Tunable photonic crystal lens with high sensitivity of refractive index

Sheng Gao; Yushen Dou; Qiucui Li; Xunya Jiang

doi:10.1364/OE.25.007112

1. Introduction

Photonic crystals (PhCs), known as the semiconductors of electromagnetic waves, are attracting increasing attention due to their ability to control light propagation on a wavelength scale and their promising applications in optical integrated circuits [1,2]. Due to their rich and engineerable dispersion properties, many new phenomena have been have been explored in theoretical and experimental works, such as negative refraction [3], super-prism [4–6], slow light [7] and super-collimation (SC) [8–11]. Meanwhile, it is highly desirable to obtain some degree of tunability for PhC devices [12–19]. As an elementary device in integrated photonic/optical circuits, special attention has been addressed on the tunable PhC lens in the last decade, which could be widely used on beam shaping and auto-focusing [12]. Generally, there are two methods to realize the tunable PhC lens. One method is to change the lattice structure mechanically [12–14], such as using the MEMS-based actuators. This method has the weaknesses of slow speed, complex structure and low stability [15]. The other method is to change the refractive index by thermo-optic effect [16–18] or electro-optic effect [19,20] which seems more promising since it is much quicker and more stable. However, even for the index-changing method, there are still many difficulties to overcome for realizing practical tunable PhC lenses. The greatest difficulty is to achieve index change large enough to generate usable difference of PhC dispersion properties. Hence, some extreme conditions, such as very high temperature [16] or very strong external electric field [18], have to be applied, which greatly reduce the application potentials of the PhC lens. Furthermore, even under such extreme conditions, a large focal-length tuning range is still hard to achieve.

In this paper, based on the frequency sensitive super-collimation (FSSC) phenomenon proposed in [11], a PhC lens is designed whose focal length can be tuned in a large range by modest refractive index change which is relatively easy achieved by thermo-optic effect or electro-optic effect. A theory of such tunable PhC lenses is developed. In the numerical experiments, we show that, with 0.2% index change, the focal length can be tuned from 28a (a is lattice constant) to 240a. The mechanism of such strong tunability is from the fact that the curvature of equi-frequency-contours (EFCs) is very sensitive to the modest frequency change around the FSSC. Since this FSSC phenomenon is from the average effect of band structure, our design is quite robust against local defects as demonstrated in SC experiments [9, 21]. It could also be realized in 3D PhC slabs [22] and compatible with the semiconductor on-chip technology. With all these prominent properties, such lenses have great potentials for photonic/optical circuits and will greatly expand the applications of the tunable micro-lenses.

2. Model and theory

Our model is similar as that proposed in [11], the two-dimension (2D) rectangular lattice PhC consist of silicon rods ( $n_{S i} = 3.4$ ) in air with the aspect ratio β = b/a = 2 and the rod radius r = 0.3a, where a and b are the lattice constants in x and y directions, respectively, as shown in Fig. 1(a) inset. In this work, only the transverse electric modes (H polarization) are considered. The photonic band structure and EFCs are calculated by means of the plane-wave-expansion (PWE) method [23] and are shown in Fig. 1(a). There is a flat EFC corresponding to SC at frequency $ω_{s c} = 0.386 \frac{2 π c}{a}$ crossing the whole Bloch zone. Since the curvature of EFCs changes considerably as the frequency deviated very little from $ω_{s c}$ , it is corresponding to the FSSC phenomenon. Physically, the high sensitivity is from tuning the Van Hove singularities to SC frequency by reducing the symmetry of lattice, so that the group velocity around FSSC is very low [11]. For the frequencies on two sides of the SC frequency, the PhC acts as a convex lens or a concave lens for an incident beam composed of plane waves with different wave-vectors, as shown by Poynting vectors (black arrows) in Fig. 1(a) [4]. Near the SC frequency $ω_{s c}$ , the curvature of a EFC around the $Γ Χ_{1}$ axis can be written as $κ_{0} = \frac{\partial^{2} k_{x}}{\partial k_{y}^{2}} |_{k_{y} = 0} = - \frac{ω_{y y}}{ω_{x}} |_{k_{y} = 0}$ , where $ω_{y y} = \frac{\partial^{2} ω}{\partial k_{y}^{2}}$ , $ω_{x} = \frac{\partial ω}{\partial k_{x}}$ and the dispersion relation can be written as $k_{x} = k_{x 0} + \frac{κ_{0}}{2} k_{y}^{2}$ near the $Γ Χ_{1}$ axis in Fig. 1(a) [11, 24]. In contrast, for common 2D isotropic material, the dispersion relation is $k_{x} = \sqrt{k^{2} - k_{y}^{2}} \approx k - \frac{1}{2 k} k_{y}^{2} = k - \frac{c}{2 n ω} k_{y}^{2}$ when $k_{y} < < k$ , where $k$ is the wave-vector in the material. In Fig. 1(b), the $κ_{0}$ vs frequency in the PhC, silicon and air are compared. Obviously, $κ_{0}$ of the PhC changes dramatically in a very small frequency range from positive to negative, while others of common bulk materials keep almost constant. It is the effect of the FSSC phenomenon [11]. It will be illustrated later that $κ_{0}$ plays an essential role in determining the focal length of the tunable lens, so the FSSC phenomenon is useful in the tunable PhC lens design.

