Exploring the optical beam shifts in monolayers of transition metal dichalcogenides using Gaussian beams

Accepted Manuscript Exploring the optical beam shifts in monolayers of transition metal dichalcogenides using Gaussian beams Akash Das, Manik Pradhan

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S0030-4018(18)31140-4 https://doi.org/10.1016/j.optcom.2018.12.082 OPTICS 23750

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Optics Communications

Received date : 26 September 2018 Revised date : 18 December 2018 Accepted date : 25 December 2018 Please cite this article as: A. Das and M. Pradhan, Exploring the optical beam shifts in monolayers of transition metal dichalcogenides using Gaussian beams, Optics Communications (2018), https://doi.org/10.1016/j.optcom.2018.12.082 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Exploring the optical beam shifts in monolayers of transition metal dichalcogenides using Gaussian beams Akash Das1 and Manik Pradhan1,2* 1

Department of Chemical, Biological and Macro-Molecular Sciences, S. N. Bose National Centre for Basic Sciences, Salt Lake, JD Block, Sector III, Kolkata-700106, India 2

Technical Research Centre (TRC), S. N. Bose National Centre for Basic Sciences, Salt Lake, JD Block, Sector III, Kolkata-700106, India

26th September 2018 (Submission date) 18th December 2018 (Revised manuscript submission date) Figures: 10

*Corresponding author: Email: [email protected]

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Abstract: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

We have extensively studied Goos-Hänchen (GH) and Imbert-Fedorov (IF) shifts for reflection of a fundamental Gaussian beam using transfer matrix method. By considering a dielectric slab coated with monolayer of transition metal dichalcogenides (TMDC), we theoretically investigate the potential role of four different TMDC monolayers (WS2, WSe2, MoS2, and MoSe2) on the spatial and angular GH and IF shifts for reflection of the light beam that has not been explored previously. We find the nature of GH and IF shifts to be explicitly dependent on the mode of polarization of light beam. In case of partial reflection of light, both GH and IF shifts acquire moderate magnitude. In contrary, giant negative spatial GH shifts are examined for total internal reflection. Our analysis revealed that the typical characteristics of GH and IF shifts are significantly affected by the complex surface conductivity of TMDC monolayers and consequently the shifts are found to differ for different TMDC monolayers. We also present a comparison of the beam shifts for the monolayer TMDC-coated surfaces with the corresponding bulk TMDCs. Finally, we address the most significant question of how the GH and IF shifts depend upon the wavelengths of incident light, in particular, establishing the role of optical conductivity in beam shifts.

Keywords: Goos-Hänchen shift, Imbert-Fedorov shift, Transition metal dichalcogenides, Gaussian beam, Surface Optics, Light matter interaction.

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1. Introduction: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

The phenomena of reflection and refraction of electromagnetic field from a planar surface are governed by Snell’s law and Fresnel formulas [1, 2] producing the simplest expressions for plane waves. But, for a beam of light with finite transverse extent, the electromagnetic field is resolved into group of plane wave components (forming an envelope in the k-space) each of which undergoes reflection individually following the laws of specular reflection. The nature of the reflected beam can be observed by summing up the plane wave components. However, depending upon the nature of interaction of light beam with the surface, non-specular reflection may manifest as longitudinal shift in the plane of incidence [3-5] or transverse shift orthogonal to the plane of incidence [6-9] which are known as Goos-Hänchen shift (GH) and Imbert-Fedorov shift (IF), respectively [10-14]. Although these effects are typically very small, their magnitude can be significantly increased by means of weak measurement techniques [8, 15-20]. However, being in the sub-wavelength domain, these shifts affect the modes of optical waveguides and microcavities [21, 22]. Hence, over the past few decades, GH and IF shifts have received enormous applications in a variety of fields, e.g. bio-sensors, optical heterodyne sensors, measurement of beam angle, beam displacement, refractive index, film thicknesses, characterization of permittivity and permeability of different materials [23]. Recently, there has been immense research interest in 2-Dimensional (2D) materials such as graphene,

hexagonal

boron

nitride,

topological

insulators

and

transition

metal

dichalcogenides (TMDC) [24-26] owing to their unique and versatile electrical, optical, chemical and mechanical properties. Among these extraordinary properties, TMDCs exhibit relatively high carrier mobility, ultrafast carrier dynamics, photoluminescence and electroluminescence, ultrafast nonlinear absorption, second and third harmonic generations, as well as indirect-to-direct band gap transition as bulk TMDC decreasing to monolayers [27]. TMDCs include a large family of layered materials, which can be represented by the

