Parity time-symmetric vertical cavities: intrinsically single-mode regime in longitudinal direction.

We explore a new class of distributed feedback (DFB) structures that employ the recently-developed concept of parity-time (PT) symmetry in optics. We show that, based on PT-symmetric pure reflective volume gratings, a vertical surface-emitting cavity can be constructed. We provide a detailed analysis of the threshold conditions as well as the wavelength and angular spectral characteristics using the Kogelnik coupled-wave approximation, backed up by an exact solution of the Helmholtz equation. We show that such a PT-symmetric cavity can be configured to support one and only one longitudinal mode, leading to inherently single-mode lasing.

Analysis of unidirectional non-paraxial invisibility of purely reflective PT-symmetric volume gratings.

We study the diffraction produced by a slab of purely reflective PT-symmetric volume Bragg grating that combines modulations of refractive index and gain/loss of the same periodicity with a quarter-period shift between them. Such a complex grating has a directional coupling between the different diffraction orders, which allows us to find an analytic solution for the first three orders of the full Maxwell equations without resorting to the paraxial approximation. This is important, because only with the full equations can the boundary conditions, allowing for the reflections, be properly implemented. Using our solution we analyze unidirectional invisibility of such a grating in a wide variety of configurations.

Analysis of PT-symmetric volume gratings beyond the paraxial approximation.

We study the diffraction produced by a PT -symmetric volume Bragg grating that combines modulation of refractive index and gain/loss of the same periodicity with a quarter-period shift between them. Such a complex grating has a directional coupling between the different diffraction orders, which allows us to find an analytic solution for the first three orders of the full Maxwell equations without resorting to the paraxial approximation. This is important, because only with the full equations can the boundary conditions, allowing for reflections, be properly implemented. Using our solution we analyze the properties of such a grating in a wide variety of configurations.

Novel optical characteristics of a Fabry-Perot resonator with embedded PT-symmetrical grating.

We explore the optical properties of a Fabry-Perot resonator with an embedded Parity-Time (PT) symmetrical grating. This PT-symmetrical grating is non diffractive (transparent) when illuminated from one side and diffracting (Bragg reflection) when illuminated from the other side, thus providing a unidirectional reflective functionality. The incorporated PT-symmetrical grating forms a resonator with two embedded cavities. We analyze the transmission and reflection properties of these new structures through a transfer matrix approach. Depending on the resonator geometry these cavities can interact with different degrees of coherency: fully constructive interaction, partially constructive interaction, partially destructive interaction, and finally their interaction can be completely destructive. A number of very unusual (exotic) nonsymmetrical absorption and amplification behaviors are observed. The proposed structure also exhibits unusual lasing performance. Due to the PT-symmetrical grating, there is no chance of mode hopping; it can lase with only a single longitudinal mode for any distance between the distributed reflectors.

Distributed Bragg reflector structures based on PT-symmetric coupling with lowest possible lasing threshold.

A new approach towards the design of optimized distributed Bragg reflector (DBR) structures is proposed by taking advantage of recent developments related to the concept of parity-time (PT) in optics. This approach is based on using unidirectional gratings that provide coupling between co-propagating modes. Such couplers with PT symmetric gratings can provide co-directional mode coupling occurring only in one direction. This specific coupling property is achieved through a combined contribution of superimposed index and gain/loss modulations with same grating periodicity, but shifted with respect to one another by a quarter periods. Based on the transfer matrix approach, the transmission and reflection properties of the structure are modeled. One of the unique characteristics of the structure is very low lasing threshold. Such low threshold can be achieved by 100% reflectivity of the both Bragg grating mirrors, and by releasing the amplified signal in one single direction through a PT symmetric grating assisted co-directional coupler. Besides the lasing applications, the proposed structure can be implemented as an optical memory unit of replicating any input optical waveform.

Resonant cavities based on Parity-Time-symmetric diffractive gratings.

We explore a new class of Distributed Feedback (DFB) and Distributed Bragg Reflector (DBR) structures that employ the recently-developed concept of Parity-Time (PT) symmetry in optics. The approach is based on using so-called unidirectional Bragg gratings that are non diffractive (transparent) when illuminated from one side and diffracting (Bragg reflection) when illuminated from the other side, thus providing a uni-directional Bragg functionality. Such unusual property is achieved through diffraction through a grating having periodic variations in both, phase and amplitude. DFB and DBR structures traditionally consist of a gain medium and reflector(s) made via periodic variation of the (gain media) refractive index in the direction of propagation. As such structures are produced in a gain material. It becomes just possible to add periodic amplitude modulation in order to produce the unidirectional Bragg functionality. We propose here new and unique DFB and DBR structures by concatenating two such unidirectional Bragg gratings with their nonreflective ends oriented outwards the cavity. We analyze the transmission and reflection properties of these new structures through a transfer matrix approach. One of the unique characteristics of the structure is that it inherently supports only one lasing mode.

Free space diffraction on active gratings with balanced phase and gain/loss modulations.

