1. Introduction

Magnetic fluid (MF) is a kind of homogeneous colloidal dispersion [1] of very fine magnetic nanoparticles (usually 3–15 nm in diameter) dispersed in a suitable liquid carrier with the aid of surfactant coated on the surface of the particles, which will prevent the particles from sticking to each other due to van der Waals attraction. The sizes of the magnetic particles are so small that the thermal energies are comparable to their gravitational ones [2] and then the sedimentations are avoided. MFs possess both the fluidity of liquids and the magnetism of magnetic nanoparticles. The conventional optical properties of MFs including linear birefringence, linear dichroism, and Faraday effect have been investigated for several decades [3–5]. The latest study of optical properties of MFs have been extended to magnetically tunable optical scattering [6,7], field dependent refractive index [8–10] and optical transmission [11,12], tunable magnetic photonic crystals [13–16], etc. Several potential MF-based optical devices have been proposed and demonstrated, for instance, MF tunable optical gratings [17,18], MF optical switch [19,20], MF optical modulator [21–24], MF optical capacitor [25], MF optical limiters [26,27], and MF sensors [28–32].

On the other hand, conventional electromagnetic current transformers (EMCTs) are unable to meet the requirements of current measurement devices in the new generation of power system due to their numerous defects, e.g., small dynamic range, saturated magnetic circuit, insulation problems, large size, inflammable and explosive, severe electromagnetic interference, and large measurement errors [33]. Then, the optical current transformers (OCTs) based on Faraday effect are proposed [34,35]. Though the performance of OCTs can surpass those of EMCTs, OCTs have low signal to noise ratio (SNR) owing to their high sensitivity to environmental disturbance. Moreover, OCTs usually need complex polarization maintaining devices to attain high performance. If the optical fiber is applied to the OCTs system (which is widely used at present), their sensitivity, linearity, and stability [36–38] may be reduced owing to the degradation of polarization state caused by the inherent birefringence of the sensing medium and the additional birefringence effect caused by various factors, e.g., change of outside temperature and pressure. Besides, multimode fiber as a sensing medium will lead to complex mode coupling and great depolarization phenomenon. As a result, the emergent light will turn out to be elliptically polarized light, which will decrease the performance of the OCTs.

Under such circumstances, MFs with magnetically sensitive properties are used as sensing media to develop OCTs. In 1980, Presley and Rex developed highly-reliable sensors based on the magnetic hydrodynamics (MHD) of MF, which are widely used in the aircraft or the tank gun stabilization system [39]. Recently, many researchers have developed various MF-based sensors and the related patents are published [40–43], which are widely used in the field of aerospace, national defense, and military. These resolve the various testing problems in the peculiar, complex, and harsh conditions.

In this work, the magnetic field sensing system based on V-shaped groove filled with MFs is developed, which does not necessarily need linearly polarized light. The V-shaped groove structure is relatively simple when compared with other sensing cells and the sensing sensitivity of the system can be tuned by adjusting the vertex angle of the V-shaped groove. Besides, the system can operate at any incident angle. Simultaneously, there is no inherent linear birefringence accompanying with the system because the sensing medium is in colloidal state. These make the MF-based OCT superior than other EMCTs and OCTs. Detailed theoretical derivation and simulation of the sensing system are conducted and verified by the experimental results.

2. Operating Principle

It is well-known that magnetic particles within the MFs are dispersed randomly in the carrier liquid under zero external magnetic field owing to the Brownian motion [44–46]. When the external magnetic field strength exceeds a critical value (denoted by $H_c$), the magnetic particles will agglomerate to form magnetic chains which are distributed in the MF along the field direction owing to the magnetic attraction [47]. This variation of structure results in the refractive index ($n_{\mathrm{MF}}$) change of MF. For the transverse field configuration (the applied magnetic field is perpendicular to the propagation direction of the incident light), $n_{\mathrm{MF}}$ will decrease monotonously with the increase of magnetic field [48]. When the strength of magnetic field is very large, $n_{\mathrm{MF}}$ will reach a saturated value due to the saturated state of the agglomeration at high field [49,50].

