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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

Electronically reconfigurable superimposed waveguide long-period gratings.

The perturbation to the refractive index induced by a periodic electric field from two systems of interdigitated electrodes with the electrode-finger period l is analyzed for a waveguide with an electro-optically (EO) active core-cladding. It is shown that the electric field induces two superimposed transmissive refractive-index gratings with different symmetries of their cross-section distributions. One of these gratings has a constant component of an EO-induced refractive index along with its variable component with periodicity l, whereas the second grating possesses only a variable component with periodicity 2l.