Resonant cavities based on Parity-Time-symmetric diffractive gratings

Mykola Kulishov; Bernard Kress; Radan Slavík

doi:10.1364/OE.21.009473

1. Introduction

Semiconductor distributed feedback (DFB) lasers are extremely popular today due to their unique properties such as narrow spectral linewidth and ruggedness. In such lasers, a periodic variation of the geometry produces an effective diffractive grating that is characterized by periodic changes of both refractive index (phase grating) and of the gain/loss (amplitude grating). When designing DFB lasers, both (phase and amplitude) gratings are considered [1, 2]. For example, just a small amount of gain-coupling (amplitude grating) in addition to the main refractive index structures (phase gratings) significantly enhances the side modes suppression and thus provides the much desired single mode operation [3].

Multiple studies have been published on the influence of the compounded phase and amplitude gratings and their relative amplitude and phase relationships on operation characteristics of DFB lasers. In particular, Cardimona et al. [4] studied the influence of the relative phase φ between the amplitude and phase gratings and found that operating characteristics can be profoundly altered. For example, the output bandwidth can be narrower compared to that of purely amplitude grating and the possible range of output frequencies could be increased [4].

The phase grating is characterized by modulation of the real part of the refractive index along the z-axis of $n (z) = n_{0} + n_{1} \cos (K z)$ and the amplitude grating characterized by the imaginary part (gain/loss coefficient), $α (z) = α_{0} + α_{1} \cos (K z + φ)$ . Here n₀ is the average refractive index in the grating area; n₁ is the amplitude of the index modulation, α₀ is the average absorption coefficient (α₀ >0) or gain coefficient (α₀ <0), and α₁ is the magnitude of the periodic amplitude modulation, K = 2π/Λ, where Λ is the grating period. In the text, we refer to a ‘compound grating’ as a grating that is compounded by both periodic phase and amplitude modulations.

In virtually all reports published, researches focused on achieving the best possible DFB operation characteristics and left outside of their attention the case when the compounded grating has amplitude grating shifted by exactly a quarter period in respect to phase grating, i.e. φ = ± π/2. Indeed such structures cannot be used for DFB lasing for the simple reason. It has been shown [5–7], that such compounded grating does not reflect light from one of its sides due to its complete optical unidirectionality. Interestingly enough this particular phase condition, φ = ± π/2, creates the structures that recently attracted considerable attention for their unidirectional reflective properties [5, 6] or, in another classification, for their optical Parity-Time (PT) symmetry [7]. This class of physical systems was originally studied in quantum theory, first by Bender and Boettcher [8]. However, much of the reported progress remained theoretical. Recently it was suggested that the concept of PT-symmetry could be physically achieved within the framework of classical optics [9]. An optical system obeys PT symmetry if its complex refractive index distribution $n (\vec{r}) = n_{Re} (\vec{r}) + j n_{Im} (\vec{r})$ and satisfies this following condition, the real index profile must be an even function of position while the gain/loss must be odd. This is indeed achieved when the phase and amplitude gratings are shifted by a quarter of the period in respect to one another, which is represented by φ = ± π/2.

One can use the well-known transfer matrix method [6] to perform a theoretical analysis of such grating. For a compound uniform grating composed of both phase and amplitude modulations within a single mode waveguide, the electric fields of the forward E_A(z) and backward E_B(z) propagating waves with the complex propagation constant $\tilde{β} = β + j χ$ ( $χ = 2 π α_{0} / λ$ ) at the grating input z and its output z = L ends are related through the transmission matrix [6]:

[\begin{matrix} E_{A} (z + L) \\ E_{B} (z + L) \end{matrix}] = [\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}] [\begin{matrix} E_{A} (z) \\ E_{B} (z) \end{matrix}]

with the elements are defined as

\begin{array}{l} M_{11} (L, z) = [\cos h (γ L) + j \frac{σ}{γ} \sin h (γ L)] е^{- j (σ - \tilde{β}) L}, \\ M_{12} (L, z) = j \frac{κ_{Re} - κ_{Im}}{γ} \sin h (γ L) е^{- j (σ - \tilde{β}) (L + 2 z)}, \\ M_{21} (L, z) = - j \frac{κ_{Re} + κ_{Im}}{γ} \sin h (γ L) е^{j (σ - \tilde{β}) (L + 2 z)}, \\ M_{22} (L, z) = [\cos h (γ L) - j \frac{σ}{γ} \sin h (γ L)] е^{j (σ - \tilde{β}) L}, \end{array}

