Single-mode operations of two rib widths.

## Abstract

In this chapter, the quantum process of the Goos-Hänchen (GH) spatial shift is first derived out, then the coherence between spatial and angular shifts in the GH effect in the quantum state is discovered and a function of digital optical switch is developed. It is found that a waveguide corner structure always makes the reflected guide-mode have both the GH spatial and angular shifts when the incident beam is in the vicinity area of critical and Brewster angles. Meanwhile, these two GH shifts have the interesting coherent distributions with the incident angle, and only in the common linear response area the two GH shifts are mutual enhancing, then a mini refractive index modulation (RIM) of guided mode at the reflecting interface can create a great stable jump of reflected beam displacement at an eigenstate. As a result, on 220 nm silicon-on-insulator (SOI) waveguide platform, with a tapered multimode interference (MMI) waveguide a 5.0 × 10 18 cm − 3 concentration variation of free carriers can cause a digital total 8–25 μm displacement of the reflected beam on the MMI output end, leading to a 1 × N scale digital optical switching function. As a series of verifications, the numerical calculations, finite difference time domain (FDTD) simulations and experiments, are sustainable to the quantum GH shifts.

### Keywords

- silicon photonics
- silicon-on-insulator
- waveguide corner
- coherent Goos-Hänchen spatial and angular shifts
- eigenstate of total GH displacement
- and digital optical switch

## 1. Introduction

In 1947 a novel spatial shift phenomenon of the reflected beam at an interface of two media was discovered by Goos and Hänchen, which was later referred to as the Goos-Hänchen (GH) effect [1]. Then, in 1948 it was theoretically modeled by V. K. Artmann by providing an Artmann equation [2]. But, at that time no special attention was ever paid to this interesting phenomenon until 1972 when Chiu and Quinn further testified the simple delay process of optical scattering caused by this optical GH effect because Chiu and Quinn’s work exactly presented the two polarization-determined parallel reflected beams in both theory and experiment [3]. In the past decades, based on the different media and technologies a variety of GH shift phenomena were studied in which their essential applications have been shown out [4, 5, 6, 7, 8]. At the condition of plane waves, the first impressive research in theory was done by Wild and Giles in [4]. Later, in 1986 Lai’s team performed the more detailed theoretical investigation for the GH effect with Gaussian beam and obtained the more sustainable results, and furthermore in 2002 they extended their research to the more regular case—the complex expression for both the positive and negative electromagnetic materials at the reflection beam side and the frustrated total internal reflection (FTIR) was defined [5, 6]. These typical theoretical achievements in both plane and Gaussian waves have built a powerful fundamental for research on the GH behaviors of guided modes in a waveguide corner. Thus, in the past years, the establishments in the following three interesting topics relating to the GH effect have stimulated intensive research on new theories, materials and functionalities: (i) the integrated optical technologies due to a broad landscape of applications; (ii) the great spatial and angular shifts of Gaussian and quasi-Gaussian beams; and (iii) the control of GH effect in a micro-cavity by using the coherent light into the atomic energy levels [7, 8, 9, 10, 11]. With the special waveguide structures and corner-mirror materials, some novel phenomena of GH effect and the other involved effect such as Imbert-Fedorov shifts were also investigated and established [12, 13, 14].

In fact, the quantum mechanism supporting to theoretical research in the spatial shift of GH effect was established by Steinberg and Chiao in [15]. Then, in 2013 we theoretically demonstrated this quantum effect of the GH spatial shift with the consistent solutions of Maxwell equation and Schrödinger equation, and then proposed a digital electro-optic (EO) switching regime on semiconductor platform where a new metal-oxide-semiconductor (MOS)-capacitor type EO modulation method is also proposed and analyzed to realize an effective free-carrier dispersion (FCD) effect based refractive index modulation (RIM) [16]. Meanwhile, as a typical EO modulation scheme of semiconductor material, the FCD-based RIM of silicon-on-insulator (SOI) waveguide has been attracting the broad and intensive research attentions in the field of photonic integrated circuit (PIC) components and applications in the past decade [17, 18]. The similar part of all these FCD-RIM based switching schemes is that the FCD effect is imposed to a section of waveguide channel to cause an optical phase change, so their common intrinsic drawback is the FCD-induced optical absorption. As a result, there have not been any real accomplishments of product development reported so far. Consequently, the digital operations for the high-speed optical switching devices with advanced FCD-RIM schemes and device structures are expected for high-speed PIC devices [16, 17, 18, 19].

