Enhanced opposite Imbert–Fedorov shifts of vortex beams for precise sensing of temperature and thickness

: Imbert–Fedorov (IF) shift, which refers to a tiny transverse splitting induced by spin-orbit interaction at a reflection/refraction interface, is sensitive to the refractive index of a medium and momentum state of incident light. Most of studies have focused on the shift for an incident light beam with a spin angular momentum (SAM) i.e., circular polarization. Compared to SAM, orbital angular momentum (OAM) has infinite dimensions in theory as a new degree of freedom of light and plays an important role in light-matter coupling. We demonstrate experimentally that the relative IF shifts of vortex beams with large opposite OAMs are highly enhanced in resonant structures when light refracts through a double-prism structure (DPS), in which the thickness and temperature of the air gap are precisely sensed via the observed relative IF shifts. The thickness and temperature sensitivities increase as the absolute value of opposite OAMs increases. Our results offer a technological and practical platform for applications in sensing of thickness and temperature, ingredients of environment gas, spatial displacement, chemical substances and deformation structure.


Introduction
The specular reflection and refraction of light beam are well-known behaviors of light in geometric optics.While the non-specular reflection effects emerge from the spin-dependent excitation of electromagnetic waves at an interface, exhibiting spatial shifts named as Goos-Hänchen (GH) [1] and Imbert-Fedorov (IF) effects [2,3].
The most interesting feature of the IF effect is the spin-related shift in transverse plane and is regarded as one kind of spin Hall effect of light induced by spin-orbit interaction between the spin angular momentum (SAM) and orbital angular momentum (OAM) of light [4].For a paraxial propagating light beam, the SAM is ±1ћ determined by the polarization helicity for right-and left-hand circular polarizations, while the OAM includes extrinsic and intrinsic OAMs given by r  P (r and P are the transverse position and momentum of a beam) and lћ (l = 0, ±1, ±2, … is the topological charge (TC) for a symmetrical left-and right-helicity vortex beam), respectively [5,6].
The traditional IF or spin Hall effect of beam shifts is induced by the SAM of a Gaussian laser beam, and it is relatively small and was measured via weak measurements [7,8].This kind of tiny shift shows applications in sensing of displacement, refractive index [9][10][11].For example, Zhou et al. realized the experimental observation of the spin Hall effect of light on a nanometal film via weak measurements, and their precision for the thickness of the nanometal film is 10 nm [9].Researchers also measured the change of the refractive index by measuring the spin Hall effect via weak measurement [10,11].Meanwhile, the increasing of IF shift and its measurement sensitivity, e.g., using surface plasmon resonance technology [12][13][14] and waveguide technology [15], attract wide interests for temperature and displacement sensors, however these theoretical proposals are not yet demonstrated in experiments.Besides the SAM effect, the intrinsic OAM of structured light beams can be manipulated via controllable values of l (e.g., l = 10) [16], also allowing for enhancing the spin-orbit coupling with high angular momentum [17,18].The IF shift associated with OAM was first experimentally investigated using vortex beams with TE and TM polarizations [19].Furthermore, it was theoretically revealed [20] and experimentally demonstrated [21] that the observed GH and IF shifts are linearly proportional to OAM.The studies of IF shifts in various interfaces were also concerned to show the enhanced effect of OAM on shifts [22][23][24][25][26][27][28].
The enhancement of IF shifts with OAM brings the possibility for its direct observation and supplies a convenient platform for enriching sensor devices.To control the TC for getting higher values of l and to design special resonant structures are experimentally reasonable to enhance the IF shift.Motivated by this consideration, in this work we propose to measure the relative IF shift that is the difference between the IF shifts of two vortex beams with l and -l TCs in the same polarization, giving an alternative way for greatly improving the sensitivity of the IF shift sensor.This difference of the IF shifts induced by opposite OAM beams we proposed is much larger than the difference of the same-OAM IF shifts induced by TE-and TM-polarized beams.Thus, we experimentally investigate the IF shift using a high-order vortex beam in a double-prism structure (DPS), which both lead to the enhancement of the relative IF shift.As we all know that the trajectory of geometric optics does not exist at all, here, measuring the relative IF shift of two vortex beams with l and -l TCs provides us a convenient and effective method to determine the change in the sensed structure.In our experiment, we have observed that the relative difference between the IF shifts of l and -l vortex beams is sensitive to both the air gap d and its temperature T of the DPS, showing the sensing ability for multiple parameters of the system.We have achieved the detected temperature sensitivity 33.388 μm/ °C for |l| = 20.On the other hand, because of the existence of multiple resonance regions in the DPS, the extended sensing range becomes feasible through using different resonant regions, which can further enhance the relative IF difference by choosing the steeper changing resonance peaks.We believe that the enhanced IF shifts induced by spin-orbit interaction in this work can be applied to spatial displacement and temperature sensors.

