# Optical Effects Induced by Bloch Surface Waves in One-Dimensional Photonic Crystals
## 4 The Goos-Hanchen Effect Driven by the BSW  
The Goos-Hanchen effect has been known since 1947 [30], when [PERSON] and [PERSON] first discovered the deflection of the propagation of a reflected light beam from the trajectory predicted by geometric optics (Figure 6) during total internal reflection from the surface of the glass prism. After this, the Goos-Hanchen effect was discovered and extensively studied for electromagnetic waves propagating in various media [31; 32; 33; 34; 35], including surface plasmon polaritons at the boundary of a metal film [36]. A Goos-Hanchen shift significantly increases at resonances of the angular dependence of the reflection coefficient phase. The Goos-Hanchen shift \(D\) can be obtained as: \(D=-(\lambda/2\pi)(\partial\varphi/\partial\theta)\), where \(\lambda\) is the wavelength, \(\varphi\) is the phase of the complex reflection coefficient, and \(\theta\) is the angle of incidence [37]. At the critical angle, the reflection coefficient varies rather smoothly and the values of the Goos-Hanchen effect turn out to be small. In contrast, it is possible to obtain significant values of the Goos-Hanchen shift at the angular vicinity of the Brewster angle [38], the edge of the photonic band gap in photonic crystals [39] and especially near the resonance of surface plasmons on a metal film [40]. Since the spectral-angular resonances of the BSW can be chosen to be substantially narrower than the resonances of surface plasmons in the visible spectral range, this caused an interest in the detection and research of the Goos-Hanchen effect near the resonance of the BSW. The study was performed in [26; 28]. Multilayer structures consisting of 9, 11, 13 and 15 ZrO\({}_{2}\)/SiO\({}_{2}\) bilayers with refractive indices of 1.9 and 1.46 were chosen as one-dimensional photonic crystal samples. A Kretschmann scheme of an attenuated total internal reflection prism configuration was used to couple the incident \(s\)-polarized light to the BSW at the photonic crystal-air interface. The Goos-Hanchen effect caused by the BSW was obtained in two ways: from the experimentally measured BSW resonance spectra, and by fluorescence microscopy in the far field. The microimages shown in the Figure 7 made it possible to observe both the scattered image of the BSW and the fluorescence image of the spots of the beams incident and reflected from the photonic crystal. Both estimates gave results that are satisfactorily consistent within the experimental error. In fluorescent images, it is seen that with a single \"inc\" spot of the incident beam, the reflected beam \"ref \" is divided into two parts, shifted relative to each other. Analogous to the interpretation of the Fano resonance in the reflectance of a photonic crystal during the BSW excitation, these two parts  
Figure 6: Illustration of the Goos–Hanchen effect. The lateral shift D of the totally reflected beam (\(n_{1}>n_{2}\)) is observed. Reprinted from [PERSON], [PERSON], [PERSON], JETP Lett. 91, 382 (2010) with permission of [PERSON] [28]. are associated with the resonant part of the incident beam that passes into the BSW, and the nonresonant one that is totally reflected from the surface. The distance between these two parts can be regarded as the Goos-Hanchen shift caused by the BSW. It has been shown that the BSW-induced Goos-Hanchen shift significantly depends on the Q-factor of the BSW resonance. The maximum shift observed in Ref. [26] was 66 \(\upmu\)m or 125 wavelengths of incident light, that is, at least two times larger than the effect observed for surface plasmons [36; 40]. Later, in higher-quality photonic crystal systems, even larger values of the Goos-Hanchen shift were obtained [41].