and

γ = {((κ_{Re} - κ_{Im}) (κ_{Re} + κ_{Im}) - σ^{2})}^{1 / 2}

, and mismatch factor

σ = \tilde{β} - π / Λ = \tilde{β} - β_{B}

. The coupling coefficients

κ_{Re}

and

κ_{Im}

are proportional to the overlap between the spatial mode distributions of the waveguide and the variable components of the refractive index n(z) and amplitude (gain/loss) coefficient α(z) defined earlier. We also see that the non-diagonal matrix elements depend on the location of the grating origin z. Based on the transfer matrix elements, the transmission and reflection coefficients for forward ( + left) and backward (- right) signal incidence can be expressed as

t^{(\pm)} = \frac{\exp (\pm j (σ - \tilde{β}) L)}{\cos h (γ L) - j (σ / γ) \sin h (γ L)}

r^{(\pm)} = \frac{j ((κ_{Re} \pm κ_{Im}) / γ) \sin h (γ L)}{\cos h (γ L) - j (σ / γ) \sin h (γ L)}

2. Concatenation of two gratings

Concatenation of two Bragg gratings with a spacing d between them and the same resonance wavelengths produces a DBR structure with a portion of non-modulated waveguide between the two gratings as a cavity, Fig. 1 . A similar architecture can be applied to DFB by replacing the non-perturbed section with a phase shift, e.g. by setting d = Λ/2n₀

Fig. 1 Geometry for DBR/DFB Fabry-Perot structure formed by two gratings, M1 and M2, with PT-symmetry and non-reflective ends placed outwards the cavity.

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General laser architecture is formed by two mirrors, one of which has lower than 100% reflection, to form the lasing cavity. In DFB lasers, the mirrors are formed by reflective Bragg gratings which act also as spectral filters, and provide the cavity feedback over specific wavelength range. In a DFB laser, the gain is distributed all over the DFB structure, hence the term “distributed”. As the Bragg grating reflects only a narrow band of frequencies, it produces mainly a single longitudinal lasing mode. In a DBR structures, the cavity is formed by two separate reflecting Bragg gratings. The distance between the reflectors can here be substantial, and therefore generally produce multiple longitudinal resonance modes simultaneously. In general, the amplifying section (gain medium) would be located in between the Bragg gratings. In our case, where the gratings are active structures, it is more difficult to draw the line between DBR and DFB structures. We would suggest a simple criterion: if the structure shown in Fig. 1 produces a single longitudinal mode, it would be qualified as DFB structure; otherwise (supporting two or more mode modes) we entitle is as DBR.

Light transmission and reflection through such DBR/DFB structure can be presented as an infinite sum of reflections from the both gratings M1 and M2 and its transmission/reflection can be expressed in the following form:

t^{(Σ)} = t_{1}^{(+)} t_{2}^{(+)} \exp (j \tilde{β} d) \sum_{n = 0}^{\infty} {(r_{1}^{(-)} r_{2}^{(+)} \exp (j 2 \tilde{β} d))}^{n} .

r^{(Σ)} = r_{1}^{(+)} + r_{2}^{(+)} t_{1}^{(+)} t_{2}^{(-)} \exp (j \tilde{β} d) \sum_{n = 0}^{\infty} {(r_{1}^{(-)} r_{2}^{(+)} \exp (j 2 \tilde{β} d))}^{n} .

For particular condition where

| r_{1}^{(-)} r_{2}^{(+)} \exp (j 2 \tilde{β} d) | < 1

, the transmission and reflection can be expressed as:

t^{(Σ)} = \frac{t_{1}^{(+)} t_{2}^{(+)} \exp (j \tilde{β} d)}{1 - r_{1}^{(-)} r_{2}^{(+)} \exp (j 2 \tilde{β} d)},

r^{(Σ)} = r_{1}^{(+)} + \frac{r_{2}^{(+)} t_{1}^{(+)} t_{2}^{(-)} \exp (2 j \tilde{β} d)}{1 - r_{1}^{(-)} r_{2}^{(+)} \exp (j 2 \tilde{β} d)},

by using the expansion formula:

\sum_{n = 0}^{\infty} x^{n} = \frac{1}{1 - x}

here, we focus on the so called ‘PT-symmetry breaking point’ case for which

n_{1} = α_{1}

or

κ_{Re} = κ_{Im} = κ / 2

[6]. As follows from Eq. (2) in this case the compounded grating asymmetry reaches its maximum, and the grating becomes truly ‘unidirectional’, as

t^{(\pm)} = \exp (\pm j β_{B} L) \exp (j σ L)

,

r^{(+)} = - j κ L \sin (σ L) \exp (j σ L) / (σ L)

and

r^{(-)} = 0

. Practically,

r^{(-)} = 0

means that there is no reflection from the left side of left grating M1. (Fig. 1), and the compounded grating becomes invisible for any optical signal launched into the grating from that side. At the same time, the signal launched into the compounded grating from the other side experiences strong reflection and even amplification (when

κ L > 1

) without any depletion of the transmitted wave.

