Efficient optical sensing based on phase shift of waves supported by a one-dimensional photonic crystal

sensors Article Efficient Optical Sensing Based on Phase Shift of Waves Supported by a One-Dimensional Photonic Crystal Roman Kaˇnok * , Petr Hlubina * , Lucie Gembalová and Dalibor Ciprian   Citation: Kaˇnok, R.; Hlubina, P.; Gembalová, L.; Ciprian, D. Efficient Optical Sensing Based on Phase Shift of Waves Supported by a One-Dimensional Photonic Crystal. Sensors 2021,21, 6535. https:// doi.org/10.3390/s21196535 Academic Editors: Aitor Urrutia, Pablo Zubiate, Nerea De Acha Morrás and Diego Lopez-Torres Received: 29 July 2021 Accepted: 28 September 2021 Published: 30 September 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Department of Physics, Technical University Ostrava, 17. Listopadu 2172/15, 708 00 Ostrava-Poruba, Czech Republic; [email protected] (L.G.); dalibor[email protected] (D.C.) *Correspondence: [email protected] (R.K.); petr[email protected] (P.H.); Tel.: +420-732-851-207 (R.K.); +420-597-323-134 (P.H.) Abstract: Interferometric methods of optical sensing based on the phase shift of the Bloch surface waves (BSWs) and guided waves (GWs) supported by a one-dimensional photonic crystal are presented. The photonic crystal, composed of six SiO 2 /TiO 2 bilayers with a termination layer of TiO 2 , is employed in the Kretschmann configuration. Under resonance condition, an abrupt phase change is revealed, and the corresponding phase shift is measured by interferometric techniques applied in both the spectral and spatial domains. The spectral interferometric technique employing a birefringent quartz crystal is used to obtain interference of projections of p - and s -polarized light waves reflected from the photonic crystal. The phase shifts are retrieved by processing the spectral interferograms recorded for various values of relative humidity (RH) of air, giving the sensitivity to the RH as high as 0.029 rad/%RH and 0.012 rad/%RH for the BSW and GW, respectively. The spatial interferometric technique employs a Wollaston prism and an analyzer to generate an interference pattern, which is processed to retrieve the phase difference, and results are in good agreement with those obtained by sensing the phase shift in the spectral domain. In addition, from the derivative of the spectral phase shifts, the peak positions are obtained, and their changes with the RH give the sensitivities of 0.094 nm/%RH and 0.061 nm/%RH for the BSW and GW, respectively. These experimental results demonstrate an efficient optical sensing with a lot of applications in various research areas. Keywords: photonic crystal; interferometry; spectral domain; spatial domain; Bloch surface waves; guided waves; Kretschmann configuration; relative humidity of air 1. Introduction Dielectric structures composed of alternating stratified media, referred to as the onedimensional photonic crystals (1DPhCs) or the Bragg reflectors, are interesting for their optical properties. Due to a periodic modulation of the refractive index (RI), regions of abandoned light frequencies—the photonic band gaps—exist and within them, light is not allowed to propagate through the structures [ 1 , 2 ]. Thus, the 1DPhCs have a high reflectivity and are widely used as reflective coatings and filters [ 3 ]. The 1DPhCs can also act as planar waveguides [ 4 ] and the guided waves (GWs) can be used in sensing applications [ 5 – 7 ]. Last but not the least, the Bloch surface waves (BSWs) propagating along the interface of the 1DPhC with an external medium are widely used in sensing, including the angular [ 8 – 11 ] or wavelength [ 12 – 17 ] interrogations. Thus, the BSW-based sensors extend mature technologies applied in sensing that have several applications in different fields of biology [ 18 ], physics [ 18 ], and chemistry [ 18 – 21 ]. The