Spatial filtering with photonic crystals

Transcription

1 Spatial filtering with photonic crystals Seminar I-First year Author: Mateja Pršlja Mentor: Prof. dr. Irena Drevenšek Olenik Ljubljana, November 2017 Abstract Photonic crystals are well known for their applications in frequency filtering, in which certain wavelengths of light are reflected from the structure and can³t propagate through it. A step forward is spatial or angular filtering, in which optical rays incident at specific angles can³t be transmitted. There exist two different regimes, the first is filtering with band-gaps, i.e. the so-called Bragg regime, and the second is filtering without band-gaps, i.e. the so-called Laue regime. For improving the bandwidth of filtration it³s very useful to introduce a chirp, which is associated with gradual increase of the characteristic length of the structure. If the structure is filled with an anisotropic optical material, such as a liquid crystal, the effect of filtering can be regulated also by polarization state of the optical beam. 1

2 Contents 1 Introduction 2 2 One dimensional photonic crystals Frequency filtering Chirped structures Spatial filtering Two dimensional photonic crystals Spatial filtering with angular band-gap Spatial filtering without angular band-gap Chirped structures Three dimensional photonic crystals Spatial filtering Woodpile photonic crystal filled with liquid crystal Conclusion Appendix Introduction In photonic technology Gaussian optical beams are usually used, because of their high spatial quality (or spatial coherence), meaning that the beam diverges minimally and in principle retains its cross-sectional profile. However, with travelling through optical components (amplifier, nonlinear material, etc.) beam³s spatial quality can worsen, as it gains variations (noise) in spatial intensity, which can be reduced (cleaned) with spatial filters. Figure 1: Conventional spatial filter. Reproduced from [1]. A conventional filter consists of two focusing lenses and a pinhole of specific diameter placed between them. The overlap between the pinhole and the central part of the beam gives a pass to clean and blocks the noisy angular components of spatial spectrum (see Fig. 1). In spite of good effectiveness, this filter is³t useful in micro-structures (micro-lasers etc), because the focal distance of the lenses is too large(order of centimetres). Therefore, a new method of spatial filtering was recently proposed, namely filtering with photonic crystals (PhCs). The simplest way to understand PhCs is by the analogy with solid crystals. The structural periodicity of the atoms in the crystal lattice creates a periodic potential for propagating electrons. With the use of quantum mechanics, one can show that there 2

3 Figure 2: Schematic drawings of 1D, 2D and 3D PhCs. Regions of different dielectric permittivity are represented with different colours. Reproduced from [3]. exist some energy band-gaps, in which electrons can³t exist. If we exchange atoms with dielectric materials, periodic potential with periodic dielectric permittivity and electrons with photons, we construct PhC. Similar to energy band-gaps in solid crystal, we obtain frequency band-gaps in PhC, which mean that light of specific frequencies can³t propagate through it. The basic structure is one dimensional (1D) PhC, in which dielectric permittivity ε is changing only in one direction. However, if it changes in two or three dimensions (2D, 3D), we have 2D or 3D PhCs (see Fig. 2). They are very useful for controlling light propagation and besides the frequency band-gaps, they can also have angular band-gaps. This means that there exist some directions, along which light of specific frequency can³t propagate through the crystal. In order to explain this phenomenon, I will at first present frequency and then spatial filtering in 1D, 2D and 3D PhCs. I will end with Appendix, in which numerical calculations of spatial filtering are briefly presented. [1, 3, 4] 2 One dimensional photonic crystals 2.1 Frequency filtering With repeated piling of two dielectric materials one obtains periodic modulation of dielectric permittivity only in one direction and consequently the structure of a 1D PhC. Immediately, even when there exists only a very small contrast, there appears a photonic band-gap, which bandwidth is expanding with increasing contrast. The average frequency of the band-gap is ω m = πc m, (1) d where m is the order of the gap (m = 1,2,3) and d is the period of the structure. The width of the lowest gap ω 1 is given approximately as ω 1 c π n, (2) λ where n is refractive index difference between the two media. Optical frequencies within the band-gaps are reflected from the structure, i.e. the reflectivity R for this frequencies is 1. Optical transmission T = 1 R defines frequencies, which are propagating through the crystal. [1, 3] A simplified model, which is used to calculate T or R considers electric field as [1] E = (A(z)e i(ωt kz) +B(z)e i(ωt+kz) ) e, (3) 3