In a traditional DFB/DBR structures, the incident light is predominantly reflected without penetrating deep into the amplifying grating structure. The penetration depth is inversely proportional to the grating strength, and therefore in traditional designs for a fixed gain level of amplification media (given, for example, in dB/cm), the maximum gain decreases with the grating reflectivity strength. The weak modulation improves the threshold but the grating length must also be increased to obtain sufficient reflectivity, increasing also the laser resonator length. Below we will show that this situation is very different for a cavity built up by PT-symmetric gratings.

As mentioned previously, there is no incident light reflection for the PT-symmetric grating provided the light is launched from the non-reflective side (left side in Fig. 1). When two such gratings are concatenated with the reflecting ends separated by a non-modulated portion of the waveguide as it is shown in Fig. 1, the unidirectional behavior can be used to build DFB/DBR structures that differ significantly from those used traditionally. For example, a signal could be launched into the cavity made by the PT-symmetric gratings and remain trapped there while emitting a portion of the signal each time it is reflected back and forth from internal reflective ends of the gratings.

For the sake of simplicity, in our analysis we consider both PT-symmetric gratings to be identical with the same coupling coefficients, length and propagation constants. Under such assumptions, Eqs. (5) and (6) are reduced to the following forms:

t^{(Σ)} = \exp (2 j (β_{B} + σ) L) \exp (j \tilde{β} d) \sum_{n = 0}^{\infty} {(j κ L \frac{\sin (σ L)}{(σ L)} \exp (j (σ L + \tilde{β} d))}^{2 n}

r^{(Σ)} = j κ L \frac{\sin (σ L)}{(σ L)} \exp (j (3 σ L + 2 \tilde{β} d)) \sum_{n = 0}^{\infty} {(j κ L \frac{\sin (σ L)}{(σ L)} \exp (j (σ L + \tilde{β} d))}^{2 n} .

In these sums, each term represents an echo of the input signal in the cavity impulse response. The echo exits the cavity at each return trip. Each echo is also delayed from the previous one by the cavity round-trip time. The shapes of successive echoes are changing, as in temporal domain each echo is a convolution of the preceding echo with a triangular impulse response of the two gratings. We recall that the Fourier transform of a triangular temporal function is a squared sinc-function. If the signal undergoes a net gain at each round trip, our compounded grating structure will therefore lase.

Considering negative round-trip net gain (below threshold) which occurs – following Eqs. (10) and (11) - for $| κ L \exp (j (σ L + \tilde{β} d) \sin (σ L) / (σ L) | < 1$ , the transmission and reflection coefficients can be written in the following compact form:

t^{(Σ)} = \frac{\exp (j \tilde{β} (2 L + d))}{κ^{2} L^{2} \sin^{2} (σ L) / {(σ L)}^{2} \exp (2 j (σ L + \tilde{β} d) + 1},

r^{(Σ)} = \frac{j κ L \sin (σ L) / (σ L) \exp (j (\tilde{β} d + 3 σ L))}{κ^{2} L^{2} \sin^{2} (σ L) / {(σ L)}^{2} \exp (2 j (σ L + \tilde{β} d) + 1} .

It is worth mentioning that the same transmission and reflection characteristics can be obtained using the transfer matrix approach:

M_{Σ} = M_{2} (L, L + d) \times I \times M_{1} (L, 0)

, where M₁ and M₂ are the transfer matrix for the left and right compounded gratings, and