3

formula MX2, where M is a transition metal element, and X is a chalcogen atom, one layer of 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

M atom being sandwiched between two layers of X atoms, e.g. WS2, WSe2, MoS2, and MoSe2. The monolayer (ML) of these materials is of paramount importance as they are direct bandgap semiconductors [28] in sharp contrast to the multi-layered TMDCs which are indirect band gap in nature, thus making them suitable for applications in optoelectronics as transistors, emitters and detectors. Moreover, the presence of strong spin-orbit coupling [29] combined with the absence of inversion symmetry of TMDC monolayers suggests the possibility of novel applications of these new class of materials for valleytronics, optoelectronics, and nanotechnology [30]. In the last decade, the GH and IF shifts have been observed in various systems such as graphene [15, 31-38], photonic crystals [39], and metamaterials [40, 41]. But it would be interesting to see how these optical beam shifts phenomena occur in 2D TMDC, an area of research has yet to be reported. In the present study, we have, therefore, formulated a detailed theoretical model of the GH and IF shifts for fundamental Gaussian beam impinging upon dielectric surface coated with a monolayer of TMDC that has not been previously explored. In our study, we have taken four different monolayers of TMDCs under consideration i.e. WS2, WSe2, MoS2, and MoSe2, specifically due to their salient features and extraordinary applications in modern research. We have established a general theoretical model exploiting Transfer Matrix Method (TMM) to describe the beam shifts and both the spatial as well as angular shifts have been determined in case of partial reflection (PR) and total internal reflection (TIR) from the interface. We observed substantial changes in the angular dependence of Fresnel’s reflection coefficients in presence of different TMDC coatings. Being dependent on the reflection coefficients, GH and IF shifts also lead to finite values which might be positive, negative or zero depending on the conditions of PR and TIR. In the previous works of beam shifts, large shifts were predicted in

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close proximity of pseudo-Brewster’s angle and the critical angle of incidence [42, 43]. In 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

this work, we also inferred nontrivial behavior of beam shifts near these special angles in presence of monolayer TMDC coating. Interestingly, giant negative spatial GH shifts are observed for the four different TMDCs under consideration in case of TIR which might be of immense interest in contemporary research. We have also predicted the polarization dependence of the GH and IF shifts of four different ML TMDCs for a light beam of wavelength spanning from 400 to 1200 nm that has not been investigated previously. Subsequently, we have also compared the nature of beam shifts for the monolayer TMDCs to the corresponding bulk structures which show significant dissimilarity. Finally, we have also studied how the maximum value of the optical beam shifts depend upon the wavelength of the incident light beam covering a wide wavelength range of the electromagnetic spectrum. The observed results are remarkable which has not been examined till now, to the best of our knowledge. This gives us the opportunity to select a desired region of the electromagnetic spectrum to observe appreciable beam shifts for a particular material. Besides, it is already well-known that the optical properties of 2D materials justify their macroscopic character and their optical response is determined by their macroscopic surface conductivity [44, 45]. Consequently, we have shown in the present study that the typical behavior of the beam shifts for the different materials can be attributed to the dependence of the Fresnel reflection coefficient on the complex optical conductivity. The remainder of this article is organized as follows. In Sec. 2 we review the mathematical formalism to describe the GH and IF beam shifts experienced by a Gaussian light beam and evaluate their theoretical expressions. Transfer Matrix Method (TMM) for calculation of optics in layered media has been applied. In Sec. 3 we have presented the results and discussed the angular dependence of Fresnel reflection coefficients in both partial reflection

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as well as total internal reflection cases. Finally, the main conclusions are drawn and an 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

outlook towards future work is given in Sec. 4.

Fig. 1. (a) Schematic representation of Goos-Hänchen and Imbert-Fedorov Shift from the surface under consideration, (b) Geometry of the coordinate system for reflection of the light beam from the surface.