Scheme and theory
In Fig. 1(a), it displays the schematic of the DPS that consists of two identical isosceles rightangle prisms with an air gap in the middle.The relative permittivity of the two prisms is 2946 . 2 3 1

   
, which can be derived from the measurement of the total internal reflection.
The thickness of the air gap with permittivity being 1 2 


is initially measured at d = 4.16 μm.A monochromatic beam with wave vector k  is incident on the DPS with an angle  of incidence on the interface between the first prism and air gap.The critical total internalreflection angle of light inside the prism is  c  41.312 °, which can be measured precisely.
Although the displacements of light occur at the reflection and transmission interfaces of the air gap, the actual measurements are operated in the transverse plane (vertical to the axis of propagation) outside the second prism.The transmission coefficient of light in this DPS is expressed as where is the z component of the wave vector for 2 2 x j k k   in the jth layer with j = 1, 2, 3, otherwise is the x component of the incident wave vector inside the first prism with for TE polarization.Here, we only consider the TE-polarized cases, and similar results for TMpolarized cases can be analyzed and achieved in the same method.For vortex beams with TE polarization, the OAM induced IF shift can be theoretically calculated from the following expression [19][20][21][29][30][31] where l is the TC of vortex beams.Note that the IF shift in our cases is only contributed from the OAM effect, and higher value of l leads to larger IF shift.There is no contribution from the SAM effect due to the linear polarization of incident light.From Eq. (2), it is clear that the IF shifts for vortex beams with opposite TCs are totally opposite, and their difference can be enhanced in the resonant structures.Thus, we use this difference between the IF shifts for the vortex beams with + l and -l TCs to sense the change of the thickness and temperature.The experimental setup is shown in Fig. 2. A linearly-polarized laser beam 1 with wavelength of 632.8 nm passes through a half-wave plate (HWP1) and a polarizer (P).The combination of HWP1 and P is used to control the beam intensity and polarization.After the beam expander (BE), the beam diameter of the expanded light is about 3.5 mm and the expanded beam is well collimated.Then it passes through the beam splitter (BS) and is incident on the phase-only spatial light modulator (SLM) for generating different vortex beams with controllable TC l.The vortex beam is reflected by the SLM and then is reflected by BS.The Laser beam 2 with wavelength of 520 nm passes through the circular aperture (CA).The CA is used for removing the stray light.This beam is always adjusted to be well collimated with the propagation axis of the vortex beam (generated from the Laser beam 1) when they are coaxially propagating after the BS.We use the transmissivity curves of two light beams with different wavelengths to determine the thickness of the air gap.This double-wavelength method is more accurate than the single-wavelength method.Then the polarizations of these beams are tuned by HWP2.We use the TE-polarized beam as the incident beam to generate and measure the IF shift.Note there are similar results for the cases of TM polarization (see the Supplemental Material).The beams are focused by using the lens L (with the focal length f = 125 mm) onto the interface of the air gap inside the DPS, which is placed on the controllable rotation platform to provide high precision with 1.8″ angular resolution and 36″ angle-positioning accuracy for precisely adjusting the angle of incidence.The role of this lens L here is to provide the waist position of the vortex beams at the place of the air gap of the DPS.Thus, one can safely neglect the effect of the beam's wavefront curvature on the IF shifts due to the propagation effect of vortex beams.The transmitted beams are detected by the beam quality analyzer (BQA) (Ophir SP928, imaging CCD unit 1928×1448 pixels with each pixel size 3.69 μm  3.69 μm), and their intensity distributions are real-timely analyzed by the commercial beam-profiling software (BeamGage).The BeamGage software is capable to determine