Figure 1 illustrates the cross-section view of the system used to realize the magnetic field sensing. $d_0$ is the thickness of the glass slide; $n_0$, $n_1$ and $n_{\mathrm{MF}}$ are the refractive indices of the outside medium, glass slide, and MF, respectively; $\theta$ is the angle of incidence on the interface between the outside medium and the glass slide; $2{\theta }_0$ is the included angle of the V-shaped groove; ${\theta }^{\prime }$ (${\theta }_0^{\prime }$) is the refraction angle on the interface between the glass slide and the MF (between the glass slide and the outside medium); $h_0$ and $h_1$ are the heights of the incident point (denoted as $A$) and the emergent point (denoted as $B_1$ and $B_2$ corresponding to the situations at different magnetic fields) on the glass slide; $Y_1$ and $Y_2$ are the heights of the emergent light on the detecting plane corresponding to the situations at different magnetic fields; $L$ is the distance along $z$ direction between the vertex of the V-shaped groove and the detecting plane; $L_2$ is the distance along $z$ direction between $B$ and the vertex of the V-shaped groove. $L_1$ is the distance along $z$ direction between $B$ and the detecting plane. The solid and dashed lines correspond to the situations at different external magnetic fields.

 

Fig. 1. Cross-section view of the magnetic field sensing system based on V-shaped groove filled with magnetic fluids.

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The propagation of the incident light passing through the index-tunable MF will be influenced when the externally applied magnetic field is changed. In our proposed sensing system (see Fig. 1), the direction of the emergent light is related to the strength of the externally applied magnetic field. Then, measuring the location of the emergent light on the detecting plane can be used to meter the strength of the externally applied magnetic field.

3. Theoretical Analysis

A. Analytical Expression

From Fig. 1, the following relationships can be obtained according to the Snell’s law:

Equations (1) and (2) show that the glass slides are just like a bridge between the outside medium and the MF, so the thickness of the glass slide ($d_0$) is ignored at first for the sake of the simplification (in the later context, the influence of $d_0$ will be further considered). Accordingly, the modified Snell’s laws for this system are

Then, the analytical expression for $y$ (the height of the emergent light on the detecting plane when ignoring $d_0$) can be derived as (see Appendix A for detailed mathematic operations)

Actually, the emergent light will have an offset after passing through a parallel glass plate. For this reason, the height of the emergent light on the detecting plane may also depend on the thickness of the glass slide. So, it is necessary to consider the influence of $d_0$ on the height of the emergent light on the detecting plane for improving the accuracy and precision of the potential sensor. The analytical expression for the practical offset ($\mathrm{\Delta }y$) on the detecting plane is obtained as (see Appendix B for detailed mathematic operations)

Therefore, the actual height of the emergent light on the detecting plane can be expressed as

B. Simulation Results and Discussion

Figure 2 simulates the height of the emergent light ($Y$) as a function of refractive index of MF ($n_{\mathrm{MF}}$) under various different parameters based on Eqs. (5)–(7). From Fig. 2, we can conclude that the sensitivity of the sensing system (the slope of the $Y\text{-}{n}_{\mathrm{MF}}$ curve) changes with the included angle of the V-shaped groove ($2{\theta }_0$), the distance along $z$ direction between the vertex of the V-shaped groove and the detecting plane ($L$) and the type (refractive index) of the outside medium ($n_0$). While it does not change with the incident angle ($\theta$), the height of the incident light on the glass slide ($h_0$), the material type (refractive index) of the glass slide ($n_1$), and the thickness of the glass slide ($d_0$).

 

Fig. 2. Height of the emergent light ($Y$) as a function of refractive index of MF ($n_{\mathrm{MF}}$) under several different parameters: (a) different $2{\theta }_0$, (b) different $L$, (c) different $n_0$, (d) different $\theta$, (e) different $h_0$, (f) different $n_1$, and (g) different $d_0$.

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In Fig. 2, the refractive index of MF ($n_{\mathrm{MF}}$) rather than the actual magnetic field ($H$) is used as the variable. This is assigned to the multiple-parameter dependent property of $n_{\mathrm{MF}}$. For instance, $n_{\mathrm{MF}}$ is not only related to $H$, but also related to ambient temperature $T$. The detailed expression for $n_{\mathrm{MF}}(H,T)$ has a complex Langevin-function-like form [51]. Moreover, even under constant $H$ and $T$, $n_{\mathrm{MF}}$ also depends on many other factors, such as the type and concentration of the MF, the thickness of the sample. Consequently, $n_{\mathrm{MF}}$ is employed as the variable, which is proportional to the magnetic field. For any specific situation, $n_{\mathrm{MF}}$ can be replaced with the actual value of $H$. And then, the purpose of magnetic field sensing is obtained.