In this chapter, by starting with the eigenstates of guided modes we first investigate the correspondences between the Fresnel equation and Schrödinger equation of guided modes and the coherence of the spatial shift and angular shift of the reflected guided modes under the GH effect. Then, the quantum processes responding to an incident angle with the eigenstates of guided modes of the MMI waveguide are discussed. Furthermore, the total displacement versus the concentration change of free-carrier holes is analyzed to discuss the performance of FCD-RIM based digital optical switch within the SOI-CMOS submicron waveguide corner structure. Furthermore, the sustainable numerical calculations, professional software simulations and experiments are discussed. Finally, the conclusions are given.

## 2. Spatial and angular shifts in Goos-Hänchen (GH) effect

### 2.1. Coherence between the spatial and angular shifts in GH effect

Figure 1 shows the schematic relationship of incident, reflected and transmitted beams across the interface of two optical media with the refractive indices n_{1} and n_{2}, respectively. A common definition of GH shift is the effective spatial shift

where

Eq. (1) tells that the phase change of reflected beam is a key parameter in the GH spatial shift, and this GH spatial shift must involve with a time delay no matter it is the light-wave traveling process of classical physics or the quantum mechanics. As demonstrated below, the optical phase change with incident angle is theoretically demonstrated to be either a quantum photonic process or the classical lightwave traveling mechanism, which is controlled by the delay time of the photon beam tunneling process in penetration depth [15, 16]. We know, in quantum mechanics theory, an electron beam in semiconductors can have a quantum-mechanical electron-wave transportation process to a potential barrier, so similarly a photon beam can also have a quantum-mechanical light-wave transportation process to a photon tunneling with a quantum effect. In 1990s, Steinberg and Chiao of the UC Berkeley did an experiment of passing through and blocking back of photon particles by using an optical system of two prisms with an air gas, then they found that the delay time of transmitted beam is shorter than the value calculated by the ray optical trajectory method, which is also referred to as an FTIR phenomenon. The FTIR phenomenon produces a quantum change of optical phase and was equivalently depicted as in Figure 2, where

where

If we use φ replace

From Ref. [15], as shown in Figure 2, we have the delay time of transmitted beam during producing the shift

Then, by comparing Figure 2 with Figure 1, the delay time in Eq. (3) can be understood as the phase change of the reflected beam after it suffers from a GH shift. Then, setting

Substituting Eq. (2) into Eq. (4) yields

Then, we obtain

We know

Thus, the term

### 2.2. Duality of spatial and angular shifts in GH effect

In 1980s the scientists of the United States predicted, it is reasonable for the reflected beam to display a small angular deviation from the law of specular reflection [20, 21]. Until two decades later, in 2006 and 2009, the experimental results of the angular shift under the GH effect were observed in microwave and optical domains, respectively [9, 22]. In the optical experiments, the general role of GH spatial and angular shifts is modeled with both the phase and amplitude of reflected beam, then with Figure 1, by setting a multimode interference (MMI) with

where

Then, with the propagation constant of the input guided mode

where

For the TE- and TM-mode, the amplitude reflection coefficients

where

### 2.3. Understanding to the quantum spatial and angular shifts in GH effect

We know all the guided modes of an MMI waveguide are the eigenstates of quantum physical process and the corresponding effective indices are the eigenvalues of all the refractive indices of optical beam at the phase velocities [23, 24]. So, if we set the input waveguide channel as a single-mode,

For a quantum system, if

Comparing (13) with (12a) and (12b) yields the conclusions that the TE-mode corresponds to the case that the effective mass is independent of x, while the TM-mode corresponds to the case that the effective mass is dependent of x. Hence, we can find the following correspondences for qth mode:

It turns out that the angular shift defined by Eq. (10b) is a quantum selection from multiple eigenstates of the reflection angle under the GH effect. It turns out from relation (14a) that the distance z of light wave traveling corresponds to the time t of photon beam tunneling, while in quantum mechanism, there is no variation of particle velocity, so this corresponding relation is very reasonable. It turns out from relation (14b) again that the propagation constant relates the effective wave number of the guided mode in the traveling-through channel, while the Plank constant