BSW states exist within the band gap of a truncated 1DPhC. Both the GWs and BSWs cause a phase jump of incident light, similarly to the surface plasmon resonance (SPR). Therefore, sensors based on the phase detection of waves supported by a 1DPhC are feasible [ 22 – 27 ]. In addition, they represent an alternative to relative humidity sensors based on resonances of surface Sensors 2021,21, 6535. https://doi.org/10.3390/s21196535 https://www.mdpi.com/journal/sensors Sensors 2021,21, 6535 2 of 17 plasmons [ 16 ], BSWs [ 16 ], whispering gallery modes [ 28 ], guided modes [ 4 , 7 , 29 ], photonic crystal modes [30], and lossy modes [31,32]. One of the advantages of the BSWs supported by 1DPhCs is the fact that also s - polarized light can be used for their excitation, depending on the structure geometry. Since the structure is composed of dielectric materials, absorption is very low and resonance dips in the reflectance spectrum are narrow. In addition, because of its chemical stability, sensing using the 1DPhCs can be adopted in aggressive environments. Although the sensing using the reflectance evaluation in the spectral domain is more often reported owing to a simple set-up, the phase interrogation has some substantial advantages even if the method is more complex. One of them is that resonance phase peaks are narrower than the resonance reflection dips. Moreover, in many cases the resonance reflection dips cannot be resolved and an interferometric method to measure an abrupt phase change overcomes the limitation. In this paper, two optical interferometric sensing methods based on measurement of the phase shift of the BSWs and GWs supported by a 1DPhC are presented. The methods, as alternatives to original approaches [ 33 – 39 ], are applied in both the spectral and spatial domains, and as an analyte, moist air of a varied relative humidity (RH) is used. For the 1DPhC under test we show that the phase shifts of both the BSW and GW can be resolved using the spectral method. On the contrary, only the phase shift of the GW can be resolved using the spatial method. At a specific wavelength, the phase shift is determined as a function of the RH. The sensor performance is evaluated in terms of sensitivity, and in the case of the BSWs, achieved sensitivity to the humidity is as high as 0.029 rad/%RH. Similarly, in the case of the GWs, the sensitivity reaches 0.012 rad/%RH. Moreover, to show an advantage of the spectral method, derivative of the phase shift is performed, and peak position is tracked as a function of the RH, giving the sensitivity to the RH as high as 0.094 nm/%RH and 0.061 nm/%RH for the BSW and GW, respectively. The paper is organized as follows. The first part is focused on the material characterization. In the second part, computational tools used in theoretical model are introduced. The third part is focused on a band structure of an infinite 1DPhC. Then, the theoretical results are presented. In the following part, an experimental set-up used in recording the interferograms is described. The last but one part is focused on experimental results obtained by techniques applied in both the spectral and spatial domains. In the final part, conclusions are presented. 2. Theoretical Model 2.1. Material Characterization The multilayer structure under study is shown in Figure 1a, and it represents a 1DPhC consisting of six SiO 2 /TiO 2 bilayers and a termination layer of TiO 2 . The 1DPhC is deposited on a glass substrate and employing an immersion oil, the substrate is attached to a coupling prism made of BK7 glass in the Kretschmann configuration. In Figure 1b, a detail image of the structure profile obtained by a scanning electron microscope (SEM) is shown, revealing different layer thicknesses. To characterize the thin layers, the variable angle spectroscopic ellipsometry (VASE) measurement