For convenience and simplification (while without loss of generality), air ($n_0=1$) is chosen as the outside medium ($n_0$ has a very slight influence on the sensitivity of the sensing system). This is the most cases for practical applications. The glass slide’s refractive index ($n_1$) and thickness ($d_0$) are set as 1.5 and 1 mm, respectively, which do not influence the sensitivity of the sensing system. Because the sensitivity of the sensing system is independent of the incident angle $\theta$, $\theta$ is adjusted to be equal to the included angle of the V-shaped groove ($2{\theta }_0$) for simplification. Afterwards, on the basis of Taylor’s formula, Eqs. (5)–(7) can be simplified as (see Appendix C for detailed mathematic operations)

Equation (8) indicates that the relationship between the height of the emergent light ($Y$) and the refractive index of the MF ($n_{\mathrm{MF}}$) is linear, which can also be seen in Fig. 2. This is favorable for practical application to sensors.

To evaluate the accuracy of the simplified expression [Eq. (8)], simulations about the height of the emergent light ($Y$) as a function of refractive index of MF ($n_{\mathrm{MF}}$) are implemented with the simplified [Eq. (8)] and exactly analytical expressions [Eq. (7)], which are shown in Fig. 3(a). From Fig. 3(a), it is obvious that the curves for the simplified expression almost overlap with those for the exactly analytical expression. The quantitative difference (denoted as $\mathrm{\Delta }$) between the simplified [Eq. (8)] and exactly analytical expressions [(Eq. (7)] is shown in Fig. 3(b). From Fig. 3(b), we can see that the approximation error between the simplified [Eq. (8)] and exactly analytical expressions [Eq. (7)] is so small that we can ignore it, especially when the included angle of the V-shaped groove is very small. Therefore, the simplified expression $Y=2{\theta }_{0}L{n}_{\mathrm{MF}}+h_0-3{\theta }_{0}L$ and the derived linear relationship between $H$ and $n_{\mathrm{MF}}$ are reliable.

 

Fig. 3. (a) Comparison of the $Y\text{-}{n}_{\mathrm{MF}}$ curves calculated with the simplified [Eq. (8)] and the exactly analytical expressions [Eq. (7)] and (b) Difference ($\mathrm{\Delta }$) between the two sets of equations as a function of $n_{\mathrm{MF}}$

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4. Experimental Verification

The MF we utilize for experimental investigation in this work is oil-based MF with saturation magnetization of 100 Oe and viscosity of $9\mathrm{mPa}·\mathrm{s}$, which is provided by the Ferrotec Corporation. The magnetic nanoparticles are ferrite materials with average diameter of about 10 nm. The MF was injected into a V-shaped groove made of cover glasses and glass slides as shown in Fig. 4. The included angle of the V-shaped groove can be adjusted through changing the number of the cover glasses. Figure 5 shows the experimental diagram for investigating the proposed sensing system. A highly stable He-Ne laser generating a visible light of 632.8 nm is employed, whose electric vector is perpendicular to the magnetic field direction. The V-shaped groove sample is placed in the gap between two poles of the electromagnet which generates a uniform magnetic field (with nonuniformity of less than 0.1%) paralleling the sample surface. The strength of the external magnetic field is adjusted by tuning the magnitude of the supply current. The combination of lenses with long and short focal lengths is utilized to improve the collimation effects of the laser beam. The movable diaphragm at the detecting plane can be moved in two dimensions to let the emergent light pass through it. The transmitted light after the diaphragm is recorded by a digital power meter.

 

Fig. 4. Schematic drawing of the V-shaped groove.

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Fig. 5. Schematics of experimental setup for studying the sensing system based on the V-shaped groove filled with MF.

