coupled resonator. The blue curves in the left column are the theoretically fitted curves for the output intensity. (a1) the original input pulses N =0. (b1) The sub-pulse of N =1 in output pulse train, (c1) N =2, and (d1) N =3. The right column represents the slowly varying electric field amplitude calculated using the same parameters as the left column. The parameters x 2 = 0.98 , y2 = 0.89 were used in the
of the input pulse transformed into oscillatory structures as N increased. The temporal duration of the earlier oscillation peak became shorter as N increased (Fig. 4(d1)), which is one of the characteristic features of the coherent 0 π pulse [17]. The blue curves in the left column represent the fitted curves to the experimental data. We used Fourier space analyses based on Eqs. (4) and (6). The input pulse was Fourier-transformed once, and the response function was multiplied. The result was then inverseFourier-transformed, and the transmitted pulse profiles were obtained. The green curves in the right column correspond to the electric field amplitude calculated using the same parameters as in the left column. In Fig. 5, the pulse area |θ (N )| and pulse energy W (N ) are plotted as a function of N , as defined by Eq. (7), in which the pre-factor ηN in Eq. (6) was compensated. The pulse area decayed exponentially to 0π, obeying the small-area pulse theorem represented by Eq. (12). The observed decay rates were Γ =−0.37/stage, which is in good agreement with the theoretically calculated value of Γ =−0.37/stage based on Eq. (12) using parameters of x 2 =0.98 and y2 =0.89. In contrast to the pulse area, the pulse energy did not exhibit exponential decay, as shown in Fig. 5. In an ideal system in which there is no energy dissipation, the pulse energy does not decay, while the pulse area decays to zero. In an atomic system, this occurs when the pulse duration is much shorter than the phase relaxation time T2 and the energy relaxation time T1. In cavity systems, this occurs when the pulse duration is much shorter than the cavity decay time Q / ω . In real systems, however, there are inherent dissipations leading to greater or less energy decay (which is not critical to the pulse area decay). This energy decay rate differs from that of simple absorption described by the Lambert–Beer law. The pulse energy did not exhibit exponential decay in Fig. 5.
calculations. All intensities were normalised with respect to the maximum of the input pulse.
5. Discussion
(a1)
0
(b1) N =1
0 1
(c1) N =2
0 1
(d1) N =3
Electric field (Normalized)
Intensity (Normalized)
N =0 0 1
(a2)
-0.5 1
(b2)
0 -0.5 1
(c2)
0 -0.5 1
(d2)
0 -0.5
0
400
800
0
1000
2000
Time(ns)
A single-stage resonator corresponds to a single atom, or at best, a thin atomic gas sheet, in which electromagnetic waves interact only once with the atom rather than propagating over a distance. To emulate the propagation effect in atomic gases, we need to prepare a serial array of resonators. In a previous experiments with the dynamic recurrent system [20], the development of 0 π optical pulses was successfully demonstrated over a large number of recurrence times as the excess loss due to the coupler, i.e. C1 in Fig. 1(a) was absent. Dependences on the pulse shape, pulse duration, and the parameters of the resonator were systematically investigated. In the present system, owing to the inevitable coupling loss, which is relevant to the pre-factor ηN in Eq. (6), it is not easy to observe the output pulses in the large N regions. However, in contrast to the dynamic recurrent system in Ref. [20]., the present system can be recognised as a coupled resonator. This system exhibits the characteristic comb spectral structures relevant to the under- and over-coupling conditions. In traditional coupled resonator systems, the first resonator has a moderate Q-value and the second resonator has a ultra-high Q-value [6–8,10]. In the present sizeimbalanced coupled resonator, therefore, the combination of the Qvalues was inverted with respect to that of the traditional ones. So far, various types of coupled resonators have been proposed in many different material platforms. They were conceived as a new technology in integrated devices. The size-imbalanced coupled resonator may also have unique features for demonstrating phenomena, including optical procedures [22], cavity coherent transient effects, and fast and slow light.
Log[Abs[Pulse area]] Log[Pulse energy]
0
-0.5
-1 -1.5 0
3 1 2 N* (Number of resonator)
Fig. 5. Pulse area |θ (N )| (solid green squares) and the pulse energy W (N ) (solid blue circles) as a function of N , obtained from the experimental data in Fig. 4.