2. Theory: We consider a monochromatic paraxial electric field, which characterizes a well collimated Gaussian light beam, impinging upon a monolayer (ML) TMDC-coated dielectric surface, as illustrated in Fig. 1. Here, the TMDC monolayer is modeled as an infinitesimally thin, local two-sided surface characterized by a complex surface conductivity    (where  is the frequency of the incident light beam in radian) following the previous works for graphene [46, 47]. For convenience, we can define three coordinate systems: the laboratory frame

ˆ y, ˆ zˆ x,

attached to the dielectric interface and oriented with zˆ axis pointing towards the dielectric, and two auxiliary frames xˆ k , yˆ k , zˆ k  attached to the incident ( k  i ) and the reflected ( k  r ) field, respectively [11, 14, 31, 42, 48]. In the angular spectrum representation, the electric field of the incident beam at the air side of the interface ( z  0 ) can be written as [49]: 6

2 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Ei  r     e  U, V,   E   U, V,   ei UXi  VYi  WZi dUdV

(1)

1 

where  is the polarization index (   1, 2 corresponds to p and s-polarization, respectively). Also, U  k.xˆ i k 0 , V  k.yˆ i k 0 and W  1  U2  V2 are the dimensionless components of the wave vector k in the incident frame, connected to the wave vector in the laboratory frame by:

ˆ x  yk ˆ y  zˆ k 02  k 2x  k 2y k  k 0  Ux i  Vyi  Wzi   xk

(2)

Moreover, in Eq. (1), E  (U, V) is the vector spectral amplitude which determines the shape and polarization of the beam and can be written as: E  U, V   A  U, V    U, V  , with

A  U, V  being the angular spectrum of the incident field and   U, V  the polarisation



function given by    U, V,    eˆ   k  .fˆ , with eˆ  k representing the polarisation vectors attached to the single plane wave components of the incident beam. The polarization unit

eˆ 2  k zˆ  k and eˆ 2  , where the ‘×’ symbol ˆe2  k zˆ  k

basis vectors e  have been chosen as: eˆ1 

denotes the standard vector cross product in basis set in

3.

3





. Thus eˆ1 ,eˆ 2 , kˆ forms a complete orthogonal

Conventionally, a plane wave whose electric field is parallel to either eˆ 1 or eˆ 2

is referred to as either TM (p-polarization) or TE wave (s-polarization), respectively. The unit complex vector fˆ  a p xˆ i  a s exp  i yˆ i (with a p , a s , 

) fixes the polarization state of the

beam. After reflection, each plane wave component of the Fourier spectrum (Eq. 1) undergoes modification by Fresnel’s law.



 

Therefore, eˆ  k eik.r  r k eˆ  k eik.r

7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65



 

ˆ where r k is the Fresnel reflection coefficient and k  k  2zˆ z.k Thus, the electric field in the reflected frame can be expressed as: 2 

E r  r     A   U, V,   ei  UXr  VYr  WZr dUdV

(3)

1 

where A  U, V,   eˆ   U, V,     r  U, V    U, V,   A  U, V  When a 2D material is deposited upon a dielectric surface, the Fresnel’s reflection coefficients are modified and by applying the Transfer Matrix Method (TMM) for layered media [50], these reflection coefficients can be obtained as: rs () 

rp () 

cos   n 2  sin 2   ()

(4a)

cos   n 2  sin 2   ()

n 2 cos   n 2  sin 2  1  () cos 

(4b)

n 2 cos   n 2  sin 2  1  () cos 

where θ is the angle of incidence, n is the refractive index of the substrate, and   

c0

,σ

is the complex surface conductivity of the TMDC under consideration and  1  is the  c0  impedance of vacuum. The Fresnel reflection coefficients can be expressed as:

r  R  exp  i 

(5)

where R  is the amplitude and   is the phase of the reflection coefficient (with  p,s ). After some mathematical steps [14, 31, 42, 48], we can arrive at dimensionless spatial    and angular   GH and IF shifts which can be specified by the following expressions:

k 0  GH  w p

p 

 ws

s 

(6a)

8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

 w p a s2  w s a p2  k 0  IF   cot   sin   2 w p w s sin    p  s   a pa s  

(6b)

  ln R p  ln R s  GH    w p  ws     

(7a)

IF 

w p a s2  w s a p2 a pa s

where w  

cos  cot 

(7b)

a 2 R 2 is the fractional energy contained in each polarization state. a R 2P  a S2 R S2 2 P

