Reflection phase of photonic bands in finite bi-directional 1D photonic crystals using an effective medium approach

Qiuling Zhao; Jing Liu; Dong Gao; Kai You; Xia Wang; Ho Ming Leung; Tsz Kit Yung; Ranran Zhang; Wing Yim Tam

doi:10.1364/OSAC.1.000332

1. Introduction

Photonic crystals (PCs) exhibit unique property where light wave with frequencies inside the bandgaps of the PCs cannot propagate through, analogous to the bandgaps for electrons in semiconductors [1]. Early studies had focused mainly in achieving complete bandgaps in the hope of controlling the flow of photons for replacing electronic devices with optical counterparts [2–7]. The study of PCs has since diversified into many areas such as waveguides [8,9], band edge lasings [10–13], harmonic generations [14,15], etc. One application of PCs is phase manipulations in waveplates [16]. Phase can be readily measured in the microwave regime. However, it is only recently that the reflection phase of photonic bandgaps in PCs are measured for the visible range using a simple Fabry-Perot (FP) interference method [17–20]. Reflection phase in the photonic bands of PCs has not attracted much attention partly because for finite (thin) PCs the reflection is dominated by fast FP interference oscillations which play very little role in the study of PCs as compared to the photonic bandgaps. However, in view of practical applications using finite size PCs in phase manipulations, we report the study of reflection phase using an effective medium approach [21–23] to analyze the FP reflection in the photonic bands of bi-directional 1D photonic crystals. We model the effective refractive index of PC by a Fano-like dispersion and obtain reflection phase in good agreement with direct calculation using Transfer Matrix method (TMM) in the photonic bands not too close to the band edges. Our approach enables measurements of effective refractive index, and hence the reflection phase of more complex PCs in which direct calculations may be complicated. This could open up new opportunities in phase manipulations using PCs.

2. Effective medium photonic crystal model

2.1 Band structure and effective dielectric

We consider a simple 1D binary PC composed of dielectric layers A and B with refractive indexes n_A and n_B (n_A > n_B) and thicknesses d_A and d_B, respectively. For infinite PC with unit cell thickness Λ = d_A + d_B, the band diagram (dispersion) can be calculated using the Block’s Theorem as:

\cos [Κ Λ] = \cos (\frac{n_{A} ω d_{A}}{c}) \cos (\frac{n_{B} ω d_{B}}{c}) - Δ \sin (\frac{n_{A} ω d_{A}}{c}) \sin (\frac{n_{B} ω d_{B}}{c}),

where

Δ = \frac{n_{A}^{2} + n_{B}^{2}}{2 n_{A} n_{B}}

, K is the Block wave number, c is the speed and ω is the frequency of light in free space. Equation (1) gives the band structure (bands and bandgaps) for the PC. Furthermore, one can define an effective refractive index for the bands of the PC as n_eff (ω) = Kc/ω [21–23]. Figure 1(a)

Fig. 1 (a) Band diagram of 1D PC: bands (blue) and bandgaps (red) curves. (b) Effective refractive index (green circles). The blue circles and red curve are fits to the effective refractive index with and without the PC terms. The inset shows residuals of the fits with (blue circles) and without (red circles) the bandgap contributions. (PC parameters: n_{A, B} = n_{A0, B0} (1 + n₁/λ²), for n_A0 = 1.4659, n_B0 = 1.2741, n₁ = 0.0054, d_A = 166 nm and d_B = 34 nm and Λ = 200 nm.)

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shows the band diagram, covering the first three bandgaps, for a PC with Cauchy dispersions for both dielectrics as given in the figure caption. The corresponding effective refractive index is shown as green circles in Fig. 1(b) for the bands. Note that as K is ill defined in the bandgaps thus n_eff has no value in the bandgaps. However, it turns out that n_eff can be well approximated for the photonic bands by a Fano-like expression with “resonances” inside the bandgaps as:

n_{e f f} = n_{0} {1 + \frac{n_{1}}{λ^{2}} + \sum_{i} \frac{C_{i} (\frac{1}{λ} - \frac{1}{λ_{i}})}{D_{i} + {(\frac{1}{λ} - \frac{1}{λ_{i}})}^{2}}},

where n₀, n_1, C_i, and D_i are parameters and λ_i is the “resonance” wavelength of the i-th bandgap. The first two terms in Eq. (2) correspond to un-stratified dielectric medium (no_PC slab) while the summation terms are contributions from the PC.