the location of the beam center in terms of the normalized first-order moment of the measured intensity distribution [32], which gives both GH and IF shifts.Here we should point out that the distance between the DPS and the BQA is only about 8 cm, which can further avoid other effects like the displacement induced by the angular IF shifts (despite nearly zero).Here, we are only interested in IF shifts since GH shifts for different vortex beams in our cases are much smaller than IF shifts, which in turn makes the experiment become more feasible and efficient to detect and separate IF shifts.We have also found that both the change laws of IF shifts for TE and TM polarizations are similar with each other except that the amplitude for TM cases is a little smaller.Thus, in the following contents, we only discuss the results for TE polarization.Note that, the angular IF shifts in our scheme are very tiny, which theoretically induces an additional displacement about 10 -3 m (much smaller than the spatial IF shifts of vortex beams) and cancels each other thus having no contribution to the relative IF shifts we measured.Meanwhile, please see the Supplemental Material for the details of the DPS installation process, the double-wavelength method to determine thickness of the air gap, the experimental results of the IF shifts for the TE and TM polarizations, the comparison of the relative GH and IF shifts for vortex beams with opposite TCs, and the theoretically estimated magnitude of the angular IF shifts.
Since the geometrical light beams are fictitious, we have experimentally measured the relative difference between the IF shifts of the transmitted vortex beams with l and -l.Thus, we have obtained the difference of . In Fig. 3, it displays the IF shifts of vortex beams with l and their difference.In experiments, when the air gap of the DPS is fixed, we first used the vortex beams with l=10 to demonstrate the dependence of the IF shifts with the angle  of incidence.The angle  changes as the rotating platform rotates at a constant speed, meanwhile the BQA is trigged synchronously to capture every intensity distribution corresponding to every θ and it also analyzes automatically their beam-centroid positions.Then, we have measured the IF shift Y10 for vortex beams with l =10 as a function of θ when θ increases from 39.45 ° to 41.25 °, and the sampling interval of θ is 15″.In order to overcome the random fluctuations (albeit very small), we still repeated every measurement 5 times and obtained the average beam-centroid positions taken average from 5 realizations.Through the SLM, we have changed the value of l and have repeated the above process.Finally, we have achieved the relationship between l Y  and θ, as shown in Fig. 3.Note that, in our experimental data there are the statistical fluctuation errors in the range of ±1.5 μm, which is roughly matched with the size of each pixel of the CCD.Here, we emphasize that the amplification factor is the maximum not at the angles of resonances but at the angles where the absolute slopes of the transmission curve are maximal at both sides of the resonance peaks.In Figs.3(c) and 3(d), the relative difference is linearly proportional to the value of |l| at two different angles at both sides of a resonance peak, and the large amplification factors (the slope of the linear-fitted red lines) are achieved, showing 12.855 m/TC and -19.535 m/TC, respectively.More information on the amplification factor is further explained in the Supplemental Material.It turns out that using vortex beams with larger |l| is more beneficial for measuring the relative difference because of enhanced opposite IF shifts.Note that the relative difference between GH shifts for opposite vortex beams in our cases are much smaller (almost independent of the value of TCs, see the Supplemental Material) since the wide-width vortex beams are used.The inset of Fig. 3(d) also shows the transmission intensity distribution for vortex beams with l = 10 and l = -10.In the Supplemental Material, we have further pointed out that the relative difference l Y  here becomes positive as the angle of incidence approaches the resonant angles and it becomes negative as the angle is away from the resonant angles.The change of IF shifts for vortex beams becomes maximal when the transmission curve changes quickly, whose underlying nature is mainly due to that the IF shifts of vortex beams here originate from the angular GH shifts [21] that are proportional to the slopes of the transmissivity or reflectivity curves.