Similarly, we can also calculate the GH and IF shifts for total internal reflection (TIR) from the interface. In this case, the reflection coefficients are modified as prescribed before and can be calculated by the Transfer Matrix Method [50]:

rs () 

rp () 

n cos   1  n 2 sin 2   n()

(8a)

n cos   1  n 2 sin 2   n()

cos   n 1  n 2 sin 2  1  () cos 

(8b)

cos   n 1  n 2 sin 2  1  () cos 

Further calculation [31] provides adimensional spatial and angular GH & IF shifts for TIR case which gives identical expressions (Eq. 6, 7). 3. Results & Discussion: We shall discuss the behavior of GH and IF shifts separately for the two cases i.e. partial reflection (PR) and total internal reflection (TIR). Prior to this, we need to evaluate the significant parameters that have considerable influence on the beam shifts. From Eq. (4) and Eq. (8), it is explicit that the monolayer (ML) TMDC-coating on the dielectric surface affects the Fresnel reflection coefficients by introducing a part containing complex optical 9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

conductivity      of the material and thus it is important to understand beforehand the wavelength dependent optical conductivity of all the ML TMDC-coated materials under consideration. However, calculation of the optical conductivity requires the knowledge of dielectric permittivity of the material. There are several techniques to determine the permittivity of ML TMDCs, such as the Lorentz model [44], the Hybrid Lorentz-DrudeGaussian model [51], the variational dielectric Kramers-Kronig function based on Lorentz [45], the self-consistent method [52], and the surface current model [53]. In the present work, we have considered the Lorentz model as it is a good approximation [54] producing consistent fittings to the results of experimental reports [45, 55]. In case of TMDC monolayer materials, the relative electric permittivity can be expressed as a superposition of N Lorentzian functions:

 r   

N N    fk f kE  1  2  1  2 2 2 E 0 k 1 k    i k k 1 E k  E  iE k

(9)

where f k , k and  k are the oscillator strength, resonance frequency, and spectral width of the kth oscillator, respectively. The values of the model parameters for the four TMDC monolayers under investigation were taken from Ref. [44]. The complex surface optical conductivity of the TMDC monolayers can now be easily calculated as [44, 45]:

S  E   r  E   ii  E   i0h eff 1  r  E  

(10a)

S ih eff  1  r  E   0 c

(10b)

where 0 

1 e2 is the universal dynamic conductivity,   is the fine structure constant, 137 4

c  3 108 m / s is the velocity of light in free space and h eff is the effective thicknesses of

TMDC monolayers. 10

Spectral peaks can be observed at wavelengths specific to the particular TMDC monolayers. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Observing this wavelength dependence of optical conductivity, we expect the reflection coefficients to rely upon wavelengths. For convenience, we shall work at a particular wavelength of 633 nm for further analysis unless specified.

Fig. 2. Angular dependence of modulus ( R  ) and phase (   ) of the reflection coefficient for p-polarization (upper panel) and s-polarization (lower panel) in PR case.

As depicted beforehand, GH and IF shifts are dependent upon the reflection coefficients. Hence, the behavior of the modulus (R) and phase (  ) of the reflection coefficients for four different ML TMDC-coated (WS2, WSe2, MoS2, and MoSe2) surfaces are shown in Fig. 2. Clearly, the presence of TMDC monolayers alters both the modulus as well as phase of the reflection coefficients if we compare to the corresponding uncoated surface (shown by dotted curves in the figures). But the four TMDC coated surfaces do not exhibit appreciable changes among themselves in their behavior of reflection coefficients and thus we may expect a 11

similar response in GH and IF shifts for the four specific TMDC monolayers, differing only 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

slightly. The reflection coefficients for s-polarization, R S and S increase smoothly with the angle of incidence (  ) though the change in S is negligible. However, for p-polarization, R P attains minimum at Pseudo-Brewster’s angle ( PB ) (but R P (  PB )  0 for uncoated surface). Again,  P undergoes a sharp jump from 0 to  in the vicinity of Pseudo-Brewster’s angle (