І = d i a g [\exp (j \tilde{β} L), \exp (- j \tilde{β} L)]

is the diagonal matrix describing the signal transfer between the gratings. The second grating should start with the same phase as the first grating. However, this condition will never be met, if we use the same expression for the phase and amplitude distributions, i.e., cos(Kz) ± jsin(Kz), for the second grating as we used for the first one. In order to have both gratings in phase for any arbitrary d, the expression for the second grating should be defined cos(K(z-d)) ± jsin(K(z-d)). The phase factor of the second grating Kd = 2πd/Λ will compensate the phase term 2(σ-β)d = 2πd/Λ in the off-diagonal elements of the second matrix, and therefore the right transmission of the structure in Fig. 1 can be described by the resulting transfer matrix with the following elements:

\begin{array}{l} M_{11}^{(Σ)} = \exp [j \tilde{β} (2 L + d)], \\ M_{12}^{(Σ)} = j κ L \frac{\sin (σ L)}{σ L} \exp (- j (σ - 2 \tilde{β}) L) \exp (j \tilde{β} d), \\ M_{21}^{(Σ)} = - j κ L \frac{\sin (σ L)}{σ L} \exp (j (3 σ - 2 \tilde{β}) L) \exp (j \tilde{β} d), \\ M_{22}^{(Σ)} = (κ^{2} L^{2} \frac{κ^{2} \sin^{2} (σ L)}{σ^{2} L^{2}} \exp (2 j (σ L + \tilde{β} d)) + 1) \exp (- j \tilde{β} (2 L + d)) . \end{array}

Therefore, the transmitted wave is defined by T(2L + d) $= M_{11}^{(Σ)} - M_{12}^{(Σ)} M_{21}^{(Σ)} / M_{22}^{(Σ)}$ , and the reflected one by R(0) $= - M_{21}^{(Σ)} / M_{22}^{(Σ)}$ . As it can be easily verified, this leads to the same expressions as in Eqs. (12) and (13). The transfer matrix approach provides us with the equations that are correct only for negative round trip-trip gain, while Eqs. (10) and (11) are in principle – valid for both, negative and positive round-trip gain. Unfortunately, as our model does not include gain saturation, it cannot be used for describing steady-state lasing. However, the gain saturation results mainly in reduction of the gain factor, which does not change the physics discussed in the article.

Typical plots of reflection and transmission spectra are shown in Fig. 2 . The dashed (blue) curves represent the geometry when both gratings are touching (d = 0), and solid (red) curves represent the condition for which we have a half period distance (d = Λ/2) or half period shift between the gratings. When the grating are touching or are separated by a multiple of Δ (d = m Λ, where m = 0, 1, 2, 3,…) the space between the gratings produce a double-peak reflection and transmission spectra, as well as the single peak spectra when d = (m + 1/2)Λ. The number of the resonance peaks is increased as the cavity length d between the gratings is getting larger, as it is shown in Fig. 3 . All spectra exhibits amplification even with zero average gain level (α₀ = 0) not only in the gratings, but also in the cavity between the gratings.

Fig. 2 Reflection and transmission spectra of the pair of unidirectional Bragg gratings L = 1 mm oriented as shown in Fig. 1 and separated by d = Λ/2 (red solid curves) and d = 0 (blue, dashed), with Λ = 0.5 µm. The spectra are shown for different values of the grating strength: below the threshold (a, b) and at the threshold (c, d).

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Fig. 3 Reflection and transmission spectra of the pair of unidirectional Bragg gratings L = 1 mm oriented as shown in Fig. 1and separated by d = 20000Λ (blue dashed curves) and d = 20000.5Λ (red, solid) with Λ = 0.5 um.

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In what follows, we analyze key features of the DFB/DBR structures made by PT-symmetric diffractive gratings in respect to the threshold condition. The threshold condition is described as a non-trivial solution without any signal at both inputs as the boundary conditions. In the transfer matrix formalism this implies $M_{22}^{(Σ)} (ω) = 0$ , i.e.

κ L \frac{\sin (σ L)}{σ L} \exp (j σ (L + d) + j \frac{π d}{Λ}) = \pm j .

Let us consider the average level of gain/loss equal to zero (α₀ = 0,

\tilde{β} = β

). Subsequently, the mismatch factor is taking real values. The first solution is the one for which the phase-matching condition is satisfied, i.e.