## 3. A digital optical switch principle with the waveguide corner

### 3.1. Device concept and condition for the digital optical switch

With the SOI-based waveguide corner mirror (WCM) structure comprising semiconductor (silicon) waveguide channel material, oxide (SiO_{2} or SiON) corner mirror material and metal material for electrodes, a new regime of 1xN scale digital optical switches was proposed as shown in Figure 3. This special regime of digital optical switches composed of a single-mode input waveguide, a tapered multi-mode corner waveguide, a corner mirror and N single-mode output waveguides is proposed, where N is an integer. In this device regime, the single-mode output waveguide channels are all in conjunction with the output end of the tapered MMI corner waveguide [15, 16]. Then, the free-carrier concentration variation at the interface area of mirror and waveguide is controlled to realize the FCD-RIM of silicon of this WCM structure [16, 17, 18, 19].

In optical switching operations, as shown in Figure 3, all the possible angles of the reflected guided mode are no longer the same due to an angular shift of the GH effect. Namely, the angular shift in this project makes the reflection angle different from the input angle and then creates a great total displacement at the output end of tapered MMI waveguide with a combined effect of the spatial shift and angular shift as defined by Eq. (8).

### 3.2. Discussion for the mode distributions of MMI waveguide

In this 1xN type digital EO switch shown in Figure 3, the tapered MMI waveguide is a critical part as it not only forms a reflecting interface with the corner mirror, but also transfers the reflected optical beam from a single-mode input waveguide to its corresponding single-mode output waveguide. Thus, at the input end this tapered MMI waveguide is required not only to have the corresponding eigenstate guided mode to match the eigenstate reflected-mode of the GH effect, but also have the length to maintain the single-mode state of reflected guided mode from the input waveguide to the corresponding output waveguide.

With the consideration for both the GH spatial and angular shifts in the FTIR process of WCM structure, the tapered MMI waveguide was set to meet the conditions at its input end so that it can produce a sufficient separation between two adjacent output modes at its output end. As explained above, the tapered MMI waveguide is assumed to contain N modes with the numbers q = 0, 1, …, N-1, the GH angular shift defined by Eq. (10b) is a quantum selection for the eigenstate reflection angle of a reflected guide-mode to match one of the multiple guided modes of tapered MMI waveguide at its input end which has an effective index

In the MMI waveguide a specified factor is the beat length of the two lowest-order modes (i.e., the two key modes)

The 1-mode length is required to be

For the qth mode eigenstate guided mode, the lateral wavenumber

Accordingly, the propagation constant spacing of any high order mode from the fundamental mode is expressed as

### 3.3. Matching condition between the RIM modulation and the guided mode

In the FCD-RIM process of an SOI-waveguide device, if the concentration of free-carriers in silicon waveguide has a variation for the guided mode, at λ = 1550 nm the RIM of silicon is defined by the refined Drude-Lorentz model as [18, 19].

where

where the reflection coefficient

We know, there are more than two guided modes in the input end of an MMI waveguide and any independent guided mode has an independent

## 4. Simulations and analyses for the FCD-based modulation performance

In Section 2.2, we theoretically have proved that both the GH spatial and angular shifts form a total displacement of reflected beam at the output end of MMI waveguide. So, the performance of this digital optical switch is based on the dual GH shifts.

### 4.1. Analysis for both the spatial and angular shifts of GH effect

By selecting the CMOS-compatible SOI waveguide and taking the silicon layer thickness as H = 220 nm, ridge height as h = 130 nm, silicon refractive index as n_{1} = 3.43, and SiO_{2} refractive index as n_{2} = 1.46, then for the rib widths: 350 and 400 nm, with the finite-difference time-domain (FDTD) software we obtain the effective indices and the corresponding effective widths of the guided modes at λ=1550 nm and x-polarization as depicted in Table 1.