PB ). These effects are clearly reflected in the nature of spatial as well as angular GH and IF shifts as depicted in Fig. 3. The spatial GH shifts are dependent on the slope of  vs  graph as obvious from Eq. (6a). Therefore, dimensionless spatial GH shift ( k 0  GH ) which is zero for PR in uncoated surfaces, as was originally formulated by Goos and Hӓnchen [3], now becomes nonzero in presence of ML TMDC coating. This is quite a remarkable observation in our study. Due to a negligible change in the slope of S vs  graph, we observe a non-zero spatial shift for s-polarization

(k 0 SGH ) but it is quite small in magnitude [Fig. 3 (b)]. However, k 0  SGH increases monotonically with increasing angle of incidence. But, for p-polarization, the sharp jump of P  P in the vicinity of PB , gives rise to a large spatial GH shift ( k 0  GH ) as depicted in Fig. 3

(a). Thus, we reckon nonzero spatial GH shifts for both s and p-polarization of incident light even in case of PR. Subsequently, to discuss the spatial IF shifts, knowledge of  P  S  is important as is clear from Eq. (6b). If we carefully look at Fig. 3 (b) and (d), we can find out  P  S  to change from  to 2 in the vicinity of PB . Thus, we expect a nonzero spatial IF shift which is

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depicted in Fig. 3 (c) and (d) for linear polarization (450) (LP) and circular polarization (CP) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

state, respectively in contrast to zero IF shift for the uncoated surface.

Fig. 3. Angular dependence of spatial GH shift (upper panel) and IF shift (lower panel) for PR case.

Adding to that, we can infer from Fig. 3 that the four ML TMDC-coatings on dielectric surfaces have almost congruent effects. The slight change in the nature of shifts can be attributed to the magnitude of optical conductivity of the particular TMDC monolayer at the specific wavelength considered. The values of optical conductivity at 633 nm for the selected four TMDC monolayers are adjacent to each other. On that account, the spatial shifts do not differ appreciably for the four different TMDC monolayers. As the optical conductivity of TMDC monolayers is dependent on the wavelength of incident light, we should expect modification in the response of GH and IF shifts if we change the wavelength of the incident beam of light. This will be clear in the later part of this section. 13

We can also compare the optical response of the beam shift for the TMDC coated surface 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

with the graphene-coated ones [31]. Giant negative GH shift was observed for graphenecoated dielectric, but the presence of TMDC produces nonzero GH shifts which are smaller in magnitude than the former.

Fig. 4. Angular dependence of modulus ( R  ) and phase (   ) of the reflection coefficient for p-polarization (upper panel) and s-polarization (lower panel) in TIR case.

We shall now focus on TIR condition where light is incident from the dielectric side of the interface. The angular dependence of modulus (R) and phase (  ) of the reflection coefficients for four different TMDC-coated surfaces in TIR case are depicted in Fig. 4 along with the uncoated surface (dotted curve). We can clearly observe the difference in the behavior of reflection coefficients in presence of TMDC coating compared to the uncoated one. For incident angles greater than the critical angle ( C ), both S and  P sharply decreases 14

from 0 to  . This finite negative slope of  vs  curve gives rise to finite negative spatial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

GH shifts for both s and p-polarizations of the light beam as shown in Fig. 5.

Fig. 5. Angular dependence of spatial GH shift (upper panel) and IF shift (lower panel) for TIR case.

P On that premise, both of the spatial GH shifts, k 0  SGH and k 0  GH attain large negative values. P However, we also notice the small positive value of k 0  GH at Pseudo Brewster’s angle which

is due to the positive slope of  P vs  curve in the vicinity of PB . However, in the present study, the observation of giant negative spatial GH shift is also a notable observation in case of TIR condition. Next, to discuss the IF shift we evaluate the value of  P  S  . It is clear from Fig. 3. that

 P  S  value increases from a value close to 15

 to a value close to 0 in the vicinity of the

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

critical angle ( C ). On that ground, we anticipate negative spatial IF shift in presence of ML TMDC coating for both diagonal linearly polarized    00  (LP) and circularly polarized light    900  (CP) [Fig. 5]. Moreover, the response of the four different TMDC coated surfaces does not differ appreciably from each other which is due to the close proximity of the optical conductivities of the ML TMDCs at this particular wavelength (633 nm). As a consequence, the spatial shift in case of TIR can significantly be enhanced by choosing an appropriate wavelength. The behavior of the maximum spatial shifts for different wavelengths has been discussed in the later part of this section. We can also compare the present findings to the graphene-coated surfaces [31]. Giant spatial shifts are observed in both the cases, though the positive peak in the vicinity of pseudoBrewster’s angle was not observed for graphene-coated surfaces. Moreover, the dip in the angular dependence of IF shift ( k 0  lin IF vs  plot) in the vicinity of PB is a notable difference from the graphene-coated surface.