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Figure 6 displays the experimental sensing property of the system at several included angles of the V-shaped groove: (a) $2{\theta }_0=1.42°$, (b) $2{\theta }_0=1.77°$, (c) $2{\theta }_0=2.49°$, (d) $2{\theta }_0=2.84°$, (e) $2{\theta }_0=3.2°$, and (f) $2{\theta }_0=3.55°$. From Fig. 6, we can see that the height of the transmitted light decreases with the current (to which the external applied magnetic field is linearly proportional) under certain included angle of the V-shaped groove. The slight fluctuations between the experimental and Langevin fitting data are also observed. These may be originated from the instability of the experimental setup, the variation of the ambient temperature, and the light power. The variation range of the height of the transmitted light $\mathrm{\Delta }Y$ (the difference between the maximum and minimum values of the Langevin fitting curve) increases with the included angle of V-shaped groove, which is more evident in Fig. 7.

 

Fig. 6. Sensing property of the system at several included angles of the V-shaped groove: (a) $2{\theta }_0=1.42°$, (b) $2{\theta }_0=1.77°$, (c) $2{\theta }_0=2.49°$, (d) $2{\theta }_0=2.84°$, (e) $2{\theta }_0=3.2°$, and (f) $2{\theta }_0=3.55°$ and the strength of magnetic field $H$ as a function of current $I$.

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Fig. 7. Variation range of the height of the transmitted light $\mathrm{\Delta }Y$ as a function of the included angle of the V-shaped groove $2{\theta }_0$.

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The physical mechanism in charge of the magnetic field sensing is the magnetically tunable refractive index of the MF. The electric susceptibility ($\chi$) of MF is dependent on the magnitude of the applied magnetic field and the relative direction between the electric vector of the incident light and the magnetic field. In our experiments, the electric vector is perpendicular to the magnetic field and $\chi$ will decrease with the increase of magnetic field [48,52–54]. So $n_{\mathrm{MF}}$ will decrease with the external magnetic field according to $n=\sqrt{{\epsilon }_r}=\sqrt{1+\chi }$. Hence, the larger the magnetic field strength is, the smaller the electric susceptibility ($\chi$) of MF is and the smaller the refractive index of MF is. When the magnetic field strength is very low, the agglomeration would not happen and then the height of the transmitted light will almost not change with the applied magnetic field (current) as shown in Fig. 6. While the strength of the external magnetic field is large enough, the refractive index of MF will decrease into the saturated value, so the height of the transmitted light will not (or very slightly) change with the applied magnetic field (current) further as shown in Fig. 6.

From the above theoretical analysis and experimental results, it can be concluded that it may be convenient to get the strength of the externally applied magnetic field to a certain extent by measuring the height of the transmitted light. In our experiments, the variation range of the transmitted-light’s height, dominated by the tunable range of the refractive index of MF, is not very large due to the low viscosity and small saturation magnetization of the MF. However, it may be possible to improve the variation range of the sensing system through optimizing the correlative parameters (e.g., concentration, viscosity and types of the MF; wavelength of the incident light).

5. Conclusions

In summary, the V-shaped groove sensor based on the tunable refractive index of MF has been studied analytically and numerically. The operating principle of the sensor is analyzed in detail. The linear sensing property of the sensing system is obtained. The variation range of the height of the emergent light depends on the included angle of the V-shaped groove and the maximum variation range is around 3 mm in our experiments. Some parameters of the sensor can be optimized to get better results. The sensor based on this principle may lay a solid foundation for designing photonic sensing devices exploiting the tunable refractive index of MF.

Appendix A

The simplified diagram for the V-shaped groove is showed in the Fig. 8 where the thickness of the glass slide has been ignored. In this diagram, $y$ is the height of the transmitted light in the detecting plane. ${\theta }_0-{\theta }^{\prime }$ is the horizontal angle of the refraction light within the MF.

 

Fig. 8. Diagram of geometric optical path ignoring the thickness of the glass slide.