Rib width, W_{r} (nm) |
350 | 400 |

Effective index at E_{x}, N_{eff} |
2.3368 | 2.4114 |

Effective mode-width at E_{x}, W_{eff} (nm) |
~350 | ~400 |

As shown in Figures 3 and 4, the GH effect happening on the FTIR interface has the changes of two physical parameters—the position and angle of reflected beam, which both strongly depend on the incident angle θ when it is close to the Brewster critical angle

In our previous work, the switching scale potential of digital optical switch scheme was only based on the GH spatial shift. However, as analyzed above, in a real FTIR process the reflected beam not only has an anti-trajectory rule phenomenon of reflecting position, but it also predicts a non-specular reflective angle shift, thus as a continuous research project, these two special photonic phenomena in the FTIR process are both dependent on one quantum state of photonic tunneling process. In the same manner, when the input single-mode waveguide has the rib widths of 350 and 400 nm, with the data used for Figure 5 and Eq. (10b) we further obtain the simulations for the dependences of the GH angular shift Θ of reflected mode on the incident angle θ as shown in Figures 6(a) and 6(b), respectively. Note from Figures 5 and 6 that, under the quantum GH effect, both the spatial and angular shifts of reflected beam have the sharp responses to the incident angle and are mutually influenced. However, the difference between these two shifts is that the spatial shift Δ is plus while the angular shift Θ is minus, which can be manipulated to realize a switching function with the WCM structure.

By selecting the rib width of 350 nm, the dual dependences of spatial and angular shifts on the incident angle is shown in Figure 7. Two significant interesting points should be found in Figure 7 as that (1) in a vicinity of Brewster critical angle

### 4.2. Analysis for the mode distribution at the input end of MMI waveguide

In order to meet the mature fabrication technique for future experiments, we select the CMOS-compatible 220 nm standard SOI-PIC platform with 130 nm rib height, at λ = 1550 nm the refractive indices for core and cladding layers are 3.43 and 1.46, respectively. For the input waveguide having a rib width of 350 nm, at the input end of MMI waveguide as shown in Figure 8(a), we calculate the optical field distribution of reflected guided mode as depicted in Figure 8(b). Note that the mode size is approximately 2.00 μm, so at the input end of the MMI waveguide the smallest center-center spacing between two adjacent ports is 2.00 μm, then total width should be set in the range from 2.00 to (2.00 + 11.4) μm.

### 4.3. Investigation for the total GH displacements at the MMI output end

In accordance with the simulations of both the spatial and angular shifts of GH effect shown in Figures 5 and 6, we determine the input end width

Note from Figure 9 that the sharp response range is the common linear response area of the two GH shifts in Figure 6, which is in the range of incident angle from 38.91 to 38.95° and labeled as a consistent change area, and in the other two incident angle areas, namely, it is smaller than 38.91° and greater than 38.96°, there are two the moderate response ranges of

From Figures 5 and 6 we know the sharp-response area of GH effect to the incident angle is a consistent area between the spatial shift and the angular shift under this GH effect, and Figure 9 proves that the total displacement is co-contributed by the spatial and angular shifts. So, as a result, in the incident angle range from 38.83 to 39.16° it presents a sharp linear response area to the incident angle for a given MMI width at its input end. Therefore, for this 350 nm rib waveguide we first select the incident angles as 38.7, 38.9 and 39.1° in the sharp-response area of GH effect in Figure 9 that are all very close to the critical angle and then at the wavelength of 1550 nm obtain the dependence of the total displacement of the reflected beam on the free-carry hole concentration change (HCC) for the two given MMI waveguide width values at its input end: 2.5 and 5.0 μm as shown in Figure 10(a) and (b), respectively. In this process, as described above, the HCC activates an RIM of silicon material at the reflective interface between the silicon material of waveguide and the silicon-dioxide of corner mirror as defined by the Drude-Lorenz Eq. (21). In fact, Figure 10 presents one significant attribute of the GH-effect based total displacement of reflected beam when the HCC is changed from 0 to

Similarly, by selecting three incident angles: 39.1, 39.3 and 39.5° that are in the moderate area of Figure 9 for the incident angle greater than the critical angle we obtain the total displacement dependence of the reflected beam on HCC as shown in Figure 11 where (a) and (b) are still for the two MMI waveguide input-end widths: 2.5 and 5.0 μm, respectively. Note from Figure 11 that the HCC of

## 5. Verification for the GH shifts

### 5.1. Simulations for the GH spatial shift with finite difference time-domain (FDTD) software

As early as the end of 1990s, we had started to investigate the optical output performance of waveguide corner mirrors with the theoretical modeling, numerical and professional software simulations and published the achievements in 2009 [27]. Based on the numerical calculations and finite difference time-domain (FDTD) simulations for the optical power transfer efficiency of SOI waveguide corner mirrors, an SOI rib waveguide structure is designed as shown in Figure 12(a), then in order to specially simulate the impact of the GH shift on the optical power transfer efficiency, a waveguide corner mirror (WCM) structure is designed to have one input single-mode waveguide channel and one output single-mode waveguide channel as shown in Figure 12(b), where the reflecting mirror is created with a step fabrication of deep etching as shown in Figure 12(c).