In Fig. 4, it experimentally shows the dependence of the relative IF shifts of the vortex beams with l = 10 on the thickness of the air gap in the DPS at different angles of incidence.
Here the surrounding temperature of the DPS is tried to keep unchanged at T = 21 °C.In Fig. one can sense the small change of d from the large change of Y  , and in principle, the larger the linear amplification factor, the higher the precision for achieving the change of d.Here, from the above data, we can estimate that the precision for thickness (displacement) sensing can be less than 4 nm at |l| = 10.Thus, one can always choose a suitable value of  , which can lead to a linear relation between Y  and d, to obtain the information of the thickness of d through measuring the value of Y  .In practice, one can first set the region of interest on the thickness d, then choose a suitable angle of incidence to match the range of d within the linear working region, finally measuring Y  determines thickness d in the linear region.Unlike the method based on surface-plasmon resonance, which has usually only one measuring range near the angle of surface-plasmon excitation, here in the DPS there exist multiple resonant regions for realizing the enhanced opposite IF shifts for vortex beams with opposite TCs, leading to a wider sensing range for measuring the change of d.Now let us turn to consider the effect of the surrounding temperature T on the relative IF shifts.To investigate the effect of temperature T, the thickness of the air gap inside the DPS is initially adjusted to be the same value, which is confirmed by the double-wavelength method.Note that the thickness of the air gap here is initially tuned at d = 5.37 μm in all situations, and a slightly increase in thickness leads to the narrowing effect of resonant peaks.Due to the change of T, we believe that the thickness of air gap will vary too.Since the thermo-optic coefficient for silica is 1×10 -5 K -1 [33], the temperature dependence of prisms' refractive index is negligible.Thus, the linear change of ΔY with temperature is mainly resulted from the effect of temperature on the thickness d.Here, the DPS is placed in a heating device, and a thermal sensor connected to a computer is attached to the DPS for confirming the surrounding temperature of the DPS.The precision of the thermal sensor is 0.125 °C.The temperature is displayed and recorded in real time on the computer. , so that the precision for temperature sensing is less than 0.045°C at |l| = 20 in this structure.Such linear dependence shows us that using the differences between opposite IF shifts of vortex beams in the DPS can be a possible tool as a temperature sensor in microstructures.
Here we use the above slope, which can also be defined by , to estimate the sensitivity of the temperature sensor.From Fig. 6, it shows the effect of OAM on the sensitivity S of the temperature sensor at  = 40.344°.Here the thickness is initially tuned at  Finally, we have noted that there are some theoretical studies on the application of the IF shifts in sensing [12][13][14][15][16][17].For example, in Ref. [13], without OAM, the temperature sensitivities of 0.79 cm/K and 188 μm/K were theoretically predicted with the GH shift and IF shift, respectively.In Ref. [14], the sensitivity of the IF shifts was theoretically found to be enhanced by increasing the incident vortex charge, and the maximal sensitivities for the GH and IF shifts can theoretically reach 3.01 × 10 3 μm/RIU, 4.63 × 10 3 μm/RIU with l = 1.Yet all these theoretical predictions have not been demonstrated in experiments.Here we proposed and experimentally realized to use the relative IF shifts of vortex beams with opposite TCs for the thickness and temperature sensing in the DPS, and such method could be naturally extended to other resonant structures like photonic crystals.We believe that our work paves the route to use the IF shifts in sensing devices.