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From Fig. 8, the following expressions are obvious:

(A1)$$h_1=h_0+(L_2+h_{0}tg{\theta }_0)tg({\theta }_0-{\theta }^{\prime }),$$

(A2)$$h_1=L_{2}tg(\frac{\pi }{2}-{\theta }_0),$$

(A3)$$L_1=L-L_2,$$

(A4)$$y=L_{1}tg({\theta }_0^{\prime }-{\theta }_0)+L_{2}tg(\frac{\pi }{2}-{\theta }_0).$$

According to the triangular transformations and combining Eqs. (3) and (4), the following equations are obtained:

(A5)$$tg({\theta }_0-{\theta }^{\prime })=\frac{tg{\theta }_0-tg{\theta }^{\prime }}{1+tg{\theta }_{0}tg{\theta }^{\prime }},$$

(A6)$$tg{\theta }^{\prime }=\frac{\sin {\theta }^{\prime }}{\cos {\theta }^{\prime }}=\frac{n_0\sin \theta }{\sqrt{n_{\mathrm{MF}}^2-n_0^2{\sin }^2\theta }},$$

(A7)$$tg({\theta }_0^{\prime }-{\theta }_0)=\frac{tg{\theta }_0^{\prime }-tg{\theta }_0}{1+tg{\theta }_0^{\prime }tg{\theta }_0},$$

(A8)$$\sin {\theta }_0^{\prime }=\sin 2{\theta }_0\sqrt{\frac{n_{\mathrm{MF}}^2}{n_0^2}-{\sin }^2\theta }-\cos 2{\theta }_0\sin \theta ,$$

(A9)$$\cos {\theta }_0^{\prime }=\sqrt{1-[{\sin }^{2}2{\theta }_0(\frac{n_{\mathrm{MF}}^2}{n_0^2}-{\sin }^2\theta )+{\cos }^{2}2{\theta }_0{\sin }^2\theta -\sin 4{\theta }_0\sin \theta \sqrt{\frac{n_{\mathrm{MF}}^2}{n_0^2}-{\sin }^2\theta }]}.$$

Combining Eqs. (A1)–(A9), the height of the transmitted light on the detecting plane when ignoring the thickness of the glass slide can be expressed as

(A10)$$\begin{gathered} y=[L-\frac{(P+tg{\theta }_0\sin \theta )h_0+h_{0}tg{\theta }_0(Ptg{\theta }_0-\sin \theta )}{(P+tg{\theta }_0\sin \theta )tg(\frac{\pi }{2}-{\theta }_0)-(Ptg{\theta }_0-\sin \theta )}] \\ \times \frac{\sin 2{\theta }_{0}P-Q-tg{\theta }_0\sqrt{1-({\sin }^{2}2{\theta }_0{P}^2+Q^2-\sin 4{\theta }_0\sin \theta P)}}{\sqrt{1-({\sin }^{2}2{\theta }_0{P}^2+Q^2-\sin 4{\theta }_0\sin \theta P)}+\sin 2{\theta }_{0}tg{\theta }_{0}P-Qtg{\theta }_0} \\ +\frac{h_0(P+tg{\theta }_0\sin \theta )+h_{0}tg{\theta }_0(Ptg{\theta }_0-\sin \theta )}{(P+tg{\theta }_0\sin \theta )tg(\frac{\pi }{2}-{\theta }_0)-(Ptg{\theta }_0-\sin \theta )}tg(\frac{\pi }{2}-{\theta }_0). \end{gathered}$$

Appendix B

The offset within the MF (denoted as $D$) between the transmitted lights when ignoring and considering the influence of thickness of the glass slide is shown in Fig. 9. In Fig. 9, $d$ is the offset within the MF when $n_{\mathrm{MF}}$ is equal to the refractive index of outside medium ($n_0$). $D$ is the practical offset within the MF when considering and ignoring the influence of thickness of the glass slide. The solid and dash lines represent the optical path when ignoring and considering the influence of thickness of the glass slide, respectively.

 

Fig. 9. Offset within the MF between the transmitted lights when ignoring and considering the influence of thickness of the glass slide.

