Giant nonlinear Faraday rotation in iron doped CdMnTe

Hernando Garcia; Sudhir Trivedi

doi:10.1364/OME.513751

1. Introduction

Research on Semimagnetic semiconductors started more than 40 years ago [1]. The majority of research has focus on the linear optical, linear magnetooptical, and band structure properties of these semiconductors [2–5]. Cd_1-xMn_xTe has applications in magnetic field sensors, solar cells, photonic devices, and optical isolators [6–8]. Lately, Cd_1-xMn_xTe was identified as a potential material for X-ray and Gamma-rays detectors [9–10].

CdMnTe is the result of replacing the Cd cations by Mn ions in the CdTe crystal structure. It has been found that in Cd_1-xMn_xTe as the content of Mn is increased, the energy gap increased, to the point where CdMnTe becomes transparent in the visible. CdMnTe crystallizes in a zincblende structure. Its magnetic and optical properties can be attributed to the introduction of the Mn^2 +sd⁵ atom. The presence of the Mn^2 +sd⁵ give rise to an exchange interaction between the sp electrons and the d electrons producing a large Zeeman splitting of the electronic levels [11]. This large spin effect produces huge Faraday rotations, bound magnetic polarons, and magneto induced metal-insulator transitions.

In the case of Semimagnetic semiconductors the nonlinear optical response of the delocalized electrons can also be affected by static magnetic fields. It has been predicted and experimentally measured that when a sample is placed in a static magnetic field and made to interact with a high-intensity laser beam, the direction of the polarization was affected by the combination of photoinduced polarization rotation (nonlinear Faraday effect) and the linear polarization rotation (linear Faraday effect) [12–13].

2. Theoretical background

The general expression for the total polarization of the system up to third order can be expressed as

(1)$$\begin{array}{l} P(\omega )= \; \; {\chi ^{(1 )}}\; (\omega )E(\omega )+ \; {\chi ^{(2 )}}({0,\omega } )HE(\omega )\\ + \; {\chi ^{(3 )}}({\omega , - \omega ,\omega } )E(\omega ){E^\ast }(\omega )E(\omega )\\ + \; {\chi ^{(4 )}}({0,\omega , - \omega ,\omega } )E(\omega ){E^\ast }(\omega )E(\omega )H \end{array}$$

In the above equation the first term corresponds to the linear case, the second term is responsible for the linear Faraday effect (LF), the third term is the intensity dependent third order process and the last term is the combination of intensity-dependent nonlinear process and nonlinear induced Faraday rotation (NLF). In the theoretical work of Frey [12] if the normal Faraday configuration is used then the nonlinear absorption can be expressed as

(2)$$I(z )= \frac{{I(0 )}}{{1 + \beta I(0 )z}}$$

were

(3)$$ \beta=\left(\frac{8 \pi^3}{n^2 c \lambda}\right) \operatorname{Im} \chi^{(3)}=\left(\frac{8 \pi^3}{n^2 c \lambda}\right) \chi_5^{\prime \prime}=2 K_1 $$

In the above equations I(0) is the on-axis intensity of the laser, $\chi _5^{\prime\prime}$ is the effective photoinduced susceptibility, c is the speed of light, n is the index of refraction, λ is the wavelength, and β is the Two-Photon Absorption coefficient (TPA). The angle of rotation is given by

(4)$$\varDelta \theta (z )= \theta (z )- \theta (0 )= VBz + \; \frac{1}{2}\frac{{\chi _{11}^"}}{{\chi _5^{\prime\prime}}}Bln({1 + {K_1}I(0 )z} )$$

The first term is the linear Faraday rotation and the second term comes from the contribution of the photoinduced rotation, where V is the Verdet constant, B is the magnetic field, z is the thickness of the sample and $\chi _{11}^{\prime\prime}$ is the magneto-electric fourth order nonlinear susceptibility [12].

3. Experimental setup

One of the simplest experimental setups used to study the nonlinear optical properties of solids is Z-scan [14]. In this configuration the sample is translated along a focused laser beam, causing the intensity at the sample to change as the sample moves across the focal point of the lens. In the Z-scan configuration, the need for a translation stage is critical. However, it is impractical to use Z-scan to study the nonlinear magneto-optical properties of the solids because typical electromagnets weigh on the order of 20 to 40 kg.

