Spatial filtering by chirped photonic crystals

We show, theoretically and experimentally, that chirped photonic crystals (where the longitudinal modulation period varies along the propagation direction) can provide a substantial spatial (angular) filtering of light beams. The chirped photonic crystals, in gapless configuration, were recorded in a bulk of glass, where the refraction index has been periodically modulated using tightly focused femtosecond laser pulses. The spatial filtering performance has been studied in detail, and the filtering efficiencies up to approximately 50$\%$ have been experimentally demonstrated.

Figure 1

Illustration of spatial filtering in a 2D PhC, in spatial Fourier domain ($k_x$, $k_y$) in configuration with (a), (b) and without (c), (d) the angular band gaps. Filtering occurs when the waves in the dark triangle zones get into resonant scattering condition with modulation with wave vectors $\vec{q}=(q_x,q_z)$ (blue arrows). Thick red arrows show the direction of filtered out angular components. The spatial spectrum (far field) of the initial beam, consisting of the central (regular) part, and of the wings (the part to be removed), is shown by bright and dark triangles.

Figure 2

Angular profiles of filtered radiation depending on the length of the nonchirped PhCs (in terms of number of periods $n$). (a) Numerical and (b) experimental results. Details are provided in the main text.

Figure 3

Fabrication setup and procedure (a). A sample of soda-lime glass is moved with respect to the tightly focused femtosecond pulse laser beam resulting in point-by-point refractive index modification in the bulk of the glass. (b) The optical microscopy image view and (c) the magnified image of the fabricated structure.

Figure 4

Experimental measurement scheme (a): The HeNe laser beam is focused into PhC. Part of the angular components is deflected to the diffraction maxima and the rest passes through. A CCD camera placed on the rotational stage measures the angular intensity profiles. (b) A part of CCD camera image of the beam behind the PhC (far field image).

Figure 5

Angular transmission profiles for varying chirp parameter $C$ for PhC sample with $n$ = 50 periods. (a) Numerical and (b) experimental results. Transverse wave number $k_x$ is normalized to transverse wave number of refractive index modulation $q_x$ in (a).

Figure 6

Numerical study of the filtering performance: (a) dependences of filtering efficiency on chirp parameter for several crystals of different length (green plot for $n$ = 30, red for $n$ = 40, blue for $n$ = 50); (b) Dependences of filtering efficiency on number of periods with different chirp parameters (black plot for $C$ = 0.0$\%$; green for $C$ = 0.1$\%$, red for $C$ = 0.2$\%$, blue for $C$ = 0.3$\%$, violet for $C$ = 0.4$\%$).

Figure 7

Numerically obtained field profiles for spatial filtering in chirped structures with higher number of periods (a) and for higher refraction index contrast (b). The parameters for (a): $n$ = 120, $s$ = 0.05, $C$ = 0.24$\%$, $d_z$ = 7.44 μm; for (b): $s$ = 0.1, $n$ = 50, $C$ = 0.53$\%$, $d_z$ = 7.2 μm. The dashed line indicates angular profile of incident beam.

Figure 8

The filtering efficiency, represented by isolines and by different colors, depending on sets of parameters. (a), (c) show dependencies on $C$ and $s$; (b), (d) show dependencies on $n$ and $s$. Parameters: (a) $n$ = 50, (c) $n$ = 120. (b) $C$ = 0.3$\%$, and (d) $C$ = 1$\%$.