4 where the amplitudes of forward and backwards waves (A(z) and B(z)) are changing along the structure as da = i ka iωb (4) dz db dz = i kb +iω A, (5) where k = k π andωisthecouplingconstantbetweenthewaves. Withdiagonalization d oftheeq. 4andEq. 5, anduseofboundaryconditions(a(z = 0) = 1andB(z = L) = 0), we obtain transmission 2 T = A(z = L) 2 = 2ξ e ξl (λ+i k)+e ξl (λ i k), (6) where ξ = ± k 2 + Ω 2 and L is the length of the structure. With this simplified model, one can obtain only first band-gap, while more complex methods calculate all of them. [1, 3] 2.2 Chirped structures In general it is difficult to construct PhCs with large value of n. However, sometimes it³s necessary to have PhCs with large spectral region of total reflection, i.e. the large band-gap. This can be achieved with gradually increasing structure³s period, which is the case in the so-called chirped structures (see Fig. 3). How broad the band-gap in such structures is, depends on the difference between thicknesses of the first and the last period. The larger the difference is, the broader band-gap is obtained. Figure 3: Chirped PhC. Reproduced from [1]. 2.3 Spatial filtering Until now we assumed that the incident light is perpendicular to the planes of the PhC. If we send a beam onto the structure at some angle α with respect to the surface normal, the fundamental band-gap is shifted to λ = 2d cos α. The corresponding Bragg condition for reflection is given as k = ω c o ncosα = q 2, (7) where q = 2π is lattice vector and n is average refractive index. d For illumination of the PhC we use monochromatic Gaussian beam, which can be decomposed by Fourier transformation into a set of plane waves with different incident angles. For every α we can, with the help of Eq. 7, calculate λ and then with Eq. 6 calculate T or R. In this way we obtain T or R as functions of the wavelength and angle, which is represented in Fig. 4. On the left side we can see the shift of the band-gap³s wavelength, if we change the angle of incident light. On the right side, we notice that the angle of the band-gap changes if we change the wavelength of incident light. Spatial filtering with 1D PhC is relatively simple, however it is still not tight enough for practice, where angular bandwidth less than 1 is needed. The tightest filtering obtained 4

5 up till now is in the range 10 [1]. Consequently, to obtain more precise control one should use 2D or 3D PhC structures. Figure 4: Optical transmission (blue) and reflection (red) of a 1D PhC structure with d = µm, N = 20 (number of periods), n 1 = 2.17 and n 2 = Reproduced from [1]. 3 Two dimensional photonic crystals 3.1 Spatial filtering with angular band-gap 2D PhCs have a structure, which is periodic along two axes and homogeneous along the third. 2D PhCs with total band-gap are difficult to be fabricated, as they require large refractive index contrast. However it³s possible to fabricate structures exhibiting band-gaps only for specific directions of light propagation. For the case of incomplete frequency band-gap, we use different interpretation of spatial filtering, so-called resonant interaction. It means that explicit angular components of incident light are resonantly reflected back. Figures 5(a) and 5(b) show, in the reciprocal space, angular dispersion curve for monocromatic light with wavelength λ, represented as a dashed circle with radius k = 2π. Grey part (α) represents angular distribution of incident beam. If specific angular λ components (black arrows within a grey part) are in resonance with crystal³s reciprocal lattice vector q, they reflect into red arrows, which corresponds to the angular band-gaps. Angles, at which light doesn³t resonantly reflect are marked with red triangle (β). The condition for existence of an angular band-gap is q > k or d < λ, (8) where d is the longitudinal period of PhC, λ is wavelength and k is wave vector of incident light. 3.2 Spatial filtering without angular band-gap The main problem of the filtering process described in 3.1 is that it requires structures with periodicity d < λ, which are difficult to be fabricated in 2D. In practice, it³s easier to build structures with large lattice distances, which enable gapless filtering with PhCs. This filtering is based on usual diffraction from a periodic structure. A condition is 5