was employed. Data obtained by the VASE were processed using the CompleteEASE software (J.A. Woollam Co., Inc., Shanghai, China) and the thicknesses of the layers were determined, as summarized in Table 1. Sensors 2021,21, 6535 3 of 17 Moist air TiO2 tT1 tS1 tS6 tT7 glass substrate immersion liquid TiO2 SiO2 SiO2 BK7 prism α θ teff z x y (a) (b) Figure 1. (a) A coupling prism with a photonic crystal under consideration. (b) SEM image of the photonic crystal. Table 1. Thin layer thicknesses obtained by the VASE. Layer Thickness (nm) Layer Thickness (nm) tT187.65 tS1120.21 tT279.09 tS2101.75 tT377.28 tS3109.24 tT480.74 tS4108 tT580.89 tS5127.3 tT676.85 tS6125.02 tT764.41 te f f 6.96 Moreover, the RI dispersions of the layers and of the substrate were also determined as a result of fitting the data obtained by the VASE. In the case of the glass substrate, the RI as a function of wavelength is expressed by Cauchy formula nsub(λ) = A−Bλ+Cλ2−Dλ3, (1) where values of constants obtained by the VASE are A= 1.51824, B= 0.19112 µm−1 , C= 0.019391 µm−2 and D= 0.07108 µm−3 , when wavelength λ is in micrometers. The RI dispersion of thin films is described by formula n2 i(λ) = A+Bλ2 λ2−C2−Dλ2, (2) where A , B , C and D are constants and i = TiO 2 , SiO 2 indicates the material. Their values obtained by the VASE for TiO 2 are A= 0, B= 4.672, C= 0.22935 µ m, D=0µm−2 . The constant values obtained for SiO 2 are A= 1.348, B= 0.756, C= 0.10683 µ m, D=0.00975 µm−2 . The obtained relations are valid in a wavelength range from 376 nm to 1700 nm. Similarly, the RI of the BK7 prism is described by a three term Sellmeier formula specified elsewhere [40], valid in a wavelength range from 0.3 µm to 2.5 µm. At the top of the termination layer, there is a rough surface. To confirm the rough surface, a square of 10 µ m × 10 µ m chosen on the surface was inspected by the atomic force microscopy (AFM) and data obtained were processed using Gwyddion software. A Sensors 2021,21, 6535 4 of 17 topography image of the 1DPhC surface corrected by a flat surface subtraction is shown in Figure 2, and the average roughness Ra= (1.302 ±0.37)nm was obtained. Figure 2. A topography image obtained by an AFM measurement. Under assumption that the average roughness is smaller than the wavelength of interacting light, the rough surface can be approximated by a layer of an effective medium. In the case of the Bruggeman effective media approximation (EMA), the dielectric constant ee f f describing the layer satisfies the equation fea−ee f f ea+2ee f f + (1−f)eb−ee f f eb+2ee f f =0, (3) where ea end eb are dielectric constants of media a and b , respectively, and f∈< 0, 1 > is a fraction of medium a in the effective medium layer. By default, set-up of the CompleteEASE software, 50% of void is assumed ( f= 0.5, eb= 1) in the models. The multilayer detection structure is sensitive to changes in RI of the external medium (analyte) of the 1DPhC. Since the change in the RI of moist air due to RH change is very low ( ∆RI ≈ 3.6 × 10 −7 obtained [ 41 ] for wavelength λ = 532 nm, temperature t = 20 ◦ C, atmospheric pressure p= 1013.25 hPa and RH change from 30% to 80%), the mechanism of sensitivity of the proposed sensor to moist air (see next Sections) has to be caused by other phenomena. One of them is adsorption of water molecules on the rough surface [ 42 , 43 ] of the 1DPhC. To gain insight into the effect, we simulate it by involving contribution of dielectric function of water in calculations of the Bruggeman EMA. It can be done using of Equation (3) recursively. First, the RI dispersion of water can be described by a four term Sellmeier formula [44] n2 w(λ)=1+ 4 ∑ n=1 Aiλ2 λ2−λ2 i , (4) where the constants Ai and λ2 i valid for temperature of 20 ◦ C are A1=5.684027565 ×10−1 , A2= 1.726177391 × 10 −1 , A3=2.086189578 ×10−2 , A4= 1.130748688 × 10 −1 , λ2 1=5.101829712 ×10−3µm2 , λ2 2=1.821153936 ×10−2µm2 , λ2 3= 2.620722293 × 10 −2 µm2,λ2 4=1.069792721 ×10 µm2. Sensors 2021,21, 6535 5 of 17 Then the dielectric function e0 e f f containing contributions of air and water is obtained using Equation (3), considering ea=n2 w and eb= 1 for fraction of medium a (water) f∈< 0, 1 > , where for f= 0 →e0 e f f =ea and for f= 1 →e0 e f f =ew . After that, the final dielectric function ee f f involving contributions of TiO 2 and previously determined effective medium is obtained, considering ea=n2 TiO2and eb=e0 e f f , with fixed f=0.5. 2.2. Matrix Formalism Interaction of electromagnetic waves with dielectric periodic structures can be effectively described by the 2 × 2 matrix method [ 45 , 46 ], assuming that the media are homogeneous and isotropic. First, a structure of N layers sandwiched between two semi-finite media is considered, as shown in Figure 3. Amplitudes of right and left propagating plane waves are represented by Ai and Bi , respectively, while the superscript 0 indicates that the wave is at the left boundary of the layer. B0 B0 B0 A0 A1 A2 B10 B10 A10 A20 AN AN+10 BN BN+10 AN0 BN0 x z dN 01 12 23 (N-1)N N(N+1) Figure 3. A structure of Nlayers under study. The corresponding right and left propagating modes can be represented as column vectors, and these vectors at the two sides of the interface ij are related via so-called dynamic matrix Dij Ai Bi=Dij Aj0 Bj0!, (5) where Dij =1 tij 1rij rij 1, (6) and rij and tij are the reflection and transmission coefficients of the ij -th interface, respectively. These coefficients are given by rij =     kix−kjx kix+kjx for s-pol. wave, n2 ikjx−n2 jkix n2 ikjx+n2 jkix for p-pol. wave, (7) and tij =     2kix kix+kjx for s-pol. wave, 2n2 ikjx n2 ikjx+n2 jkix for p-pol. wave, (8) where kix =k0[(nl)2−(n0sin θ)2]1/2 is the normal component of the wave vector of the light wave in the i -th medium. When a wave propagates through the i -th layer, a phase change kltl or −kltl is introduced, depending on whether the wave is right or left propagating, respectively. Thus, the modes on the two side boundaries of the i -th layer are related via Sensors 2021,21, 6535 6 of 17 Aj0 Bj0!=PiAj Bj, (9) where Pl=eikltl0 0 e−ikltl(10) is the propagation matrix. Putting all this together, a matrix equation that relates the wave amplitudes in substrate and superstrate can be obtained A0 B0=MAN+1 BN+1, (11) where the overall transfer matrix is expressed as M=M11M12 M21M22= [ N ∏ i=1 D(i−1)iPi]DN(N+1). (12) The complex reflection coefficient of the structure is a ratio of the reflected wave amplitude B0 to the incident wave amplitude A0 . Assuming that no light is incident from the superstrate (BN+10=0), using Equation (12) we obtain rs,p=|rs,p|eiδp,s=M21 M11 , (13) The phase difference between p - and s -polarized light waves can thus be determined as ∆(λ) = δp(λ)−δs(λ). Reflectance of the structure can be expressed as a squared modulus of the coefficient Rs,p=|rs,p|2. (14) 2.3. Band Structure To understand the shape of the reflection spectra, the band structure concept, similar to the solid-state physics, can be used. Considering an infinite 1DPhC, the periodicity leads to existence of allowed and forbidden bands—the waves at some frequencies can propagate through the 1DPhC, whereas some other cannot. Using the 2 × 2 matrix formalism described in the previous section, the transmission matrix that links electric field amplitudes at the input and at the output of one bilayer (one unit cell) of a periodic structure can be obtained. Under assumption that the RI in the 1DPhC is periodically modulated, the Bloch’s theorem can be applied, which states that a solution of the wave equation has a form of a plane wave modulated by a periodic function with the same period as the RI. In resulting eigenproblem, eigenvalues of