A theoretical analysis of asymmetrical diffraction in Raman-Nath, intermediate and Bragg diffraction regimes is presented. The asymmetry is achieved by combining matched periodic modulations of the phase and of the loss/gain of the material, which enables the breakdown of optical symmetry and redirects all resulting optical energy in only positive or only negative diffraction orders, depending on the quarter period shift directions between the phase and the loss/gain modulations. Analytic expressions for the amplitudes of the diffraction orders are derived based on rigorous multimode coupled mode equations in slowly varying amplitude approximation.

Terahertz-bandwidth high-order temporal differentiators based on phase-shifted long-period fiber gratings.

We report the fabrication of a pi-phase-shifted long-period fiber grating (LPFG) capable of operating as a terahertz-bandwidth second-order temporal differentiator. We demonstrate its operation experimentally by differentiating subpicosecond long optical pulses. A new scheme for achieving high-order photonic temporal differentiation based on LPFG filters is also proposed and demonstrated. In particular, we prepared a LPFG-based first-order differentiator that was frequency and bandwidth matched to the second-order device and demonstrated the cascadability of these devices leading to the implementation of a third-order differentiator. By also employing these devices in reflection, up to the fifth-order differentiation is demonstrated experimentally.

Long-period-fiber-grating-based filter configuration enabling arbitrary linear filtering characteristics.

The filtering scheme proposed here is based on transmission through a dual long-period-fiber-grating (LPFG) configuration and enables implementation of arbitrary spectral transfer functions using available inverse-scattering design algorithms, such as those widely used for fiber Bragg gratings (FBGs) operating in reflection. Besides the important technical advantage of operation in transmission, the proposed device can reach large spectral bandwidths that would be extremely challenging to reach by, e.g., FBG devices. The proposed concept is demonstrated by designing and fabricating a LPFG-based filter for synthesis of transform-limited 1.5-ps-long square-like pulses.

First-order loss-less differentiators using long period gratings made in Er-doped fibers.

An active-fiber-based all-optical first-order temporal differentiator with power efficiency surpassing 100% is demonstrated experimentally. It is based on a long-period fiber grating (LPFG) inscribed into a piece of highly-doped Erbium-doped fiber (EDF). The performed theoretical analysis considers effects like relative position of the LPFG with respect to the input end of the EDF and influence of the input signal power. In the design, parameters like noise characteristics and level of non-linear interaction are taken into account. The advantages of such an implementation over the setup using concatenation of a passive LPFG with an amplifier lies in reducing the unwanted nonlinearities and reducing the amplified spontaneous emission (ASE).

Design of terahertz-bandwidth arbitrary-order temporal differentiators based on long-period fiber gratings.

We show that long-period fiber grating (LPG) incorporating N-1pi-phase shifts can serve as an Nth order temporal differentiator that operates in transmission. Due to the inherent large bandwidth provided by LPGs, subpicosecond (terahertz-bandwidth) optical signals may be processed with centimeters-length devices. Design parameters for up to fifth-order differentiators are given.

Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating.

We propose and experimentally demonstrate an all-optical (all-fiber) temporal differentiator based on a simple pi-phase-shifted fiber Bragg grating operated in reflection. The proposed device can calculate the first time derivative of the complex field of an arbitrary narrowband optical waveform with a very high accuracy and efficiency. Specifically, the experimental fiber grating differentiator reported here offers an operation bandwidth of approximately 12 GHz. We demonstrate the high performance of this device by processing gigahertz-bandwidth phase and intensity optical temporal variations.

Design of high-order all-optical temporal differentiators based on multiple-phase-shifted fiber Bragg gratings.

A simple and general approach for designing practical all-optical (all-fiber) arbitrary-order time differentiators is introduced here for the first time. Specifically, we demonstrate that the Nth time derivative of an input optical waveform can be obtained by reflection of this waveform in a single uniform fiber Bragg grating (FBG) incorporating N &pi-phase shifts properly located along its grating profile. The general design procedure of an arbitrary-order optical time differentiator based on a multiple-phase-shifted FBG is described and numerically demonstrated for up to fourth-order time differentiation. Our simulations show that the proposed approach can provide optical operation bandwidths in the tens-of-GHz regime using readily feasible FBG structures.

Ultrafast all-optical differentiators.

We report the experimental realization of an ultrafast all-optical temporal differentiator. Differentiation is obtained via all-fiber filtering based on a simple diffraction grating-assisted mode coupler (uniform long-period fiber grating) that performs full energy conversion at the optical carrier frequency. Due to its high bandwidth, this device allows processing of arbitrary optical signals with sub-picosecond temporal features (down to 180-fs with the specific realizations reported here). Functionality of this device was tested by differentiating a 700-fs Gaussian optical pulse generated from a fiber laser (@ 1535nm). The derivative of this pulse is an odd-symmetry Hermite-Gaussian waveform, consisting of two linked 500-fs-long, pi-phase-shifted temporal lobes. This waveform is noteworthy for its application in advanced ultrahigh-speed optical communication systems.

Long-period fiber gratings as ultrafast optical differentiators.