Fig. 1 (a) EFCs of the 2D rectangle lattice PhC in the second band with H polarization: the refractive index n = 3.4, the radius of rods r = 0.3a and the lattice aspect ratio β = b/a = 2.0; (b) the relationship between $κ_{0}$ and the frequency ω in PhC, air and silicon, respectively.

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In order to describe the influence of the curvature $κ_{0}$ on the diffraction behavior quantitatively, we derived a theory of focal length for a Gaussian beam in PhC, based on the approximation in which PhC is considered as an effective homogeneous medium with the dispersion relationship as shown in Fig. 1(a). We suppose that a Gaussian beam is incident into the PhC from the left boundary whose magnetic field $H_{z}$ could be expressed as:

H_{z} (0, y) = H_{z 0} \exp (\frac{i 2 π y^{2}}{R a} - \frac{y^{2}}{W_{0}^{2}}),

where W₀ is the initial width of the Gaussian beam and R is the initial curvature radius of the wave front. Using plane wave expansion method [10], Hz can be expanded into a superposition of a series of plane waves whose amplitudes are

H_{z} (k_{y})

in Eq. (2):

H_{z} (k_{y}) = F {H_{z} (y)} = \frac{H_{z 0}}{\sqrt{2 (- \frac{2 π i}{R a} + \frac{1}{W_{0}^{2}})}} \exp (- \frac{R a W_{0}^{2}}{4 (R a - 2 π i W_{0}^{2})} k_{y}^{2}) .

After propagating some distance in the x direction, the beam profile can be derived by the inverse Fourier transform of the product of the initial Gaussian beam envelop and the phase propagators in the PhC:

H_{z} (x, y) = F^{- 1} {H_{z} (k_{y}) \exp (i ϕ)},

where the phase propagators are

ϕ = k_{x} x + k_{y} y = (k_{x 0} + \frac{κ_{0}}{2} k_{y}^{2}) x + k_{y} y .

After some algebraic operation, we finally get the relationship between the beam width W and the propagation distance x:

W (x) = \frac{\sqrt{(4 κ_{0}^{2} (W_{0}^{4} + {(R a / 2 π)}^{2} x^{2} - 4 κ_{0} (R a / 2 π) W_{0}^{4} x_{0} + {(R a / 2 π)}^{2} W_{0}^{4})}}{(R a / 2 π) W_{0}} .

It is easy to conclude that W_(x) will have a minimum when

x = x_{0} = \frac{(R a / 2 π) W_{0}^{4}}{2 κ_{0} (W_{0}^{4} + {(R a / 2 π)}^{2})}

according to Eq. (5). We call

x_{0}

as the focal length of the PhC lens from the common focal length definition. When

x_{0}

is positive, the beam will converge, otherwise the beam will diverge. The expression indicates that the value of

κ_{0}

plays an essential role in determining the focal length.

The basic idea of our tunable lens design is that, if we can tune the refractive index of rods in some ways (by thermo-optic or electro-optic methods), the average refractive index of PhC will shift and the SC frequency $ω_{s c}$ of PhC will shift correspondingly, then the EFC curvature of a certain frequency around $ω_{s c}$ will change considerably. For example, for a beam with frequency $ω$ , originally the EFC curvature $κ_{0} (ω)$ is determined by $ω - ω_{s c}$ . But after changing the refractive index of dielectric rods, now $κ'_{0}$ is determined by $ω - ω'_{s c}$ , where $ω'_{s c}$ is the new SC frequency. Since $κ_{0}$ is sensitive to frequency change as shown in Fig. 1(b), the focal length PhC will change dramatically with $κ_{0}$ . To describe the influence of the small index change to $κ_{0}$ quantitatively, we define the refractive index sensitivity as:

η = | \frac{\partial κ_{0}}{\partial n} |_{ω \approx ω_{S C}} | = | \frac{\partial κ_{0}}{\partial ω} |_{ω \approx ω_{S C}} \times \frac{\partial ω}{\partial n} |_{ω_{S C}} | .