Figure 1(b) shows fits to the effective refractive index using Eq. (2) with (blue circles) and without (red curve) the PC terms. It is clear from the residual plots for the fits, shown in the inset of Fig. 1(b), that the fit with the PC contributions is much better than that without the PC terms. Note that the fitted values (1/λ_i) for the “resonances” are very close to those given by the mid-gap wave numbers $\frac{1}{λ_{i}} = \frac{i}{2 (n_{A} d_{A} + n_{B} d_{B)}}$ [24] (See Table 1

Table 1. Fitting results for the n_eff and m vs 1/λ_m. The mid-gap wave numbers (in 1/μm) for the first three bandgaps are: 1.71691, 3.29525, and 4.6794 [24].

View Table

).

2.2 Finite photonic crystal

For practical applications, we consider finite PCs with two configurations: Type I PC with dense (A) and Type II PC with less-dense (B) medium facing the incident medium, here air. Furthermore, we impose inversion symmetry with unit cell consisting of 0.5d_A-d_B-0.5d_A and 0.5d_B-d_A-0.5d_B for Type I and Type II PCs, respectively. These PCs have mirror symmetry at the PC center and are symmetric for incident light from either side of the PCs, i.e. bi-directional. Figure 2

Fig. 2 Reflectance of 1D photonic crystal of 165 unit cells obtained by TMM calculations using the same parameters as in Fig. 1 for Type I (a) (green curves) and Type II (b) (red curves) PCs. The pink and blue curves are the corresponding envelopes. The insets are reflectance in expanded scales showing the FP troughs and the envelope zeros as indicated by the black arrows.

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shows the reflectance obtained by TMM calculations for 165 unit cells. (We use 165 unit cells to model the bulk PCs.) Due to the finite thickness of the PC, the reflectance shows highly oscillating FP peaks and troughs resulted from interference of the reflections from the air-PC/PC-air interfaces. Note that the reflectance is modulated by an envelope that depends on the air-PC interface. The insets in Fig. 2 show clearly that the FP troughs are superimposed with the envelope dips (zeros). Note that the FP peaks/troughs are determined mainly by the thickness of the PC, and thus they are the same for both Type I and Type II PCs, see the inset for the third photonic band for Type II PC in Fig. 2(b). However, the envelope zeros are different for Type I and Type II PCs. It turns out the envelope zeros have topological origins, similar to the bandgaps in buck PCs, and will be reported elsewhere.

The lower-right inset in Fig. 3(c)

Fig. 3 Reflectance (in green) and reflection phase normalized by 2π, (in red) from 165 unit cells TMM calculations with no_PC slab (a) and with Type II PC (b). For the no_PC simulation, we use n_A, = n_B = 1.433294 (1 + 0.0054/λ²). (c) and (d) Interference order m of the FP troughs vs wave number 1/λ_m (green circles). The blue curve in (c) and the red curve and blue circles in (d) are fits to the m vs 1/λ_m. The insets are residuals of the corresponding fits.

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shows a schematic of the reflection electric field (E_r) and phase (

φ

) for unity incident beam (E_i = 1) on a finite PC, thickness L, in air. For normal incidence, the reflection field for the bands can be written as:

E_{r} = | E_{r} | e^{i φ} = r_{12} \frac{1 - e^{i 2 δ}}{1 - r_{12}^{2} e^{i 2 δ}},

where

r_{12} = \frac{1 - n_{e f f}}{1 + n_{e f f}} = | r_{12} | e^{i ϕ_{12}}

for ϕ₁₂ = π and

δ = 2 π n_{e f f} L / λ

. From Eq. (3), the reflection intensity (reflectance R = |E_r|²) and phase (

φ

) of the PC in the photonic bands can be expressed as:

R = \frac{4 {| r_{12} |}^{2} s i n^{2} (2 π \frac{n_{e f f} L}{λ})}{{(1 - {| r_{12} |}^{2})}^{2} + 4 {| r_{12} |}^{2} s i n^{2} (2 π \frac{n_{e f f} L}{λ} + π)},

φ = t a n^{- 1} [\frac{- (1 - {| r_{12} |}^{2}) sin (2 π \frac{2 n_{e f f} L}{λ})}{(1 + {| r_{12} |}^{2}) (1 - cos (2 π \frac{2 n_{e f f} L}{λ}))}] + π .

