1 Introduction

Guided-mode resonances [1] are anomalous effects encountered when a periodically corrugated region of increased effective refractive index between two dielectric media is illuminated with a plane wave. While traditionally applied to coupling of light into waveguides [2], these resonances also permit the realization of narrow-band reflection filters [3]: with appropriate values of the frequency ω and the angle of incidence θ in the incident medium, the resonant structure can reflect all of the incident energy. For other combinations of ω and θ part of the incident energyis transmitted. Hence it is clear that (i) stationaryfields with finite spatial extent can experience partial reflection and transmission accompanied with substantial angular-spectrum modifications [4, 5], (ii) pulsed plane waves with finite temporal widths can experience phase delays because of guided-wave coupling [6, 7], and as a consequence (iii) pulsed fields with with finite spatial extent should exhibit partial reflection and transmission accompanied with strong spatial and temporal deformations. The latter type of fields are of concern here. We predict, by numerical implementations of rigorous diffraction theory, giantly enhanced Goos-Hänchen -type [8] temporal delays and pulse decompression effects as well as substantial spatial pulse deformations of initially Gaussian pulses when they interact with guided-mode resonance filters.

2 Theory

The geometry of Fig. 1 illustrates a single period of an idealized guided-mode resonance filter consisting of an infinite train of periods embedded in a uniform dielectric medium. The parameters that define the index-modulation profile are the depth h of the structure, its period d, and the refractive indices n ₁ and n ₂. In all calculations the fill factor of the grating is assumed to be 1/2. Fulfillment of the guided-mode resonance conditions demands a subwavelength-period grating such that onlythe zero-order reflected and transmitted orders propagate. The reflection and transmission coefficients for these orders are obtained straightforwardly by rigorous grating theory for any wavelength and incident angle [2, 9]. The filter produces a strong resonance reflection at the wavelength λ = 0.633 μm with a 1/e width of 0.095 nm.

The field incident upon the filter is assumed to be a paraxial, coherent, TE-polarized, spatially and temporally Gaussian pulse with

with central frequency ω ₀ = 2πc/λ_o, pulse width T ₀, and beam (waist) width ω ₀; c and λ₀ are the speed of light and the central wavelength in vacuum, and we have used the value λ₀ = 0.633 μm. This pulse can be decomposed into its (monochromatic) Fourier components

which can further be expanded in angular spectra of (monochromatic) plane-wave components

where α = (ω/v)sinθ and v is the speed of light in the incident medium. Examining the frequency domain representation of the field [ in Eq. (2)], we observe that the pulse is of type I in view of Ref. [10] since its spectral distribution is constant.

 

Fig. 1. Geometry of a guided-mode resonance filter considered in the present work. n ₁ = 1.5, n ₂ = 1.6, n _I = n _III = 1.5, d = 0.4213 μm and h = 0.1176 μm.

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The standard methods of rigorous diffraction theory can be applied to each plane-wave component, and a superposition of a discrete set of rigorous solutions provides a complete solution for the reflected and transmitted angular spectra at a given frequency. Once this procedure is repeated for a sufficientlydensely sampled set of frequencies, the temporal and spatial shape of the reflected and transmitted pulses can be constructed by inverse Fourier transforms. Such a procedure was used to construct the fields presented in the examples provided below. In the numerical work we used a spatial sampling interval of 0.001 radians and a frequencysampling period of 1.57 × 10¹¹ s^-1, which we found sufficient to guarantee satisfactory numerical convergence of the results in structures of the type presented in Fig. 1.

3 Numerical results

Figure 2 illustrates some of the numerically obtained results. Here a spatially and temporally Gaussian incident pulse with parameters ω ₀ = 0.6 mm and T ₀ = 2 ps encounters the guided-mode resonance filter with the parameters given above. The spectral width of such a pulse is 0.21 nm. The figure illustrates the time evolution of the pulse, which is split into reflected and transmitted pulses upon interaction with the filter located at z = 0; the thickness of the filter is insubstantial compared with the scale of the figures, where the spatial units are in millimeters and the time interval between the subsequent frames is 1.5 ps. When entering the filter, the incident pulse is clearlyseen to be partially guided into the filter, which is recognized as a straight vertical line in the subsequent frames at x = 0. The light originally contained in the incident pulse is partially transmitted because it contains components that do not satisfy the guided-mode resonance conditions. These components give rise to the initial transmitted pulse in (b), which effectively separates from the reflected pulse in (c). Here the spatial extensions of the incident and the reflected pulses are still essentially the same. Hence that part of the incident spatial-temporal spectrum that should be reflected totally in view of the stationary narrow-band field theory is indeed reflected. However, some spatial spreading of the the reflected field in comparison to the incident field is already observable, which is attributable to resonant coupling into the waveguide.

The situation changes rather dramatically in Fig. 2(d), where one may observe both the growth of the intensity of the reflected pulse and the emergence of new transmitted contributions that start to follow the initial transmitted pulse, however being spatially displaced. This trend is strengthened in the following frames. Both the reflected and transmitted pulses exhibit tails that result from the interaction with the filter: optical energy leaks from the waveguide practically equally to both sides as soon as the spatial extension of the incident pulse is less than the distance the waveguide modes excited by it have travelled in the waveguide.

4 Conclusions

In conclusion, we have presented, for the first time to our knowledge, a rigorous diffraction analysis of spatial and temporal deformations of pulses with finite spatial extent incident on resonant structures. The analysis revealed substantial deformations, which have consequences in engineering applications. While we were able, because of space limitations, to consider explicitly only one selected set of beam and grating parameters, the physical interpretations presented are generally applicable. Rather similar behavior as illustrated in Fig. 2 is observed also in TM polarization provided that the grating parameters are adjusted such that TM-polarized light is resonant at normal incidence and at the center frequency ω ₀. If the effective index modulation of the grating is reduced, the resonance-induced guided waves propagate further before being coupled out.

 

Fig. 2. Spatially and temporally Gaussian pulse with W ₀ = 600 μm and T ₀ = 2 ps encounters a resonance filter at z = 0. The horizontal and vertical axes are the z and x-axis, respectively. The units are in millimeters. The time interval between the figures is 1.5 ps.

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A wider incident beam or a narrower pulse spectrum lead to a higher reflectance efficiency because a larger proportion of the plane-wave components in the incident field satisfy the resonant conditions sufficiently well.

The work of T. Vallius was financed by the Graduate School on Modern Optics and Photonics. Additional support from the Academy of Finland is gratefully acknowledged.