Fig. 6. Angular dependence of angular GH shift and IF shift for PR (upper panel) and TIR case (lower panel).

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Simultaneously, we can also examine the angular GH and IF shifts which are in turn 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

governed by the angular dependence of the modulus of reflection coefficients. As shown in Fig. 6, the changes of R  in presence of TMDC is pretty small. Hence, we expect nonzero angular GH shifts, but considerably smaller in magnitude than the spatial GH shifts. Adding to that, since   0 , the angular GH shift is zero for circularly polarized light. But small angular IF shifts are observed for diagonal linearly polarized light in both PR as well as TIR conditions. Before proceeding further, we want to emphasise on the peculiar nature of the spatial [Fig. 5(a), (b)] as well as angular GH [Fig. 6(d), (e)] shifts in case of TIR. We can clearly notice that there is a sudden spike near the critical angle of incidence. This might be due to the model dependent divergence of the theory. While deriving the expressions of the beam shifts, the analysis has been approximated at the first order (in the Taylor series expansion) for the spatial shift and at the second order for the angular part. However, to know the actual origin of these spikes in the beam shifts it would be essential to incorporate the higher order terms in the calculations which is beyond the scope of the present work. Next, we have explored how the beam shifts depend on layer thickness. For this, we have compared the beam shifts for the monolayer TMDC-coated surfaces with the corresponding bulk TMDCs. It is clear from Fig. 7 that GH shifts are affected considerably by layer thicknesses. For spolarized light beam, GH shift is larger for monolayer TMDC coating than the bulk. But for p-polarized light, the effect is reversed i.e. the GH shift is larger for bulk than the monolayer. In addition, for the IF shift in the linearly polarized light beam, maximum shifts are obtained for monolayer TMDCs compared to bulk. However, IF shift for circularly polarized light is found to be unaffected by the layer thicknesses.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Fig. 7. Layer dependence of spatial GH shift (upper panel) and IF shift (lower panel) for PR case. The solid lines denote monolayer whereas dotted lines denote multilayer coatings.

Similarly, we can study the beam shifts for TIR conditions. In this case, GH shifts show no significant dependence on layer thicknesses as demonstrated in Fig. 8 (upper panel). But IF shift for linearly polarized light is larger for monolayer TMDC-coated surfaces than the bulk TMDCs. Again, IF shift for circularly polarized light remains independent of layer thicknesses similar to the case of PR. Taken together, our findings suggest that the difference in beam shifts between ML and bulk materials can be readily seen in the case of PR, but the observations are not so prominent in TIR case. This thickness dependence of the beam shifts is due to the typical behavior of optical conductivity of the material at the particular wavelength under consideration. The optical conductivity of the monolayer TMDC is significantly different from that of the bulk material [44, 45]. Therefore, it indicates that one can differentiate between the monolayer and

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the corresponding bulk TMDC by studying the optical beam shifts, particularly working in 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

the PR region.

Fig. 8. Layer dependence of spatial GH shift (upper panel) and IF shift (lower panel) for TIR case. The solid lines denote monolayer whereas dotted lines denote multilayer coatings.

Finally, we have also investigated how the beam shifts vary with wavelengths of incident light ranging from UV (Ultraviolet)-VIS (Visible) to IR (Infrared) region of the spectrum. To understand this behaviour, we have plotted the maximum of the absolute value of GH and IF shifts with wavelength. Fig. 9 depicts the maximum of the spatial beam shifts for the PR condition in monolayers of TMDCs. In case of p-polarization, the value of maximum GH shift for WS2 shows two comparatively larger peaks around 0.403 µm and 0.613 µm.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Fig. 9. Wavelength dependence of the maximum values of GH shift (upper panel) and IF shift (lower panel) for PR case in monolayers of TMDCs.