An alternate simple configuration to study the nonlinear optical properties of solid is to use an Electrically Focus Tunable Lens (EFTL) [15–16]. In this configuration, the sample is kept stationary and the focal point of the lens is changed by applying a current into the lens. This configuration allows the sample to be placed inside an electromagnet and permits the study of the linear and nonlinear magneto-optics response of the system. The experimental setup is shown in Fig. 1. As shown in the figure the laser beam passes through a half-wave (λ/2) plate to rotate the polarization of the incoming beam. After passing through the half-wave plate the beam is sent through a right-angle mirror inside the electromagnet (GMW 3470 45 mm electromagnet) that reflects the beam parallel to the magnetic field B in the Faraday configuration and passes through the sample. After the sample, the beam is sent to a second right-angle mirror and finally to a broadband cube polarization beam splitter (analyzers) where the beam is split into the s and p polarization states of the electromagnetic wave. The s and p polarized beams are sent to two identical large area photodiodes or integrated spheres, and finally, the signal is detected using Lock-In amplifiers (Stanford Research SR-830).

Fig. 1. Experimental Setup. It shows the electromagnet used which weighs 31 kg. EFTL is the Electrically Focus-tunable lens.

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4. Results and discussion

The samples preparation as well as the linear absorption are described elsewhere [17–18]. Also, the nonlinear absorption is described in [19]. The laser used in the experiment is a Mia Tai ultrafast laser generating 70 fs pulses at 80 MHz and a wavelength of 790 nm. The average power used in the experiments was 30 mW to prevent sample damage. The sample studied in this work is Iron doped Cd_0.85Mn_0.15Te. The linear absorption of the sample is shown in Fig. 2, it is worth mention that the transmittance remains flat up to the maximum wavelength of the laser (1040 nm). The bandgap is close to 1.681 eV and the energy of the laser pulses used in our experiment is 1.569 eV which is below the energy bandgap and therefore, the absorption is dominated by TPA.

Fig. 2. Linear absorption of the sample with a bandgap of the order of 1.681 eV.

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The first parameter that can be calculated is the two-photon absorption coefficient β for the case of B = 0. Because of the sample is thick, we used a generalized normalized transmittance give by

(5)$$T(f )= \frac{1}{{2{q_o}(f )}}\; \; \; \mathop \int \nolimits_{ - \infty }^\infty ln({1 + {q_o}(f )sec{h^2}(\tau )} )d\tau$$

Where it is assumed that the laser pulse has a hyperbolic secant shape, and

(6)$${q_o}(f )= \beta ({1 - R} ){L_{eff}}I(f )$$

In the above equation f is the focal length of the EFTL, R is the reflection coefficient, L_eff is the effective length of the sample L_eff= 1.9 mm, and I(f) is the on-axis intensity of the laser given by

I(f )= \frac{{2ln\left( {1 + \sqrt 2 } \right)\mathrm{{\cal E}}}}{{\pi w{{(f )}^2}{t_{\textrm{fwhm}}}}}

In the above equation $\mathrm{{\cal E}}$ is the energy per pulse, and t_FWHM = 70 fs. Figure 3 shows the experimental data for P_avg = 30 mW. The fitting is obtained by using a value for β = 35.5 cm/GW, which is in agreement with previous reported value for CdMnTe [19]

Fig. 3. Normalized transmittance (red stars) experimental values and (blue line) using Eq. (5), with a value for β = 35.5 cm/GW.

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The unnormalized transmittance of the signal after passing through the sample and the analyzer can be expressed as

(7)$${I_s} = {T_{UN}}(f )co{s^2}({{\theta_L} + {\theta_{NL}} + {\varphi_o}} )$$

for the s-polarization, and

(8)$${I_p} = {T_{UN}}(f )si{n^2}({{\theta_L} + {\theta_{NL}} + {\varphi_o}} )$$

for the p-polarization, where T_UN is the un-normalized transmittance, φ_o is the initial polarization angle of the laser beam and θ_L and θ_NL correspond to the contribution to the linear and nonlinear angle of rotation, respectively. The normalized transmittance in both cases can be expressed as Eq. (5).

Simplifying the cosine and sine function in the above equations it is easy to show that for the case of φ_o= π /4 we get for the case of p-polarization, and s-polarization to be expressed respectively as (i = p, s)

(9)$${I_i}(f )= \; {T_{UN}}(f )(1 \mp sin({2{\theta_L} + 2{\theta_{NL}}} )$$

From the above two equations one can see that the transmission at the high intensity point decreases for the s-polarization and increases for the p-polarization.

Figure 4 and Fig. 5 show a typical 3-D experimental data for the case of 30 mW of average power and magnetic fields up to 0.55 T, in increments of 0.05 T, for the case of the s-polarization, is shown in Fig. 4, and Fig. 5 shows the case of p-polarization. It is evident the trend in the increase and decreased of the transmission as the angle into the analyzer is changing, following the typical cos² and sin² dependency according to Malus’s law. Also, in the two figures it can be seen that the linear Faraday effect (low intensity points in the traces) have different magnetic field dependence compared to the high intensity points (around f = 5.18 cm). This is an indication of the intensity dependent rotation.