In Figure 12(b) the position

SOI substrate standard | Thick Si film, 4.0 μm | Thin Si film, 1.5 μm |

Rib width, W_{r} (μm) |
4.0 | 2.0 |

Rib height, h (μm) | 1.0 | 0.5 |

### 5.2. Design, fabrication and test of SOI waveguide corner mirror structure

We designed, fabricated and characterized the WCM structures in 2009-2011 and presented the achievements in 2013 [19]. Based on the fabrication condition of SOI waveguides, we selected the thin SOI substrate and then designed the

## 6. Conclusions

In this chapter, based on the coherent quantum process of the GH spatial and angular shifts, the new mechanism of substantial digital optical switches is investigated with a WCM structure. For the new regime of digital optical switches, an ideal optical refractive index modulation is FCD effect of semiconductor silicon for ultrahigh speed switching operations of pico- and nanosecond levels, but the other refractive index modulation such as the thermos-optic modulation can also be selected to realize the μs level operations. In addition, the mutual enhancing contributions of the two quantum GH shifts can be further developed to optimize the total displacement of reflected beam for digital optical switches. Therefore, this work will be helpful for research and development of high integrable and high-speed optical and photonic switches on the platform of silicon photonics.

## Acknowledgments

This work is co-sponsored by the Innovative R&D Fund of CUST/China, the Research Program of Ontario Centre for Excellences (OCE) /Canada, and the in-kind invest of D&T Photonics, a startup spun-off from University of Ottawa. Author is very grateful for Prof. Trevor J. Hall for his supervising in design and fabrication of the experimental samples, and also thanks his graduated students: Xiaoqi Li and Jia Yi for their supporting works.