σ = 0

. Therefore, the phase relation becomes:

d / Λ = \pm (2 m + 1) / 2

and the condition on the grating strength takes the following simple form

κ L = 1

This condition will only generate one lasing signal centered at the phase-matched wavelength, and the number of modes is proportional to the length of the unperturbed guide, d, between the gratings. In the case of d = Λ/2, thus yielding π-shift between the left and right gratings, only one mode can exist, such structure can be called inherently single-mode. This case is shown in Fig. 2(c) and 2(d) for the reflection and transmission, respectively. It is worth mentioning that this is actually the only condition satisfying

M_{22}^{(Σ)} (ω) = 0

, unlike the uniform gain grating case where threshold conditions are occurring at different discrete values of

κ L = π / 2

(fundamental mode), 3π/2, 5π/2… [2], the proposed structure has the only single-mode lasing threshold condition

κ L = 1

. This results in inherently single-mode operation independently on how far above the threshold it is operating. This inherently single mode behavior is taking place only for zero value of gain in the structure, i.e. for α₀ = 0. It is worth mentioning that the strength of the amplitude and phase grating parts required to reach net gain is actually π/2 times lower than in the case of the gain grating.

When d ≠ mΛ/2 the lasing condition takes place outside the grating resonance wavelength at two longitudinal modes and the threshold condition has to be found from the solution of the following equation for α₀ = 0:

κ L \frac{\sin (σ L)}{σ L} \exp (j σ (L + d)) = \pm j,

or by solving it for both the real and imaginary parts:

\begin{array}{l} κ L \frac{\sin (σ L)}{σ L} \cos (σ (L + d)) = 0 \\ κ L \frac{\sin (σ L)}{σ L} \sin (σ (L + d)) = \pm 1 \end{array}

i.e.

σ (L + d) = \pm π / 2 \pm m π

or

κ L = (m + \frac{1}{2}) \frac{π L}{(L + d)} \sin ((m + \frac{1}{2}) \frac{π L}{(L + d)})

For example, when

d = 0

or

d ≪ L

,

κ L = (m + 1 / 2) π

, m = 1, 2, 3,…, the lasing conditions can occur at multiple

κ L

values, as in the traditional DFB structures.

In the above analysis, we discussed mostly the DFB-like structures in which the distance between the two PT-symmetrical gratings is comparable to the grating period. We will now review DBR structures in which the distance d is significantly larger than the grating period. The DBR structures allows us to achieve higher round trip gains (as the laser structure is longer), narrower spectral linewidths (through longer photon cavity life time). However, such structure is prone to multimode operation (and/or mode hoping). To get better physical understanding, it is interesting to consider the two PT-symmetrical grating reflectors with complex reflectances $r_{1, 2} (ω) = | r_{1, 2} (ω) | \exp (j φ_{1, 2} (ω))$ and search for the threshold condition by setting the round trip gain equal to zero:

| r_{1} (ω) | | r_{2} (ω) | \exp (j (φ_{1} (ω) + φ_{2} (ω)) \exp (2 d (g_{0} (ω) - α_{0} (ω))) \exp (2 j β (ω) d) = 1

While the imaginary part of this equation is responsible for the lasing frequency (ω_i) selection, the real part reflects the power balance between the gain and loss in the DBR structure, which can be described in the following form:

g_{0} (ω_{і}) - α_{0} (ω_{і}) = \frac{1}{2 d} \ln (\frac{1}{| r_{1} (ω_{і}) | | r_{2} (ω_{і}) |}),

where g₀ is the total gain generated and α₀ is the compound loss in the DBR (absorption, scattering, etc.). As we can see from Eq. (22), the lowest lasing threshold will occurs when the gratings provides 100% reflection, i.e.

| r_{1} (ω) | = | r_{2} (ω) | = 1

. Obviously such arrangement is not practical because no light would be released from such resonator. However, in the case of unidirectional Bragg gratings oriented as shown in Fig. 1

| r_{1} (ω) |

and

| r_{2} (ω) |

can exceed unity, therefore reflect with amplification and provide perfect transmission

t_{1} (ω) = t_{2} (ω) = 1

at the same time. Such DBR provides the lowest possible threshold. When

| r_{1, 2} (ω) | > 1

, the logarithmic term in Eq. (22) becomes negative meaning that such unidirectional Bragg gratings provide amplifying reflection that results in lasing even when

g_{0} (ω_{і}) = 0

, i.e. there is no gain between the gratings. This low threshold is provided by the amplifying nature of the PT-symmetrical gratings. Furthermore, their amplifying behavior can be controlled by a different pump source (e.g., in semiconductor material by providing separate electrodes to control the cavity gain and the grating gain). We believe this could provide an additional flexibility and efficiency to the proposed design over traditional schemes.

3. Temporal characteristics

Another unique property of the cavity built up by PT-symmetric gratings is the temporal characteristics, for example by considering a pulse at its input and observing the transmitted/reflected signal.