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From Fig. 9, the practical offset $D$ within the MF is given by

(B1)$$D=\frac{d}{\cos \theta }\cos {\theta }^{\prime },$$

where $d$ is expressed as

(B2)$$d=\frac{d_0}{\cos {\theta }_1}\sin (\theta -{\theta }_1).$$

Combining Eqs. (B1) and (B2) and using the Snell’s law [Eq. (1)] and some triangular transformations, $D$ can be expressed as

(B3)$$\begin{gathered} D=\frac{d_0\sqrt{n_{\mathrm{MF}}^2-n_0^2{\sin }^2\theta }}{n_{\mathrm{MF}}\cos \theta \sqrt{n_1^2-n_0^2{\sin }^2\theta }} \\ (\sqrt{n_1^2-n_0^2{\sin }^2\theta }\sin \theta -n_0\sin \theta \cos \theta ). \end{gathered}$$

The actual offset on the detecting plane is shown in Fig. 10. In Fig. 10, $d^{\prime }$ is the practical offset in the outside medium when considering and ignoring the influence of thickness of the glass slide and $\mathrm{\Delta }y$ is the offset on the detecting plane. The solid and dash lines represent the optical path when ignoring and considering the influence of thickness of the glass slide, respectively.

 

Fig. 10. Offset on the detecting plane when considering and ignoring the influence of thickness of the glass slide.

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From Fig. 10, the expression of the actual offset on the detecting plane is expressed as

(B4)$$\mathrm{\Delta }y=\frac{d^{\prime }}{\cos ({\theta }_0^{\prime }-{\theta }_0)}.$$

Meanwhile, the following equalities can be obtained:

(B5)$$BE=\frac{d^{\prime }}{\cos {\theta }_0^{\prime }},$$

(B6)$$O_1{O}_2=\frac{D}{\cos (2{\theta }_0-{\theta }^{\prime })}=CD.$$

In the $\text{Rt}\mathrm{\Delta }E{O}_{1}D$:

(B7)$$\tan \angle E{O}_{1}D=(EB+BC+CD)/O_{1}D=\tan {\theta }_0^{\prime }.$$

In the $\text{Rt}\mathrm{\Delta }B{O}_{2}C$:

(B8)$$\tan \angle B{O}_{2}C=BC/O_{2}C=\tan {\theta }_1^{\prime }.$$

Combining the above equations and the Snell’s laws [Eqs. (1) and (2)], the actual offset on the detecting plane between consideration and ignorance of the influence of thickness of the glass slide can be expressed as

(B9)$$\begin{gathered} \mathrm{\Delta }y=[\frac{\sin 2{\theta }_{0}P-Q}{\sqrt{1-({\sin }^{2}2{\theta }_0{P}^2+Q^2-\sin 4{\theta }_0\sin \theta P)}}{d}_0 \\ -\frac{n_0\sin 2{\theta }_{0}P-n_{0}Q}{\sqrt{n_1^2-n_0^2({\sin }^{2}2{\theta }_0{P}^2+Q^2-\sin 4{\theta }_0\sin \theta P)}}{d}_0 \\ -\frac{d_{0}P(\sqrt{n_1^2-n_0^2{\sin }^2\theta }\sin \theta -n_0\sin \theta \cos \theta )}{\cos \theta \sqrt{n_1^2-n_0^2{\sin }^2\theta }(\cos 2{\theta }_{0}P+\sin \theta \sin 2{\theta }_0)}] \\ \times \frac{\sqrt{1-({\sin }^{2}2{\theta }_0{P}^2+Q^2-\sin 4{\theta }_0\sin \theta P)}}{\cos {\theta }_0\sqrt{1-({\sin }^{2}2{\theta }_0{P}^2+Q^2-\sin 4{\theta }_0\sin \theta P)}+\sin {\theta }_0(\sin 2{\theta }_{0}P-Q)}. \end{gathered}$$

Appendix C

Because the maximum included angle of the V-shaped groove is less than four degrees, which in radian is far less than one. According to the Taylor’s formulas of sine and cosine functions, the expression of the height on the detecting plane can be simplified as