It is clear from Eq. (4) that the FP troughs correspond to destructive interference satisfying condition: $2 π \frac{n_{e f f} L}{λ_{m}} = m π,$ for integer m. Importantly, the reflection phase can be calculated using Eq. (5) after n_effL is determined from a fit of m vs 1/λ_m. Note that our model works only for mirror symmetric PCs such that it produces the same result for incidence from either side of the PCs. Furthermore, it does not include the effect of the envelope and hence cannot differentiate which dielectric material, A or B, is facing the incident light. A correction is needed in order to capture the difference between Type I and Type II PCs.

Figure 3 shows the reflectance (in green) and phase (in red) obtained directly from the TMM calculations for (a) no_PC slab and (b) with Type II PC for the first two bandgaps. Note that the phase shows abrupt changes near the right band edges of the bandgaps at the envelope zeros as indicated by the black arrows. (Similar changes are also observed for Type I PC.) The order (m) for destructive interference, starting from 1 for the first trough, is plotted against the corresponding trough wave number for the first three bands in Figs. 3(c) and 3(d) for the no_PC slab and the Type II PC, respectively. Note that in Fig. 3(d) the envelope zeros are excluded and there is a two-order jump across a bandgap, i.e. for FP trough order m on the left-hand-side bandgap, the right-hand-side FP trough order is thus m + 2. We then fit m vs 1/λ_m by adjusting the parameters in Eq. (2) for n_eff manually. The fits are shown as blue curve in Fig. 3(c) for the no_PC slab and as blue circles in Fig. 3(d) for the Type II PC. Also shown in Fig. 3(d) is a fit to the PC data without the PC terms (red line). The fit to the no_PC slab is excellent (shown in the inset for the residual of the fit) giving parameters almost the same as those used in the TMM calculation. (See figure caption of Fig. 3 and column 3 of Table 1). The fit to the Type II PC also gives parameters very similar to those for the n_eff when the PC terms are included. The residual of the fit, shown as blue circles in the inset of Fig. 3(d), confirms the quality of the fit except very close to the band edges. The small differences in the parameters, columns 2 and 5 of Table 1, could be due to the fact that n_eff shown in Fig. 1(b) is for an infinite PC while the fit in Fig. 3(d) is for a finite PC obtained from the TMM calculation. However, the fit to the PC without the PC terms (column 4 of Table 1) is very poor showing large deviations shown as red circles in the inset of Fig. 3(d). Overall our model with the PC terms fits very well in the bands except very close to the band edges where the Fano-like expression may no longer be appropriate.

Having found the effective refractive index for the PC we can now use Eq. (5) to calculate the reflection phase for the photonic bands of the PC. Note that the effective index cannot be applied to the bandgaps and thus no reflection phase is obtained. However, reflection phase in the bandgaps can also be obtained from a glass-air-PC etalon by placing a glass plate above the PC as reported recently [18–20]. Figure 4

Fig. 4 Results of reflection phase from fits (blue dashed curves) and from TMM calculations (red curves) for no_PC (a)-(b), Type I (c)-(d), and Type II (e)-(f) PCs. The green curves are the reflectance. The black arrows indicate the envelope zeros and the bandgaps are labelled in black.