Conclusion
In summary, we have experimentally detected the relative IF shifts of vortex beams with tunable but opposite OAMs in the DPS.Our experiments have demonstrated that the relative difference between these opposite IF shifts linearly increases with the increase of TC.This method has the better advantage than that based on the difference between the IF shifts of the same TC vortex beams with different polarizations.One can use the enhanced opposite IF shifts in the DPS to realize the measurements on the change of thickness of the air gap and to achieve temperature sensing from the relative IF shifts by choosing the suitable angle of incidence.For the displacement sensor, at different incident angles, there are different linear ranges and optimal amplification factors.The displacement sensor in this work has a wide range for measuring d.For the temperature sensor, the sensitivity S can be enhanced by using larger topological charge l , and we have measured the maximal temperature sensitivity S to be 33.388μm/ °C by using the vortex beams with l =20.These results can promote the application of controlling IF shifts of vortex beams with opposite TCs in optical sensors.

The DPS installation
The process of fabricating the air gap in our DPS is as follows.Two thin polyimide films with each thickness of 6 μm are sandwiched tightly between the left and right edges of these two prisms, creating an air gap between them under considerably large pressure by using the fastened screws.The DPS is firmly fixed to the rotation platform.A slight change in the rotation angle of the screw will affect the thickness of the air gap, so we can change the thickness of the air gap by fine-tuning the rotation angle of the screw.In our experiment, the thickness d of the air gap is measured by fitting the measured transmission curve (as a function of angle of incidence) of two different wavelengths of laser beams.Here, d is always smaller than the thickness of polyimide films due to the pressure from the fastened screws.Figure S1 shows the transmissivity of two beams with λ1=632.8nm and λ2=520 nm.It is seen that the two experimental results are in good agreement with the theoretical simulation.This is the double-wavelength method used to determine the thickness of the air gap.The thickness has a great influence on the angular interval of the transmittance peaks.The greater the thickness, the smaller the angular interval between the peaks.Therefore, d is determined by optimally fitting the peak interval of experimental transmissivity data.For the better explanations, we put the transmission curve and one of the relative IF shift together in Fig. S3.Now we can clearly find the relative IF shift becomes positive as the angle of incidence approaches the resonant angles (i.e., the value of  is located at the left side of the resonant peak), and it becomes negative as the angle  is away from the resonant angles (i.e., the value of  is located at the right side of the resonant peak).According to Ref. [1] or Eq.

The theoretical simulation of the IF shifts of vortex beams
(2), one can readily obtain the relative IF shift of vortex beams with opposite TCs as , where GH  is the angular GH shift of a fundamental Gaussian beam.As we know that for transmitted or reflected light, the angular GH shifts are proportional to the slopes of the transmissivity or reflectivity curves.Thus, when the transmission curve increases quickly with the increasing angle of incidence, the relative IF shift becomes the positive maximum, in contrast when the transmission curve decreases quickly with the increasing angle of incidence, the relative IF shift becomes the negative minimum.Thus, the change of the relative IF shifts between vortex beams with opposite TCs becomes maximal when the curve of transmissivity dramatically changes, and the relative IF shifts of vortex beams are zero when the angles of incidence are located at the resonant peaks or the valleys of the transmission curve.Thus, the change of IF shifts for vortex beams becomes maximal when the transmission curve changes quickly, whose underlying nature is mainly due to that the IF shifts here originate from the angular GH shifts [1].

The negligible angular IF shift for vortex beams with different TCs
According to the work in Ref. [1], the angular IF shift is given by , where IF  is the angular IF shift of a fundamental Gaussian beam.In our case, this angular IF shift for TE polarization is given by   Figure S5 shows the theoretical amplification factors (that is the slope of the linear-fitted lines) are 41.137 m/TC for the maximum Y  condition at θ = 39.901°, and -52.991 m/TC for the minimum Y  condition at θ = 40.106°,respectively.Note that this amplification factor is strongly dependent on the angles in the resonant structures since the relative IF shifts are oscillating with the angles of incidence as shown in Fig. S2(b).In principle, if the angle of incidence corresponds to the peak or valley of the transmission curve, i.e., when the slopes of the transmission curve vs the angle are zero, in these situations the amplification factors will be zero.When the angle of incidence corresponds to the dramatical change of the transmission curve, theoretically the maximal value of Y  increases linearly as l increases.Thus, the sensitivity S of sensor can be enhanced by using larger topological charge.respectively, and the solid and dotted curves denote TM and TE polarization, respectively.We can find that the change law for the IF shifts of vortex beams with opposite TCs for TM polarization is consistent with those for TE polarization, but the amplitude for TM cases was smaller.Thus, in our work we use the vortex beams with TE polarization.Note that the thickness d here is different from that in the manuscript and was confirmed from the above double-wavelength method.Before doing the experiment, we first confirmed the thickness of air gap in the DPS when the experiment was done in different days.