the transmission matrix are related to the Bloch wave number K . For derivation, see [ 45 ] and resulting equation for p -polarized light is [ 16 ] cos(KΛ) = cos (ka⊥a)cos (kb⊥b)−1 2 n2 bka⊥ n2 akb⊥ +n2 akb⊥ n2 bka⊥!sin (ka⊥a)sin (kb⊥b), (15) where a , b and Λ are thicknesses of the layers and a bilayer thickness, respectively, ki⊥=q(niω c)2−β2 , i=a , b is a normal component of a wave vector in corresponding medium and β is a propagation constant. Equation (15) gives dependence of the propagation constant β on the angular frequency ω . Regions where cos(KΛ)< 1 are related to propagating waves (real K ). In the case of cos(KΛ)> 1, the waves are evanescent (imaginary K ). In Figure 4a, a band diagram of the infinite structure composed of SiO 2 /TiO 2 bilayers is shown. Here, reduced variables ¯ β=βΛ 2π and ¯ ω=ω c Λ 2π were used Sensors 2021,21, 6535 7 of 17 and as thicknesses of the SiO 2 and TiO 2 layers, arithmetic means of the thickness values obtained by the VASE were used ( a = 115 nm, b = 78 nm). The white region represents a photonic band gap, while the blue regions represent the allowed bands. The red crosses are related to surface waves excited on a finite 1DPhC consisting of 100 SiO 2 /TiO 2 bilayers (when substrate is glass and superstrate is air) for various angles of incidence. It can be seen that their positions are in the photonic band gap and thus they are identified as the Bloch surface waves. 00.1 0.2 0.3 0.4 0.5 0.6 0.2 0.25 0.3 0.35 0.4 0.45 ¯ β ¯ ω 0.38 0.42 0.46 0.34 0.36 0.38 0 0.2 0.4 0.6 0.8 1 0.2 0.25 0.3 0.35 0.4 0.45 Reflectance ¯ ω 0.88 0.92 0.96 0.34 0.36 0.38 (a) (b) Figure 4. ( a ) A band diagram of an infinite multilayer structure. Red crosses are related to the Bloch states. Inset shows positions of the resonance states in detail. ( b ) Reflectance of the structure with 100 bilayers as a function of ¯ ω . Light is p-polarized, angle of incidence is θ=48◦. Inset shows a shallow resonance dip related to the Bloch surface wave. In Figure 4b, the reflectance of the structure with 100 bilayers of SiO 2 /TiO 2 as a function of ¯ ω is shown. The 2 × 2 matrix formalism was used in calculating the reflectance, assuming approximate extinction coefficients for TiO 2 and SiO 2 layers of kTiO2= 1.6 × 10 −3 and kSiO2= 3.4 × 10 −4 , respectively [ 16 , 17 ]. This figure clearly shows that the Bloch surface wave resonance shows up as a shallow dip in the reflectance spectrum. 3. Theoretical Results To gain quantitative understanding of the reflection spectra, the reflectance of the 1DPhC was computed in the wavelength domain, as shown in Figure 5a, using the 2 × 2 matrix method. The extinction coefficients kTiO2 and kSiO2 given in the previous section were considered in the calculations, to enlarge resonance dips occurring in the reflectance spectra (otherwise they would be not observable). In the case of p -polarized light, three resonance dips are observed in the given wavelength region, a narrow dip approximately at λBSW = 551.5 nm corresponds to the BSW, while broad dips at λp GW1= 483.7 nm and λp GW2= 677.9 nm, respectively, correspond to guided waves. This is supported by the normalized optical field distribution of p -polarized light in the 1DPhC shown in Figure 5b. The optical field is proportional to |Hy|2 , where Hy is magnetic field component. The enhanced optical field intensity at the wavelength λBSW corresponds to character of the BSW field with exponential envelopes, while the intensities at wavelengths λp GW1 and λp GW2 are enhanced inside the structure and thus their characters correspond to guided waves. In the case of s -polarized light, two resonance dips are observed in the Figure 5a at wavelengths λs GW1= 474.5 nm and λs GW2= 735.1 nm, delimiting the borders of the photonic band gap of s-polarized light. Sensors 2021,21, 6535 8 of 17 450 500 550 600 650 