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shows the reflection phase obtained from Eq. (5) by using the results of n_eff L for the no_PC slab, Type I and Type II PCs, covering the first two bandgaps. The reflection phase obtained using our approach (blue dashed curve) for the no_PC slab agrees extremely well with the calculated value (red curve) as shown in Figs. 4(a) and 4(b). We also obtain very good agreement inside the bands for Type I and Type II PCs, except in regions enclosed by the band edges and the envelope zeros as shown in Figs. 4(c) and 4(d) and Figs. 4(e) and 4(f), respectively. Note that the reflection phases obtained from Eq. (5) are the same for both Type I and Type II PCs because they have the same reflection troughs. Furthermore, it is observed that there is a difference between the phase obtained from Eq. (5) and those obtained by the TMM calculations in the regions enclosed by the band edges and the envelope zeros where there are abrupt changes of the phase obtained from the TMM calculations as indicated by the black arrows. It is noted that the abrupt change corresponds to a π phase change, here 0.5 after normalized by 2π. This is consistent with the fact that there is a phase shift of π across an envelope zero. Thus one can make a correction by shifting the phase obtained from Eq. (5) by +/− π for the regions enclosed by the envelope zeros and the band edges. This is equivalent to choosing different branch solutions for the atan in Eq. (5).

Figure 5

Fig. 5 Results of reflection phase from fits (blue dashed curves) and from TMM calculations (red curves) for Type I (a)-(b) and Type II (c)-(d). The green curves are the reflectance in expanded scales. The black arrows indicate the envelope zeros. (e) and (f) are the phase difference Δφ of the fitting results and the TMM calculations for the no_PC slab in Figs. 4(a) and 4(b) and Type II PC in (c) to (d).

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shows, in expanded scales close to the band edges of the first two bandgaps, the corrected reflection phase for both Type I and II PCs. The agreement with the TMM calculations is very good except very close to the band edges. The agreement is better displayed in the phase difference Δφ between the fitting results and the TMM calculations as shown in Figs. 5(e) and 5(f) for the no_PC slab and the Type II PC, respectively. (The phase difference for Type I PC is similar.) Thus, our effective medium model needs additional information about the surface property of the PC in order to get good correspondence with the TMM calculations. Moreover, better agreement near the band edges could be obtained by adding more terms to the Fano-like dispersion or using other similar functions for n_eff. Our model works well for PCs with bi-directional reflection symmetry, it would be interesting to extend our approach to asymmetry PCs where surface properties at either side of the PC can be more complex.

3. Conclusion

To conclude, the reflection phase of photonic bands in finite 1D bi-directional photonic crystals is studied using an effective medium approach. We model the effective refractive index of the PC by a Fano-like dispersion and show that the peaks/troughs from the FP interference of the finite PC can be used to obtain the reflection phase of the photonic bands of the PC showing good agreement with calculations for the reflection bands except very close to the band edges. Our empirical approach could be applied, after taking the surface properties of the PC into account, to more complex PCs and thus could lead to practical applications in phase manipulations and detections.

Funding

Hong Kong RGC (AoE P-02/12).

Acknowledgments

X. Wang acknowledges support from the Key Research and Development Program of Shandong Province (2018GGX101008) and the National NSFC (11274189), W. Y. Tam acknowledges support from the College of Mathematics and Physics at QUST.

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Fitting Parameter	n_eff	No_PC slab	165 unit cells PC
			No PC	With PC
		m vs 1/λ	m vs 1/λ	m vs 1/λ
n₀	1.43426	1.433292	1.434096	1.4341
n₁	0.0054	0.0054	0.0054	0.0054
C₁	−0.00035	0	0	−0.00035
D₁	−0.00042	0	0	−0.00033
1/λ₁	1.71534	0	0	1.71525
C₂	−0.0005	0	0	−0.00051
D₂	−0.00116	0	0	−0.00083
1/λ₂	3.2937	0	0	3.2934
C₃	−0.00043	0	0	−0.00046
D₃	−0.00142	0	0	−0.00097
1/λ₃	4.67873	0	0	4.6786

Reflection phase of photonic bands in finite bi-directional 1D photonic crystals using an effective medium approach

Abstract

1. Introduction

2. Effective medium photonic crystal model

2.1 Band structure and effective dielectric

2.2 Finite photonic crystal

3. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (5)

Tables (1)

Equations (5)

OSA Continuum