Calculation of optical-waveguide grating characteristics using Green’s functions and Dyson’s equation

We present a method for calculating the transmission spectra, dispersion, and time delay characteristics of optical-waveguide gratings based on Green’s functions and Dyson’s equation. Starting from the wave equation for transverse electric modes we show that the method can solve exactly both the problems of coupling of counterpropagating waves (Bragg gratings) and co-propagating waves (long-period gratings). In both cases the method applies for gratings with arbitrary dielectric modulation, including all kinds of chirp and apodization and possibly also imperfections in the dielectric modulation profile of the grating. Numerically, the method scales as $O\left(N\right)$ where $N$ is the number of points used to discretize the grating along the propagation axis. We consider optical fiber gratings although the method applies to all one-dimensional (1D) optical waveguide gratings including high-index contrast gratings and 1D photonic crystals.

Figure 1

An optical fiber waveguide with a grating consisting of a dielectric modulation in the core (grey area) with period ${\Lambda }_G$. The incident electric mode in with propagation constant $\beta$ is partially reflected $\left(r\right)$ and partially transmitted $\left(t\right)$ by the grating. No radial symmetry is assumed in the presented theory.

Figure 2

Diagrammatic representation the zero to second order contributions to the total Green’s function Eq. (21). An incident core mode interacts with a copropagating cladding mode through the dielectric modulation.

Figure 3

Uniform LPG at the resonant wavelength. Top: The transmission of the core mode $T_{co}$ as a function of the grating length. Bottom: the uniform dielectric modulation of the grating.

Figure 4

Spectra of uniform LPGs with a uniform dielectric modulation (bottom in Fig. 3). The spectra is plot calculated for four different for grating lengths, $z_f-z_i=\frac{\pi }{10\kappa }$, $\frac{\pi }{3\kappa }$, $\frac{\pi }{2\kappa }$, $\frac{\pi }{\kappa }$ corresponding to 16, 27, 40, and 80 grating periods, respectively. At $\kappa L=\frac{\pi }{2}$ (bottom left) all core mode power is transferred into the cladding mode and back again into the core mode at $\kappa L=\pi$ (bottom right).

Figure 5

Time delay and dispersion of the uniform long-period grating with length $z_f-z_i=\frac{\pi }{2\kappa }$. The corresponding spectrum is seen in the bottom left of Fig. 4.

Figure 6

Time delay and dispersion of the uniform long-period grating with length $z_f-z_i=\frac{\pi }{\kappa }$. The corresponding spectrum is seen in the bottom right of Fig. 4.

Figure 7

Apodized LPG. The side lobes seen in the bottom left of Fig. 4 have been reduced. Inset: a schematic view of the raised Gaussian apodized dielectric modulation. The grating period has been exaggerated with respect to the grating length for clarity.

Figure 8

Linearly chirped LPG. The side lobes seen in the bottom left of Fig. 4 have been enhanced rather than reduced. Inset: a schematic view of the linearly chirped grating period of the dielectric modulation. The grating periods and the chirp have been exaggerated with respect to the grating length for clarity.

Figure 9

Diagrammatic representation of the zero to second order contributions to the total Green’s function Eq. (23) for a BG. An incident core mode interacts with its oppositely propagating counterpart through the dielectric modulation.

Figure 10

Uniform BG spectra for four different lengths, $\kappa L=1∕2$, 1, 2, 4. The Bragg rejection band is emerging for increasing $\kappa L$.

Figure 11

Group delay and dispersion for the transmission coefficient of a uniform BG with $\kappa L=1$. The corresponding spectrum is shown top right in Fig. 10.

Figure 12

Group delay and dispersion for the transmission coefficient of a uniform BG with $\kappa L=4$. The corresponding spectrum is shown top right in Fig. 10.

Figure 13

Spectrum for a raised-Gaussian apodized BG. Inset: the raised-Gaussian dielectric modulation. The grating period has been exaggerated with respect to the length of the grating for clarity.

Figure 14

Group delay and dispersion for a raised-Gaussian apodized BG. The spectrum of the grating is shown in Fig. 13.

Figure 15

Spectrum for a linearly chirped BG. Inset: the linearly chirped dielectric modulation. The chirp has been exaggerated with respect to the period of the grating for clarity.

Figure 16

Group delay and dispersion for a linearly chirped BG. The spectrum of the grating is shown in Fig. 15.