3. Optimization of the PhC lens

We calculate η for different radius r of silicon rods and aspect ratio β of PhC, shown in Fig. 2(a). The calculating range of r is from 0.26a to 0.40a, and β is from 1.0 to 1.8. We can see that η is large and approaching infinite in the dark red region. To show the origin of the large η, we also calculate $\frac{\partial κ_{0}}{\partial ω}$ and $\frac{\partial ω}{\partial n} |_{ω_{S C}}$ separately as shown in Figs. 2(b) and 2(c), where $\frac{\partial κ_{0}}{\partial ω}$ is significantly larger and changes more dramatically than $\frac{\partial ω}{\partial n} |_{ω_{S C}}$ . This can be explained by perturbation theory. It is known that $\frac{\partial ω}{\partial n} |_{ω_{S C}} \approx - \frac{ω_{S C}}{n} \times (fraction of {\int ε | E |}^{2} in the perturbed regions)$ [1]. Since the fraction of the electric-field energy inside the perturbed regions is a factor which has no specialty in the FSSC modes [11], and ranges from 0.6 to 0.9 for our cases in this work, $\frac{\partial ω}{\partial n} |_{ω_{S C}}$ will not change dramatically. So the specialty of our design is totally from very large $\frac{\partial κ_{0}}{\partial ω}$ around FSSC.

Fig. 2 (a) The SC index sensitivity for silicon rods 2D PhCs with different radii r and aspect ratio β. The unit of η is $a / 2 π$ . The SC frequency is degenerate on the left side of the dash line and nondegenerate on the right side. (b) $\log_{10} \frac{\partial κ_{0}}{\partial ω} |_{ω_{S C}}$ for silicon rods 2D PhCs with different radii r and aspect ratio β; (c) $\log_{10} | \frac{\partial ω}{\partial n} |_{ω_{S C}} |$ for silicon rods 2D PhCs with different radii r and aspect ratio β .(d) and (e) The EFCs corresponding to the two points:①r = 0.34a,β = 1.5; ②r = 0.31a,β = 1.7.

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Though η is approaching infinite in the dark red region, that region could not be utilized in practical design since the degeneracy of EFCs appears in the region. To illustrate the effect of frequency degeneracy, we calculate the EFCs of the PhC structure ① marked in Fig. 2(a) with r = 0.31a,β = 1.7. As shown in Fig. 2(d), there are two EFCs both with the same frequency $ω = 0.368 \frac{2 π c}{a}$ , which are noted by the black lines. If we use Fig. 2(d) case as our design, the incident beam with frequency $ω = 0.368 \frac{2 π c}{a}$ will excite two modes simultaneously. These two modes have different waveforms and propagation behaviors, therefore multi-beam output will occur in the PhC and the focusing lens mechanism is totally broken down. Since our goal is to design a focusing lens, we must avoid the appearance of degeneracy in our working frequency range $ω < ω_{s c}$ where all EFCs are with negative curvature in our design. Fortunately, this problem can be solved by carefully choosing structure parameters. It is found that the degeneracy of working frequency only occurs on the left side of the black dash line in Fig. 2(a). For structural parameters on the right side of the dash line on the r and β plane in Fig. 2(a), though the refractive index sensitivity η is smaller, EFCs with frequencies $ω < ω_{s c}$ are not degenerated. The case with largest η on the right of the black dash line is $η (ω_{s c}) = 2.3 \times 10^{2} \frac{a}{2 π}$ with r = 0.31a,β = 1.7, whose EFCs are shown in Fig. 2(e). The refractive index sensitivity η is 1,000-10,000 times larger than that in common silicon bulk material, 100 times larger than that in common PhC [12], three times larger than that in the original case shown in Fig. 1(a).

Besides the structural parameters, we also try different material parameters to get larger η. Here, we consider germanium whose index is 4, another kind common semiconductor material. We optimize the parameters in the range (0.26a, 0.4a) for the rod radius r and (1.0, 1.8) for the aspect ratio β. The limited parameter range without degeneracy is shown in Fig. 3(a). The optimized case is shown in Fig. 3(b), $η (ω_{s c}) = 3.3 \times 10^{3} \frac{a}{2 π}$ with r = 0.27a, β = 1.2, which is about 10,000 times larger than bulk germanium material. Besides, considering the practical application, germanium has a large absorption at telecommunication wavelength, and the modes of the PhC lens with germanium rods have a very short propagation length. In this situation, the PhC lens cannot work properly. To achieve a highly sensitive tunable lens with germanium rods, the work wavelength with low absorption should be chosen, such as 1.77μm ~15μm [25].