maximum (FWHM) of the resonance dip of R2 was δν =1.7 MHz, in good agreement with the theoretical value of 1.8 MHz calculated by Eq. (1). The free spectral range of R1 (FSR1) was much narrower than that of R2 (FSR2 ). In Fig. 3(c), the envelope function of the comb structure showed a Λ-shaped structure. The spectral width of the Λ-shaped envelope was the same as that of the naked resonance width of R2 shown in Fig. 3(a). The comb was off resonance at ω =0. These experimental observations show good accordance with the theoretically calculated curve in Fig. 3(e) and (f), which indicates that R1 and R2 were prepared in the under-coupling and over-coupling conditions, respectively. Next we examined coherent pulse propagation. The red curves in the left column of Fig. 4 show the experimental observation. Fig. 4(a1) shows the pulse of N =0, i.e. the input pulse. Fig. 4(b1), (c1), and (d1) show the pulse of N =1, 2, and 3, respectively. The long exponential tail
6. Summary We investigated the propagation of optical pulses a though a sizeimbalanced coupled ring resonator, where the first ring was large, and had a narrow free spectral range with an extremely high Q value. Theoretical analyses showed that the output light consisted of an oscillating pulse train. The pulse energy of the sub-pulses in the train 48
Optics Communications 396 (2017) 44–49
R. Suzuki, M. Tomita
[6] D.D. Smith, H. Chang, K.A. Fuller, A.T. Rosenberger, R.W. Boyd, Coupledresonator-induced transparency, Phys. Rev. A 69 (2004) 063804. [7] A. Naweed, G. Farca, S.I. Shopova, A.T. Rosenberger, Induced transparency and absorption in coupled whispering-gallery microresonators, Phys. Rev. A 71 (2005) 043804. [8] D.D. Smith, N.N. Lepeshkin, A. Schweinsberg, G. Gehring, R.W. Boyd, Q.Han Park, H. Chang, D.J. Jackson, Coupled-resonator-induced transparency in a fiber system, Opt. Commun. 264 (2006) 163–168. [9] S.E. Harris, Electromagnetically Induced Transparency, Phys. Today July 36 (1997). [10] K. Totsuka, N. Kobayashi, M. Tomita, Slow Light in Coupled-Resonator-Induced Transparency, Phys. Rev. Lett. 98 (2007) 213904. [11] F. Morichetti, C. Ferrari, A. Canciamilla, A. Melloni, The first decade of coupled resonator optical waveguides: bringing slow light to applications, Laser Photonics Rev. 6 (2012) 74–96. [12] J. Scheuer, A. Yariv, Sagnac effect in coupled-resonator slow-light waveguide structures, Phys. Rev. Lett. 96 (2006) 053901. [13] K. Liu, X.D. Yuan, W.M. Ye, Z.H. Zhu, C. Zeng, Frequency, clockwise to counter clockwise and even to odd lasing mode switching based on optical feedback mechanism, Opt. Commun. 285 (2012) 2316–2319. [14] S.L. McCall, E.L. Hahn, Self-induced transparency by pulsed coherent light, Phys. Rev. Lett. 18 (1967) 908. [15] S.L. McCall, E.L. Hahn, Self-Induced Transparency, Phys. Rev. 183 (1969) 457. [16] M.D. Crisp, Propagation of small-area pulses of coherent light through a Resonant medium, Phys. Rev. A 1 (1970) 1604. [17] J.E. Rothenberg, D. Grischkowsky, A.C. Balant, Observation of the formation of the 0π pulse, Phys. Rev. Lett. 53 (1984) 552. [18] H. Jeong, U. Österberg, Coherent transients: optical precursors and 0π pulses, J. Opt. Sco. Am. B 25 (2008) B1–B5. [19] J.F. Chen, S. Wang, D. Wei, M.M.T. Loy, G.K.L. Wong, S. Du, Optical coherent transients in cold atoms: from free-induction decay to optical precursors, Phys. Rev. A 81 (2010) 033844. [20] Y. Suzuki, M. Tomita, Development of weak coherent 0π optical pulses in a ring resonator with a dynamic recurrent loop, J. Opt. Soc. Am. B 34 (2017) 489–496. [21] K. Totsuka, M. Tomita, Slow and fast light in a microsphere-optical fibre system, J. Opt. Soc. Am. B23 (2006) 2194. [22] T. Oishi, R. Suzuki, P. Sultana, M. Tomita, Optical precursors in coupled-resonatorinduced transparency, Opt. Lett. 37 (2012) 2964–2966.
did not decay exponentially, but the pulse area decayed exponentially to 0π, independently of the pulse duration or the pulse shape, thus strictly obeying the pulse area theorem by McCall and Hahn in the linear regime. We experimentally observed the development of a coherent 0π optical pulse using the size-imbalanced coupled ring resonators. There have been continuous developments in designs in coupled resonators, demonstrating various physical effects. The 0π pulses were analyzed to describe the same physical phenomenon as optical precursors and many results from coherent transient pulse propagation in atoms [18,19]. The present coupled resonator may have unique potential for demonstration and application of phenomena including optical procedures, cavity coherent transient effects, and fast and slow light. Funding Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant number 26287091 ). References [1] K.J. Vahala, Optical microcavities, Nature 424 (2003) 839–846. [2] S. Weis, R. Rivière, Sl. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, TJ. Kippenberg, Optomechanically induced transparency, Science 330 (2010) 1520. [3] Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, KJ. Vahala, O. Painter, Coherent mixing of mechanical excitations in nano-optomechanical structures, Nat. Photonics 4 (2010) 236. [4] T. Tsegaye, E. Goobar, A. Karlsson, G. Björk, M.Y. Loh, K.H. Lim, Efficient interaction-free measurements in a high-finesse interferometer, Phys. Rev. A 57 (1998) 3987. [5] Y.D. Chong, Li Ge, Hui Cao, A.D. Stone, Coherent perfect absorbers: time-reversed lasers, Phys. Rev. Lett. 105 (2010) 053901.
49