700 750 800 0 0.2 0.4 0.6 0.8 1 Wavelength (nm) Reflectance Rs Rp 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 20 40 60 80 Distance from the substrate/1DPhC interface (µm) Normalized optical field intensity λp GW1=483.7 nm λBSW =550.7 nm λp GW2=677.9 nm (a) (b) Figure 5. ( a ) Theoretical reflectance of the 1DPhC as a function of wavelength for both p and s -polarized light. The dip at a wavelength of approximately 551.5 nm is associated with the Bloch surface wave. ( b ) Normalized optical field distribution of p-polarized light in the structure. Angle of incidence θ=41.9◦. The theoretical response of the proposed sensor can be shown for different water amount adsorption on the rough surface of the 1DPhC, although the dependence on the RH change is unknown. In Figure 6a, the phase response of the sensor to filling the rough surface with water is shown for water fractions f = 0, 0.2, 0.4, 0.6, 0.8 and 1. As can be seen, a red shift occurs for higher water fraction f . Derivative of the phase shift as a function of wavelength is shown in Figure 6b. Extreme point of the derivative is related to the so-called resonance wavelength λRat which the BSW is excited. 530 540 550 560 570 2 4 6 8 10 Wavelength (nm) Phase shift (rad) f=0 f=0.2 f=0.4 f=0.6 f=0.8 f=1 530 540 550 560 570 0 0.2 0.4 0.6 Wavelength (nm) Derivative of the phase shift (rad/nm) f=0 f=0.2 f=0.4 f=0.6 f=0.8 f=1 (a) (b) Figure 6. ( a ) Theoretical phase shift as a function of the wavelength with the increasing fraction of water f in the effective medium layer. (b) Derivative of the phase shift as a function of wavelength. Sensors 2021,21, 6535 9 of 17 4. Experimental Set-Up To measure phase shifts of waves under resonance conditions, interferometric techniques applied in both the spectral and spatial domains are employed. In Figure 7, an experimental set-up employing the spectral interferometric technique is shown. A composition of the set-up and a measurement procedure are described in the following part. A light beam is generated by white-light source WLS (halogen lamp HL-2000, Ocean Optics, Dunedin, FL, USA), guided by optical fiber OF and then it passes through collimating lens CL. The collimated beam (of diameter approximately 1 mm) passes through linear polarizer P (LPVIS050, Thorlabs, Newton, MA, USA), with optical axis-oriented 45 ◦ with respect to the plane of incidence, and both polarization components s and p are generated. An optical path difference between the components is introduced by birefringent quartz crystal BC of thickness d = 6 mm, so that interference fringes have appropriate period in resulting interferograms. The light beam then reflects from the multilayer structure which was prepared by a method of sputtering (Meopta, Pˇrerov, Czech Republic), primarily made as a Bragg reflector. BC P (45◦) CL MO A (45◦) analyte output S OF ROF PC WLS HS controller board input sensing chamber Figure 7. Experimental set-up consisting of white-light source WLS, optical fiber OF, collimating lens CL, polarizer P, birefringent crystal BC, a coupling prism with a multilayer structure and a sensing chamber, humidity sensor HS, analyzer A, microscope objective MO, read optical fiber ROF, spectrometer A and personal computer PC. A linear polarizer used as analyzer A (LPVIS050, Thorlabs) with optical axis-oriented 45 ◦ with respect to the plane of incidence projects the polarization components into one direction of polarization, so they may interfere. Then, the light beam is launched by microscope objective MO into read optical fiber ROF (M15L02, Thorlabs) and then led to the spectrometer (USB4000, Ocean Optics). As a result of the procedure, a spectral interferogram is obtained. The angle of incidence on the air/prism interface is adjusted to be α= 24 ◦ (see Figure 1). 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