The value of the imaginary part of the optical conductivity of WS2 lies very close to zero at these wavelengths which is the most plausible reason for the occurrence of these two large peaks of the maximum GH shifts for WS2. But, the nature of maximum GH shifts for other three materials WSe2, MoS2 and MoSe2 are quite similar to each other (inset of Fig. 9 (a)). But, for both p and s-polarized light, the maximum GH shifts acquire constant values as we proceed from visible (VIS) to the infrared region (IR) of the electromagnetic spectrum. Hence, we should focus on the ultraviolet (UV) and visible (VIS) region of the spectrum to study the GH shifts experimentally in case of PR. Moreover, the maximum IF shifts for the diagonal linearly polarized (LP) and circularly polarized (CP) beam also differ considerably for all the materials under consideration. But, the maximum of the IF shifts increases as we proceed to the IR region, which is opposite in nature compared to the nature of the GH shifts.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Fig. 10. Wavelength dependence of the maximum values of GH shift (upper panel) and IF shift (lower panel) for TIR case in monolayers of TMDCs.

In a similar manner, we have also studied the wavelength dependence of maximum GH and IF shifts for TIR of light beam in monolayers of TMDCs. The maximum values of GH shifts show a fascinating nature as can be seen from the Fig. 10 (upper panel). Again, the maximum shift for both GH and IF shifts decrease to a constant value in moving from VIS to IR region of the spectrum. Hence, it is better to work in the smaller wavelength region, i.e. VIS and UV range of the spectrum to obtain larger beam shifts in the TIR case. On that ground, we can infer that the maximum value of the GH and IF shifts largely depend upon the complex optical conductivity of the materials. The dependence of the optical conductivity of the TMDCs on the wavelength is clear from Eq. 9. Re  S 0  sharply decreases to zero for all the TMDCs under consideration and Im  S 0  increases to a constant negative value approaching towards zero. Since optical conductivities for all the 21

TMDCs approaches to a constant value in moving from VIS to IR region, the nature of the 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

beam shifts becomes identical for all the TMDCs in the IR region of the spectrum. 4. Conclusion: In conclusion, we have theoretically derived the spatial and angular GH and IF shifts for a fundamental Gaussian beam incident on monolayer TMDC-coated (WS2, WSe2, MoS2, and MoSe2) surfaces and have studied extensively the angular dependence of spatial together with angular GH and IF shifts in partial (PR) as well as total internal reflection (TIR) cases. The beam shifts are found to be explicitly dependent upon the behaviour of modulus and phase of the reflection coefficients of the incident light beam. The reflection coefficients are transformed by the presence of TMDC coating on the dielectric surface which causes the occurrence of nonzero GH and IF shifts. In contrary to the revolutionary work of Goos and Hӓnchen, where nonzero shifts were predicted for total internal reflection of the light beam from surfaces, we have observed non-zero spatial GH shifts even in case of PR along with TIR. Adding to that, we have also obtained nonzero IF shift too for PR case which would have been zero in absence of TMDC monolayer materials. Interestingly, giant negative spatial shifts are also recognized for TIR from the ML TMDC-coated surfaces. Most importantly, the typical behaviour of GH and IF shifts for different TMDCs may be ascribed to their optical conductivity response. The complex optical conductivity is an explicit function of the wavelength of the incident light. Consequently, we have shown the dependence of maximum beam shifts (both GH and IF) with the wavelength of incident light for ML TMDC-coatings. Thus, by tuning the optical conductivity with wavelength, maximum beam shift can be tailored which can play a pivotal role in developing new generation optical sensors and nanodevices. In addition, we have also mentioned the differences in the behaviour of the GH and IF shifts of TMDC monolayers from the graphene-coated surfaces which proves that the optical response of beam shifts in 2D 22

materials have some similarities as well as dissimilarities. We have also compared the GH 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

and IF shifts of these monolayer TMDCs to the corresponding bulk materials. This provides us the possibility to discriminate between the monolayers of these materials and the bulk by observing the beam shifts only. However, in our study, we have predicted the beam shifts for a light beam of fundamental Gaussian mode impinging upon 2D TMDCs. But, the proposed theoretical model can also be applied for studying beam shifts for different modes of incident light beams (e.g. Bessel, Laguerre-Gauss, and Hermite-Gauss) along with different types of layered structures.

Acknowledgment: A.D. and M.P. appreciate the computing facilities of S.N. Bose National Centre for Basic Sciences, Kolkata. A.D. would like to thank UGC (India) for the PhD research fellowship.

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