6 Figure 5: In Fourier space explained angular filtering in 2D PhC with a band-gap ((a),(b)), and for a gapless crystal ((c),(d)). The two cases differ in size and direction of the lattice vector q. Reproduced from [1]. q < k or d > λ and Figures 5(c) and 5(d) present two ways of this type of filtering. Black arrows that represent the incident wave vectors, which are in resonance with lattice vector q, are³t reflected, but only deviate in direction of propagation. The outgoing wave vectors are marked with red arrows. They are located not on the lower part of the circle, but on the upper. Red triangle represents light rays in crystal, which are moving forward without interruption. The disadvantage of this method is that diffracted waves can return back to the direction in which they started. Fig. 6(a) shows angular dependence of transmission depending on the number of crystal³s periods N. From the graphs for N = 8 and N = 14, it³s evident that nearly a complete filtration of some angular components occurs in the larger crystal. But once we have a complete filtration, the situation worsens with increasing N, as in this case the filtration starts to decrease, it loses its sharpness and gains some oscillations (see graphs with N = 20, 22). The two neighbouring peaks, next to the central one, represent deflected parts of angular components, which are filtered from the incident optical ray (two dips in the central peak). Fig. 6(b) presents filtration in a structure with angular band-gap, which is only improving with extension of the crystal and doesn³t show two neighbouring peaks, because filtered angular components (dips in blue spectrum) are reflected in the backward direction (peaks in red spectrum). To enhance filtering in gapless PhC, one should introduce a chirp in a structure. Figure 6: Numerical calculations of T and R depending on crystal³s thickness and incident angle of optical beam for a 2D PhC without a band-gap (a) and with a band-gap (b). Blue colour represents transmission and red reflection. A transverse period of PhC is d = 1 µm, longitudinal period is d = 6 µm for (a) and d = 0.35 µm for (b), the contrast of refractive index is n = Reproduced from [1]. 6

7 3.3 Chirped structures In the chirped structure, the longitudinal lattice distance increases for d in every new layer. Thecorrespondingchirpparameterisc = d d,whered isanaveragevalue. Chirped structures extend the span of angular filtering and also suppress back diffracted light³s components. Figure 7: Angular dependence of optical transmission as a function of the chirp parameter for a 2D PhC with N = 50. Numerical calculations (a) and experimental results (b). Reproduced from [1]. Figure 8: Numerical simulation of 1D spatial filtering of the gaussian beam with a 2D PhC. Reproduced from [1]. Fig. 7 shows numerical calculation of angular dependence of optical transmission for a chirped structure without an angular band-gap(a) and the corresponding experimental results (b). The results demonstrate the advantages of chirp introduction. Fig. 8 shows an example of 1D spatial filtering with a 2D PhC structure, which hasatransmissionwindowof 5.7,therefore much better as filtering with the 1D structure, but still not good enough for all applications. 4 Three dimensional photonic crystals 4.1 Spatial filtering In 3D PhCs refractive index is changing in space in all three directions and we obtain 2D spatial filtering. A structure with quadratic lattice in the (x,y) plane gives filtering of angular spectrum in the (x,y) plane. Since some applications prefer cylindrical symmetry, one must arrange circular geometry of the lattice. Also, in 3D PhCs it³s possible to filter with angular band-gap, if d < λ and without angular band-gap, if d > λ. To increase 7

8 the effectiveness of filtering by reducing back diffraction, we can again introduce chirp in a structure. [1] 4.2 Woodpile photonic crystal filled with liquid crystal A woodpile PhC is composed of straight dielectric logs, which are parallel in one direction and perpendicular in another. The empty spaces between them can be filled with the third material, for instance with a liquid crystal (LC), whose effective refractive index can be written as [2] n e n o n eff (θ) = n 2 e sin 2 (θ)+n 2 o cos2 (θ), (9) where θ is an angle between light polarization and the LC³s director, n e is refractive index for extraordinary and n o for ordinary polarization. The structure is shown in Fig. 9(a). It has an even number of longitudinal layers (green) and an odd number of transverse layers (orange). The LC molecules are oriented along the logs, when LC is in the nematic phase (Fig. 9(c,d)), and the structure is illuminated with x-polarized light, therefore its n eff is n 0 for odd layers (θ = 90 ) and n e for even layers (θ = 0 ). For y-polarized light the effect is opposite. But, when the LC is heated into the isotropic phase, the light³s polarization is unimportant, since n i is isotropic (Fig. 9(e,f)). Figure 9: Structure of woodpile PhC (a). Canals are filled with LC and their direction is marked with red arrows. Transverse section of PhC in xz plain (b), in yx plain for nematic phase (c,d) and in yx plain for isotropic phase (e,f). Reproduced from [2]. Figures 10(e,f,g,h) show numerically calculated spatial filtering in the far field, which is well matched with experimental measurements shown in Figures 10(a,b,c,d). Inside the central maximum there exist deflected parts of angular spectrum, presented as black lines. They depend on incident light³s polarization and the presence of the nematic or isotropic phase. When the LC is in the nematic phase and incident beam is polarized along the x-axis, the structure filters out the x- component of the incident rays at the incident angleof5 andy-componentoftheincidentraysattheangleof1.8 (Fig. 10(a)). For y-polarized rays it is just opposite, (Fig. 10(b)). The graphs (i,j,k,l) show cross sections depending on angle, where blue lines are experimental results and red lines are calculation. The graphs (c,g) are filtration for light at 45 with respect to the x,y directions, which is an average of (a) and (b). Fig. 10(d) represents filtration in the isotropic phase, where black lines are more narrow as in the nematic state. They appear at angles 1,2 and 2,8 in the x and y direction. From the picture³s symmetry it is clear, that in the isotropic phase the filtering effect is independent of incident light³s polarization. [2, 3] 8