## References

- 1.
Goos F, Hänchen HH. Ein neuer und fundamentalater versuch zur totalreflexion. Annals of Physics. 1947; 1 :333-346. DOI: 10.1002/andp.19474360704 - 2.
Artmann K. Berechnung der seitenversetzung des totalreflektierten strahles. Annals of Physics. 1948; 6 :87-102. DOI: 10.1002/andp.19484370108 - 3.
Chiu KW, Quinn JJ. On the Goos-Hänchen effect: A simple example of a time delay scattering process. American Journal of Physics. 1972; 40 :1847-1851. DOI: 10.1119/1.1987075 - 4.
Wild WJ, Giles CL. Goos-Hänchen shifts from absorbing media. Physics Review. 1982; A25 :2099-2101. DOI: 10.1103/PhysRevA.25.2099 - 5.
Lai HM, Chen FC, Tang WK. Goos-Hänchen effect around and off the critical angle. Journal of the Optical Society of America. 1986; A3 :550-557. DOI: 10.1364/JOSAA.3.000550 - 6.
Lai HM, Chan SW. Large and negative Goos-Hänchen shift near the Brewster dip on reflection from weakly absorbing media. Optics Letters. 2002; 27 :680-682. DOI: 10.1364/OL.27.000680 - 7.
Qing DK, Chen G. Goos-Hänchen shifts at the interfaces between left- and right-handed media. Optics Letters. 2004; 29 :872-874. DOI: 10.1364/OL.29.000872 - 8.
Oh GY, Kim DG, Choi YW. The characterization of GH shifts of surface plasmon resonance in a waveguide using the FDTD method. Optics Express. 2009; 17 :20714-20720. DOI: 10.1364/OE.17.020714 - 9.
Merano M, Aiello A, Exter MP, Woerdman JP. Observing angular deviations in the specular reflection of a light beam. Nature Photonics. 2009; 3 :337-340. DOI: 10.1038/NPHOTON.2009.75 - 10.
Ziauddin SQ, Quama S, Zubairy MS. Coherent control of the Goos-Hänchen shift. Physics Review. 2010; A81 :023821. DOI: 10.1103/PhysRevA.81.023821 - 11.
Wang LG, Chen H, Zhu SY. Large and negative Goos-Hänchen shift near the Brewster dip on reflection from weakly absorbing media. Optics Letters. 2005; 30 :2936-2938 - 12.
Aiello A. Goos-Hänchen and Imbert-Fedorov shifts: A novel perspective. New Journal of Physics. 2012; 14 :1-12. DOI: 10.1088/1367-2630/14/1/013058 - 13.
Bliokh KY, Aiello A. Goos-Hänchen and Imbert-Fedorov beam shifts: An overview. Journal of Optics. 2013; 15 :014001. DOI: 10.1088/2040-8978/15/1/014001 - 14.
Rechtsman MC, Kartashov YV, Setzpfandt F, Trompeter FH, Torner L, Pertsch T, Peschel U, Szameit A. Negative Goos-Hänchen shift in periodic media. Optics Letters. 2011; 36 :4446-4448. DOI: 10.1364/OL.36.004446 - 15.
Steinberg AM, Chiao RY. Tunneling delay times in one and two dimensions. Physics Review. 1994; A49 :3283-3295. DOI: 10.1103/PhysRevA.49.3283 - 16.
Sun DG. A proposal for digital electro-optic switches with free-carrier dispersion effect and Goos-Hanchen shift in silicon-on-insulator waveguide corner mirror. Journal of Applied Physics. 2013; 114 :104502. DOI: 10.1063/1.4820378 - 17.
Campenhout JV, Greens WMJ, Vlasov YA. Design of a digital, ultra-broadband electro-optic switch for reconfigurable optical networks-on-chip. Optics Express. 2009; 17 :23979-23808. DOI: 10.1364/OE.17.023793 - 18.
Reed GT, Mashanovich G, Gardes FY, Thomson DJ. Silicon optical modulators. Nature Photonics. 2010; 4 :518-526. DOI: 10.1038/nphoton.2010.179 - 19.
Sun DG, Liu P, Hall TJ. Realization for the high electro-optic modulation depth of silicon-on-insulator waveguide devices. In: Proceedings of the IEEE 2013 International Conference on Control Engineering and Information Technology (ICCIT2013); 9-11 August 2013; Nanning, China: IEEE; 2013. pp. 700-703 - 20.
Chan CC, Tamir T. Angular shift of a Gaussian beam reflected near the Brewster angle. Optics Letters. 1985; 10 :378-380. DOI: 10.1364/OL.10.000378 - 21.
Tamir T. Nonspecular phenomena in beam fields reflected by multilayered media. Journal of the Optical Society of America. 1986; A3 :558-565. DOI: 10.1364/JOSAA.3.000558 - 22.
Aiello A, Merano M, Woerdman JP. Duality between spatial and angular shifts in optical reflection. Physics Review. 2009; A88 :061801. DOI: 10.1103/PhysRevA.80.061801 - 23.
Bachmann M, Besse PA, Melchior H. General self-imaging properties in NN multimode interference couplers including phase relations. Applied Optics. 1994; 33 :3905-3911. DOI: 10.1364/AO.33.003905 - 24.
Soldano LB, Pennings ECM. Optical multimode interference devices based on self-imaging: Principles and applications. Journal of Lightwave Technology. 1995; 13 :615-627. DOI: 10.1109/50.372474 - 25.
Kawano K, Kitoh T. Schrödinger equation. In: Introduction to Optical Waveguide Analysis—Solving Maxwell’s Equations and the Schrödinger Equation. New York, USA: Wiley Interscience; 1995. pp. 615-627. DOI: 10.1002/0471221600.ch7 - 26.
Himeno A, Terui H, Kobayashi M. Loss measurement and analysis of high-silica reflection bending waveguides. Journal of Lightwave Technology. 1998; 6 :41-46. DOI: 10.1109/50.396 - 27.
Sun DG, Li X, Wong D, Hu Y, Luo F, Hall TJ. Modeling and numerical analysis for silicon-on-insulator rib waveguide corner. Journal of Lightwave Technology. 2009; 27 :4610-4618. DOI: 10.1109/JLT.2009.2025609