It is demonstrated that a single, uniform long-period fiber grating (LPFG) working in the linear regime inherently behaves as an ultrafast optical temporal differentiator. Specifically, we show that the output temporal waveform in the core mode of a LPFG providing full energy coupling into the cladding mode is proportional to the first derivative of the optical temporal signal (e.g., optical pulse) launched at the input of the LPFG. Moreover, a LPFG providing full energy recoupling back from the cladding mode into the core mode inherently implements second-order temporal differentiation. Our numerical results have confirmed the feasibility of this simple, all-fiber approach to processing optical signals with temporal features in the picosecond and subpicosecond ranges.

Ultrashort pulse propagation in uniform and nonuniform waveguide long-period gratings.

We conduct a detailed theoretical analysis of ultrashort pulse propagation through waveguide long-period grating (LPG) structures operating in the linear regime. We first consider the case of uniform LPGs, and we also investigate the effect of the typical grating nonuniformities, e.g., grating profile apodization, grating period chirping, and discrete phase shifts, on the spectral and temporal behavior of LPG structures. The two interacting modes are analyzed separately, and advanced representation tools, namely, space-wavelength and space-time diagrams (where space refers to the longitudinal grating dimension), are used to provide a deeper insight into the physics that determines the pulse evolution dynamics through the grating structures under analysis. In addition to its intrinsic physical interest, our study reveals the strong potential of LPG-based devices for optical pulse reshaping operations in the subpicosecond regime.

Nonreciprocal waveguide Bragg gratings.

The use of a complex short-period (Bragg) grating which combines matched periodic modulations of refractive index and loss/gain allows asymmetrical mode coupling within a contra-directional waveguide coupler. Such a complex Bragg grating exhibits a different behavior (e.g. in terms of the reflection and transmission spectra) when probed from opposite ends. More specifically, the grating has a single reflection peak when used from one end, but it is transparent (zero reflection) when used from the opposite end. In this paper, we conduct a systematic analytical and numerical analysis of this new class of Bragg gratings. The spectral performance of these, so-called nonreciprocal gratings, is first investigated in detail and the influence of device parameters on the transmission spectra of these devices is also analyzed. Our studies reveal that in addition to the nonreciprocal behavior, a nonreciprocal Bragg grating exhibits a strong amplification at the resonance wavelength (even with zero net-gain level in the waveguide) while simultaneously providing higher wavelength selectivity than the equivalent index Bragg grating. However, it is also shown that in order to achieve non-reciprocity in the device, a very careful adjustment of the parameters corresponding to the index and gain/loss gratings is required.

Trapping light in a ring resonator using a grating-assisted coupler with asymmetric transmission.

A recently proposed concept suggests that a matched periodic modulation of both the refractive index and the gain/loss of the media breaks the coupling symmetry of the two co-propagating modes and allows only a unidirectional coupling from the i-th mode to j-the mode but not the opposite. This concept has been used to design a ring resonator coupled through a complex grating composed of both real (index) and imaginary (loss/gain) parts according to Euler relation: n = n0 exp(-jkx) = n0 (cos(kx) - j sin(kx)). Such asymmetrical coupling allows light to be coupled into the ring without letting it out. We present a detailed theoretical analysis of the ring resonator in the linear regime, and we investigate its linear temporal dynamics. Three possible states of the complex grating leads to the possibility of developing a dynamic optical memory cell where, for example, a data modulated train of optical pulses can be stored. This data can be accessed without destroying it, and can also be erased thus permitting the storage of a new bit. Finally, the ring can be used for pulse retiming.

Ultrashort pulse propagation in grating-assisted codirectional couplers.

Ultrashort pulse propagation through grating-assisted codirectional couplers (GACCs) operating in the linear regime is theoretically investigated. For this purpose, the temporal responses of uniform GACCs to ultrashort optical pulses are calculated and the effects of varying the different physical grating parameters (e.g., length and coupling strength) on these temporal responses are evaluated. We will show that the most interesting pulse reshaping operations occur typically for the "energy receptor" mode and that depending on the length and coupling strength of the uniform perturbation one can achieve very different temporal shapes at the output of the device, including triangular pulses, square temporal waveforms as well as sequences of equalized multiple pulses. Moreover, the temporal scales of the pulses generated from a GACC are generally much shorter (in more than one order of magnitude) than those that can be generated from an equivalent Bragg grating (with the same grating length).

Theory and practice of long-period gratings: when a loss becomes a gain.

Losses of cladding modes are part of the mechanism of operation of a long-period grating (LPG) when it is used as an optical filter. We present a LPG computer simulation that accounts for these losses. On the basis of this simulation, we show that losses result in qualitatively different LPG spectral behavior. There is an optimal loss value that provides sidelobe-free, 100% power transfer from the core to the cladding mode for a uniform LPG. We obtained a simple equation that relates this optimum lose value to the LPG length and the cross-coupling coefficient. Based on the results, we propose new approaches to LPG design in a fiber as well as in waveguide platforms for fiber-optic communication and sensor applications. A design of a LPG reconfigurable filter is suggested.