(C1)$$\begin{gathered} Y=[L-h_0\frac{P\prime {\theta }_0+P\prime {\theta }_0^3}{P\prime (1-{\theta }_0^2)+2{\theta }_0\theta }] \\ \times \frac{2{\theta }_{0}P\prime -\theta -{\theta }_0\sqrt{1-(2{\theta }_{0}P\prime -\theta {)}^2}}{\sqrt{1-(2{\theta }_{0}P\prime -\theta {)}^2}+2{\theta }_0^{2}P\prime -\theta {\theta }_0}+h_0\frac{P\prime (1+{\theta }_0^2)}{P\prime (1-{\theta }_0^2)+2{\theta }_0\theta } \\ +[\frac{2{\theta }_{0}P\prime -\theta }{\sqrt{1-(2{\theta }_{0}P\prime -\theta {)}^2}}-\frac{2{\theta }_{0}P\prime -\theta }{\sqrt{{1.5}^2-(2{\theta }_{0}P\prime -\theta {)}^2}}-\frac{P\prime (\sqrt{{1.5}^2-{\theta }^2}\theta -\theta )}{\sqrt{{1.5}^2-{\theta }^2}(P\prime +\theta 2{\theta }_0)}] \\ \times \frac{\sqrt{1-(2{\theta }_{0}P\prime -\theta {)}^2}}{\sqrt{1-(2{\theta }_{0}P\prime -\theta {)}^2}+{\theta }_0(2{\theta }_{0}P\prime -\theta )}. \end{gathered}$$

where $P\prime +\sqrt{n_{MF}^2-{\theta }^2}$.

In our experiments, $2{\theta }_0$ is always less than four degrees, so $2{\theta }_0$ (in radian) $\ll 1$ is satisfied. Therefore the relationships $\theta \ll n_{\mathrm{MF}}$ or $n_1$, ${\theta }^2\ll 1$, and $\theta {\theta }_0\ll n_{\mathrm{MF}}$ are satisfied. Then Eq. (C1) can be further simplified as

(C2)$$\begin{gathered} Y=[L-h_0{\theta }_0]\times \frac{2{\theta }_0{n}_{\mathrm{MF}}-\theta -{\theta }_0\sqrt{1-{(2{\theta }_0{n}_{\mathrm{MF}}-\theta )}^2}}{\sqrt{1-{(2{\theta }_0{n}_{\mathrm{MF}}-\theta )}^2}+2{\theta }_0^2{n}_{\mathrm{MF}}-\theta {\theta }_0}+h_0+[\frac{2{\theta }_0{n}_{\mathrm{MF}}-\theta }{\sqrt{1-{(2{\theta }_0{n}_{\mathrm{MF}}-\theta )}^2}}-\frac{2{\theta }_0{n}_{\mathrm{MF}}-\theta }{\sqrt{{1.5}^2-{(2{\theta }_0{n}_{\mathrm{MF}}-\theta )}^2}}-\frac{\theta }{3}] \\ \times \frac{\sqrt{1-{(2{\theta }_0{n}_{\mathrm{MF}}-\theta )}^2}}{\sqrt{1-{(2{\theta }_0{n}_{\mathrm{MF}}-\theta )}^2}+{\theta }_0(2{\theta }_0{n}_{\mathrm{MF}}-\theta )}. \end{gathered}$$

Because the value of $2{\theta }_0{n}_{\mathrm{MF}}-\theta$ is less than $n_0$ (which is close to 1) and $n_1$ (which is close to 1.5), the Maclaurin formula and $\theta =2{\theta }_0$ (which is mentioned above) can be utilized to eliminate the radical expression. And then Eq. (C2) can be greatly simplified as

(C3)$$\begin{gathered} Y=(2{\theta }_{0}L-2h_0{\theta }_0^2+\frac{2{\theta }_0}{3})n_{\mathrm{MF}}+h_0-3{\theta }_{0}L \\ +3h_0{\theta }_0^2-\frac{4{\theta }_0}{3}. \end{gathered}$$

Equation (C3) shows that the height of the transmitted light obeys $Y=a{n}_{\mathrm{MF}}+b$, where $a$ is equal to $2{\theta }_{0}L-2h_0{\theta }_0^2+\frac{2}{3}{\theta }_0\approx 2{\theta }_{0}L+\frac{2}{3}{\theta }_0\approx 2{\theta }_{0}L$ and $b$ is equal to $h_0-3{\theta }_{0}L+3h_0{\theta }_0^2-\frac{4{\theta }_0}{3}\approx h_0-3{\theta }_{0}L-\frac{4{\theta }_0}{3}\approx h_0-3{\theta }_{0}L$.

Therefore, the final simplified form can be described as

(C4)$$Y=2{\theta }_{0}L{n}_{\mathrm{MF}}+h_0-3{\theta }_{0}L.$$

This research is supported by the National Natural Science Foundation of China (No. 10704048) and Innovation Program of Shanghai Municipal Education Commission (No. 11YZ120).

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