9 Figure 10: Experimental observation ((a)-(d)) and numerical calculation ((e)-(h)) of far field of outgoing optical beam. Figures (i)-(l) represent transmitted intensity in the x direction integrated over the y direction. Blue lines are experimental results and red lines are numerical simulations. Nematic phase and x-polarized incident beam (a,e,i), y-polarized beam (b,f,j), 45 -polarization (c,g,k), isotropic phase (d,h,l). Reproduced from [2]. 5 Conclusion I have presented only one among many possible applications of PhCs, which is spatial light filtration. The advantage of the described methods with respect to classical filtering methods are translational invariance, unified structure and small thickness ( 100 µm). 1D PhCs give wide transmission window, while 2D/3D PhCs have narrow filtration region and we obtain 1D/2D filtration. In order to gain better efficiency, structures with chirped configuration can be used. The most important implementation of PhCs as spatial filters, is their integration in micro-optical devices, e.g. integration into micro-resonators of small lasers. Consequently, the brightness of the output laser beam is increased and high spatial quality is obtained (see Fig. 11). Similar effect is also achievable with external (classical) filters, but in this case we lose the compactness and simplicity of the laser structure. [1, 5] 5.1 Appendix All presented calculations of spatial filtering are based on numerically solving Maxwell equations with the Finite Difference Time Domain (FDTD) method. It is very useful for micro-structures, where FDTD gives exact results. One starts with inserting one of the Maxwell³s equations (Eq. 10) into another (Eq. 11), which leads to wave equation (Eq. 12). H( r,t) = ε 0 ε( r) t E( r,t) (10) 9

10 Figure 11: The scheme of a micro-laser without and with inserted PhC filters (left) and experimentally measured far-fields with no filter, 3D and 2D filter (right). Reproduced from [6]. E( r,t) = µ 0 t H( r,t) (11) 2 E(r,t) ε( r) 2 E(r,t) = 0 (12) c 2 t 2 If we asume, that E( r,t) is slowly varying in space and time, E( r,t) E( r,t), ( r) λ E( r,t) ωe( r,t) and is propagating along k t 0 = nω e c z wavevector (n is average refractive index) we obtain paraxial propagation equation (2ik 0 z n(x,y,z)k2 0 )A(x,y,z) = 0. (13) The wave is propagating in z-direction, 2 = 2 x y 2 and A(x,y,z) is the amplitude of the field. Paraxial approximation is very useful for spatial filtering without band-gaps, because it neglects back-reflected waves and assumes small diffraction angles. [1] References [1] L. Maigyte, K. Staliunas. Spatial filtering with photonic crystals, Applied Physics Reviews 2, (2015). [2] C. Ho, Y. Cheng, L. Maigyte, H. Zeng, J. Trull, C. Cojocaru, D. Wiersma, K. Staliunas. Controllable light diffraction in woodpile photonic crystals filled with liquid crystal, Applied Physics Letters 106, (2015). [3] J. Joannopoulos, S. Johnson, J. Winn, R. Meade. Photonic crystals: molding the flow of light, Princeton University Press, second edition (2008). [4] ( ). id=10768 [5] L. Maigyte. Shaping of light beams with photonic crystals: spatial filtering, beam collimation and focusing, Universitat Politecnica de Catalunya (2014) [6] D. Gailevicius, V. Koliadenko, V. Purlys, M. Peckus, V. Taranenko, K. Staliunas. Photonic Crystal Microchip Laser, Scientific Reports 6, (2016). 10