Figure 4 presents the transmitted (a) and reflected (b) waveforms for a 10 ps Gaussian pulse incident on the PT-symmetric structure with the grating strength of κL = 0.9, (below threshold condition that requires κL = 1). The other parameters are L = 1 mm, Λ = 0.5 μm, and d = 10 mm.

Fig. 4 Temporal response of the pair of unidirectional Bragg gratings L = 1 mm oriented as it is shown in Fig. 1 and separated by d = 20000Λ in transmission (a) and reflection (b) (solid red curves). The input signal is depicted by the dashed blue curves.

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The solutions for the cases below and at the threshold level are given in Fig. 5 .

Fig. 5 Temporal response of the pair of unidirectional Bragg gratings L = 1 mm oriented as it is shown in Fig. 1 and separated by d = 20000Λ in reflection below the threshold (a) and at the threshold (b) κL = 1 (red curves). The input signal is depicted by the blue curves.

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As mentioned before, one of the key differences between the resonator made by two PT-symmetric gratings and a traditional DFB/DBR structures is that a signal incident on the structure fully penetrates inside the cavity while it is almost entirely reflected by a traditional DFB/DBR structure. It is thus interesting to see how the pulse that enters the cavity evolves over time. We will now analyze how the pulse evolves inside the resonator as it bounces back and forth between the two gratings. Due to the sinc-shape spectral characteristics of the grating spectral response, the pulse is periodically filtered, which transforms the original Gaussian pulse into a train of longer pulses, eventually ending up as a sinusoidal waveform.

The first seven pulses in the pulse train for different values of the input pulse duration (FWHM 8 ps – black, 9 ps – red and 10 ps blues) is shown in Fig. 6(a) . The magenta (dot-dash) curve described by harmonic function: F(t) = 0.085(1 + cos(Ωt)) with Ω = 55.3 GHz. Figure 6(b) demonstrates that this pulse train evolves into harmonic oscillations after a certain time. Thus, the number of roundtrips the pulse can survive inside the cavity is limited by the spectral response of the resonator gratings. Alternatively, this could be compensated for by shaping the gain spectral characteristics, e.g. employing some sort of non-linearity, for example, self-phase modulation [10].

Fig. 6 Temporal response of the pair of unidirectional Bragg gratings L = 1 mm oriented as shown in Fig. 1 and separated by d = 20000Λ in reflection at the threshold (κL = 1) (a) for the first seven signal replicas and (b) for the 63rd to 69th signal replicas. The black (dash) curves represent 8 ps FWHM input pulse, red (solid) curves – 9 ps FWHM input pulse and the blue (dot) curves is responsible for 10 ps FWHM input pulse. Magenta curve depicts F(t) = 0.085(1 + cos(Ωt)) with Ω = 55.3 GHz

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The structure could be used also to echo a data sequence as shown in Fig. 7 . This suggests us to use it as an optical memory that periodically echoes the stored information. The duration over which the information is stored is limited by the pulse broadening discussed previously, a limitation which is already clearly visible after 10 round trips, as shown in Fig. 7(c).

Fig. 7 Temporal response to 8-bit pulse sequence ((11011111) with FWHM 5 ps for each pulse in the train and 40 ps delay between each pulse) of the pair of unidirectional Bragg gratings L = 1 mm oriented as it is shown in Fig. 1 and separated by d = 200000Λ = 100 mm in reflection at the threshold (κL = 1): 1st and 10th replica are shown in detail.

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4. Conclusions

We analyzed a novel structure consisting of two unidirectional gratings that form a resonant cavity. First, we derived its transmission and reflection characteristics considering negative and positive net gain. We showed that for zero average gain, the structure support lasing of one and only one mode (it is inherently single-mode). Another unique characteristic when comparing it to ‘traditional’ DFB/DBR structures is that any incident signal enters fully (thus un-attenuated) into the cavity, suggesting the development of potential interesting applications such as optical memory or other signal processing functionalities. We found that the limitation of the duration over which the information could be stored inside such novel resonator structure is given by the filtering characteristics of the gratings used to form the cavity. We believe that by integrating a suitable non-linearity, this limitation could be overcome.

Although we discuss only some of the unique characteristics of these novel structures, we believe there may be also other interesting properties that could be of interest for other applications.

References and links

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Resonant cavities based on Parity-Time-symmetric diffractive gratings

Abstract

1. Introduction

2. Concatenation of two gratings

3. Temporal characteristics

4. Conclusions

References and links

Cited By

Figures (7)

Equations (22)

Optics Express