Figure 1 (Fig. 1 .
Fig. 1.Schematic of beam displacements in the DPS.(a) The structure of the DPS and the parameters of the incident light.Here k  is the wave vector of the incident light, θ is the angle of incidence at the interface between the first prism and the air gap, θ0 and θ1 are the incident and refracted angles from the outside to the first prism, respectively, 0, 1, 2, and 3 are the relative permittivities of the outside, the first prism, the air gap, and the second prism, respectively, and d is the thickness of the air gap.(b) illustrates the beam shifts in the symmetrical DPS.Yt and Yr are the IF components (vertical to the incident plane) of shifts between the actual trajectories (red) and geometrical (grey) trajectories of the transmitted and reflected light outside the DPS, respectively.

Fig. 2 .
Fig. 2. Experimental setup for measuring IF shifts.The wavelengths of Lasers 1 and 2 are 632.8nm and 520 nm, respectively.The notations are as follows: HWP1~HWP2, half-wave plates; P, polarizer; BE, beam expander; BS, beam splitter; SLM, spatial light modulator; CA, circular aperture; L, lens (with f = 125 mm); DPS, double-prism structure placed on the electrically-controlled rotating platform; BQA, beam quality analyzer (for measuring the IF shifts).Insert (a) shows the phase diagram loaded onto the SLM and (b) the corresponding field distribution with l = 10.

Fig. 3 . 1 Y
Fig. 3.The IF shifts of vortex beams with opposite TCs and their relative differences.(a) The IF shift (Yl position) of the vortex beams with l = 10 (black curve) and l = -10 (red curve).(b) The relative difference of IF shifts with opposite TCs (i.e., l l l Y Y Y     ).Here 1 Y  , 2 Y  , ...,

10 Y 10 Y
4(a), when  = 40.087°, linearly increases with d increasing from 4.8 to 5.2 μm, and the slope of the linear-fitted line is 454.741, which is a linear amplification factor for thickness sensing.When  = 40.150°, 10 Y  increases linearly with d increasing from 4.8 to 5.4 μm, and meanwhile the slope of the linear-fitted line in this case is 383.646.When  = 40.344°, 10 Y  linearly increases with d increasing from 5.2 to 5.8 μm, and the corresponding slope of the linear-fitted line is 413.630.In Fig. 4(b), linearly decreases with d increasing from 5.45 to 5.8 μm at  = 40.087° and  = 40.150°, and the slopes of the linear-fitting lines are -509.979and -450.729,respectively.These results tell us that, Y  is linearly dependent on d within certain ranges of d at different  .From the slopes (i.e, the linear amplification factors),

Fig. 4 .
Fig. 4. Dependence of the relative difference 10 Y  on the thickness d of air gap at different incident angles in the DPS.(a) The positive and (b) negative changes of 10 Y  with d, changing from 4.7 to 5.8 μm.Black dots are the cases of  = 40.087°,red dots for  = 40.150°and blue dots for  = 40.344°.Here the surrounding temperature of the DPS is kept at T = 21°C.

Fig. 5 .
Fig. 5. Dependence of the relative difference Y  on temperature T under different angles of incidence.(a) 10 Y  vs T at  = 40.087°, (b) 10 Y  vs T at  = 40.344°，and (c) Y  vs T for |l|=5, 10, 20 at  = 40.344°.Discrete points are measured data and dashed lines are linearly fitted.Here the thickness of air gap is initially tuned at d = 5.37 μm in all situations.