Fig. 3 (a) The SC index sensitivity for germanium rods 2D PhCs with different radius r and aspect ratio β and the unit of η is $a / 2 π$ . The SC frequency is degenerate on the left side of the dash line and nondegenerate on the right side. (b) The EFCs corresponding to the point ①:r = 0.27a, β = 1.2.

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4. Numerical experiments

To demonstrate the tunable lens properties of our design, we have done the numerical simulations based on the finite difference time domain (FDTD) method [26]. As shown in Fig. 4(a), a Gaussian beam with frequency $ω = 0.3642 \frac{2 π c}{a}$ is incident into a finite size PhC, $300 a \times 100 b$ , whose structural and material parameters are same as those of Fig. 3(b). The Gaussian beam source is placed at x = 0 the left side of the PhC and its H_z field distribution follows the formula of Eq. (2) with $W_{0} = 15 a$ and $R = 2 π \times 100 a$ . The large W₀ guarantees that the wave-vector y component k_y is in the range $(- 0.05 \frac{2 π}{b}, 0.05 \frac{2 π}{b})$ , so that the parabolic approximation $k_{x} = k_{x 0} + \frac{κ_{0}}{2} k_{y}^{2}$ is accurate. In Figs. 4(b)-4(d), the refractive indices of the PhC rods are chosen as 3.988, 3.992 and 3.996, respectively. Obviously, the focal length varies hugely from 28a to 240a with the small change in refractive index as shown in the Figs. 4(b)-4(d). In Figs. 5(a) and 5(b), the H_z envelop along y axes at x = 28a and x = 58a, which have been marked in Fig. 4, are shown for different refractive indices, respectively. The change of beam width can be clearly seen from Figs. 5(a) and 5(b). Our FDTD simulation indicates that the PhC lens has a great tunability of focal length with high sensitivity of refractive index.

Fig. 4 (a)The schematic diagram of FDTD simulation; (b)-(d) the monochromatic beams propagating in three different PhCs whose refractive indices of the cylinders are (b) n = 3.988; (c) n = 3.992; (d) n = 3.996, respectively.

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Fig. 5 The envelope of Hz distribution along y axes at (a) x = 28a; (b) x = 58a.

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In Fig. 6, the theoretical results of focal length from Eq. (5) (shown by continuous curves) and results from numerical experiments (shown by points) are compared. The blue and red ones are from the design in Fig. 3(b) at different frequencies $ω_{1} = 0.363 \frac{2 π c}{a}$ and $ω_{2} = 0.3642 \frac{2 π c}{a}$ , respectively. We can see that the theoretical predictions agree with numerical experiments very well.

Fig. 6 The tunable focal length. The stars are the results of the FDTD simulation and the solid line are the result of PWEM theory in section 2 while red and blue corresponding to $ω_{1} = 0.363 \frac{2 π c}{a}$ and $ω_{2} = 0.3642 \frac{2 π c}{a}$ , respectively.

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At last, we hope to emphasize the feasibility of our design. First, we note that the focal length tuning mechanism is from the averaged dispersion relation, so that they are robust against local defects. Such robustness around SC has been demonstrated in previous experiments [9, 21]. Second, the change of refractive index required by our design is so small that it could be easily realized by electro-optic effects and nonlinear effects etc. Third, as shown in other works [22], the FSSC effect has also been found in 3D PhC slab structures, so such tunable-focal-length effect can be observed in 3D PhC structures too and could be realized by semiconductor on-chip technology.

5. Conclusion

In conclusion, we have designed a tunable lens based on the FSSC phenomenon in PhCs and the corresponding theory is constructed. The focal length of the tunable lens can be one order larger with only 0.2% refractive index change. Using this tunable lens, we can control the focal length dynamically by applying thermo-optic effect, electro-optic effect or nonlinear effects which is superior to traditional methods both in speed and reliability. With high sensitivity, high speed and reliability, strong robustness and micro-size, such tunable micro-lens can be widely used in photonic circuits for signal processing, beam shaping etc. Since the lens is sensitive to refractive index change, it may also have potential in chemical or biological detection.

Funding

National Key Basic Research Project of China (NKBRPC) (2012CB927401); National Natural Science Foundation of China (NSFC) (11334015, 61475180 and 11204340); Science and Technology Commission of Shanghai Municipality (STCSM) (14JC1407600).

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Tunable photonic crystal lens with high sensitivity of refractive index

Abstract

1. Introduction

2. Model and theory

3. Optimization of the PhC lens

4. Numerical experiments

5. Conclusion

Funding

References and links

Cited By

Figures (6)

Equations (6)

Optics Express