Figures 5 ( 5 Y 5 Y
Figures 5(a) and 5(b) show that Y  is linearly dependent on temperature at two different angles.In Fig. 5(a), when  = 40.087° (at the right side of one resonant peak), 10 Y  linearly decreases with T within the range from 21 °C to 25 °C, and the slope of the linear-fitted line is -18.608μm/ °C.In Fig. 5(b), when  = 40.344° (at the left side of another resonant peak),

d = 5 .
68 μm.As the value of l increases, the slope of Y  increases, thus the sensitivity S increases too.Therefore, one can improve the sensitivity S of this temperature sensor by increasing l .In our experiment, we utilized vortex beams with the maximal value of l =20, which corresponds to the maximum S be 33.388μm/ °C as pointed out above.Of course, in practice, the increasing of the sensitivity S is limited by the quality of generated vortex beams.We use the SLM to generate the vortex beams with different TCs.The quality of vortex beams will decrease as the value of TCs increases due to the limited modulation ability of the SLM.It should be mentioned that the fluctuation of the measured data in Figs.5 and 6is mainly due to the uncontrollable temperature fluctuations, such as microscale air-flow disturbance and heat conduction in the experimental environment.

Fig. 6 . 12 Y
Fig. 6.(a) Effect of temperature on the relative difference of opposite IF shifts and (b) their temperature sensitivity S under different TCs.In (a), the black, red, blue, green, purple, orange dots denote 2 Y  , 4 Y  , 6 Y  , 8 Y  , 10 Y  , and 12 Y  , respectively.Here d = 5.68 μm and  =

Fig. S1 .
Fig. S1.Transmissivity of two beams with (a) λ1 = 632.8nm and (b) λ2 = 520 nm.The black lines are the experimental results, and the blue lines are the theoretical simulations.

Fig. S2 . 5 Y
Fig. S2.The theoretical simulation of the transmissivity and relative IF shifts Y  .(a) The transmissivity, (b) the black, red, blue, green, and purple curves denote the theoretical values of 1 Y  , 2 Y  , 3 Y  , 4 Y  , and 5 Y  , respectively.Here we take d = 4.16 μm.According to Eqs. (1-2), we can get the theoretical results of IF shifts.Figure S2(a) shows the transmissivity of the DPS. Figure S2(b) shows the theoretical simulation of

Fig. S3 .
Fig. S3.The theoretical predictions of the transmission curve (the thin black curve, left axis) and the relative IF shift △Y5 (the thick blue curve, right axis).Here △Y5 is for the relative IF shift of vortex beams with 5 TCs.

w
are the angular spread and the waist of the incident Gaussian beam.This quantity IF  is very tiny and it is less than 10 -7 rad in our cases, which induces an additional displacement about 10 -3 m (much smaller than the spatial IF shifts of vortex beams).Here we provide the theoretical spatial and angular IF shifts of a Gaussian beam with the TE polarization in the below Fig.S4.From Fig.S4, we can see that the value of the angular IF shift is very tiny (see Fig.S4(b)), and the spatial IF shift of the Gaussian beam is also much small (less than 1m), which is also much smaller than the spatial IF shifts of vortex beams with l=1, 2, and 3 (for examples, see Fig.S4(a)).

Fig. S4 .
Fig. S4.(a) Comparison of the spatial IF shifts between a Gaussian beam (black) and vortex beams with l = 1 (green), 2 (red) and 3 (blue), and (b) the angular IF shift of a Gaussian beam in our double-prism system with the air thickness d = 4.16 μm.Here the beam parameter is wo = l mm and the light beam is the TE polarization.

Fig. S6 .
Fig. S6.Comparison of experimental amplification factors at two different angles: (a) the angle of practical resonance and (b) the angle where the transmission curve decreases quickly.The amplification factors (i.e., the slopes of the linear-fitted red lines) are 0.036m/TC for the angle at θ = 40.109°(which is close to the resonant condition), and -19.535m/TC for the angle at θ = 40.344°(where transmission curve drops quickly), respectively.

Figure
Figure S7(a) shows the Y position of the transmitted beam for l = 10 and l = -10, and Fig. S7(b) shows their relative IF shifts

7 . 1 Y  , 5 Y  and 10 Y
FigureS8shows the comparison of the experimental data for the relative GH and IF shifts for vortex beams with opposite TCs.It is clear that the relative differences of GH shifts,

Fig. S8 .
Fig. S8.The experimental results of the relative differences between GH shifts 1 X  , 5 X  and