Fig. 4. Unnormalized trace for the case of the s-polarization as a function of the magnetic field in increments of 0.05 T.

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Fig. 5. Unnormalized traces for the case of the p-polarization as a function of the magnetic field in increments of 0.05 T.

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The analysis of the data is based on the initial angle of polarization. For the case of φ_o= π/4 there are four relevant points in the f-scan data, as shown in Fig. 6(a, b) and it can be shown that [20]

(11)$$\frac{{{I_{LA}} - {I_{LB}}}}{{{I_{LA}} + {I_{LB}}}} ={-} sin({2{\theta_L}} )$$

Fig. 6. Four relevant points used to analyze the data and extract all the physical properties of the sample at two different Magnetic fields (a) B = 0.2 T, (b) B = 0.5 T.

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The only limitation in this approach is the requirement that the extinction ratio of the polarizer cube be infinite, which is not in most cases. To eliminate this ambiguity, we include into the un-normalized transmission an amplitude A(f) that modified the above equation into

(12)$$\frac{{{I_{LA}} - {I_{LB}}}}{{{I_{LA}} + {I_{LB}}}} ={-} A(f )sin({2{\theta_L}} )\; $$

So, to eliminate this amplitude two TPA traces are taken with two different magnetic field values resulting in a transcendental equation that can be solve numerically. The equations are given by

(13)$$\begin{array}{l} gol = \left( {\frac{{LIp1 - LIs1}}{{LIp1 + LIs1}}} \right)\left( {\frac{{LIp2 + LIs2}}{{LIp2 - LIs2}}} \right)\\ fol = \frac{{sin({2V{L_{eff}}{B_1}} )}}{{sin({2V{L_{eff}}{B_2}} )}}\\ gol - fol = 0 \end{array}$$

Figure 6(a, b) shows the experimental results for two different magnetic fields (B₁= 0.2 and B₂ = 0.5 T). Using the experimental values for the low intensity points of the data we obtain a value for the Verdet constant to be equal V = -0.07248 deg/cm-G which is in excellent agreement with values for the Verdet constant reported before [9,18,21]. The same analysis can be applied to the case of high intensity points

In the case of the high intensity points we obtain (f = 5.18 cm)

(14)$$\begin{array}{l} goh = \left( {\frac{{HIp1 - LIs1}}{{HIp1 + HIs1}}} \right)\left( {\frac{{HIp2 + HIs2}}{{HIp2 - HIs2}}} \right)\\ foh = \frac{{sin({2V{L_{eff}}{B_1} - 2{\theta_{NL}}{B_1}} )}}{{sin({2V{L_{eff}}{B_2} - 2{\theta_{NL}}{B_2}} )}}\\ goh - foh = 0 \end{array}$$

This gives a value for the nonlinear angle of rotation equals to ${\theta _{NL}} = 0.2589\; \frac{{Rad}}{T} = 14.83\; \frac{{deg}}{T}$. Using these values, we obtain . This value is three and a half time larger than a previously reported value for undoped CdMnTe [12–13]. Figure 7 shows the graphical solution to the above equation for the case of f = 6.98 cm, f = 5.26 cm and f = 5.18 cm. It can be seen that for these focal lengths

V{L_{eff}} - {\theta _{NL}} ={-} 2.41662,\; - 2.49837,\; and - 2.67512\; rad/T\;

Fig. 7. graphical solution for the transcendental equations (Eq. (13) and Eq. (14)) the blue intersection corresponds to the low intensity point.

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The first value corresponded to the low intensity signal and gives the Verdet constant of the sample.

Table. 1 show the values of the linear and nonlinear magnetooptical properties of the sample. The experimental values are in excellent agreement with the available literature [12,18–19,21]. It is worth point here that the possible origin of this large increased in the Verdet constant and in the magneto-electric fourth order nonlinear susceptibility $\mathrm{\chi }_{11}^{\prime\prime}$, can be attributed to the presence of the Fe ions, as was discussed in [18].

Table 1. Linear and nonlinear magnetooptical parameter for Cd_0.85Mn_0.15Te:Fe at λ = 790 nm

View Table

To complement the measurement, we have performed a couple of measurements as function of the magnetic field. In this case the EFTL is focus at f = 6.98 cm (low intensity current) and data is taken as a function of the B field, then the lens is focusing at f = 5.18 cm (high intensity point) and the data is taken as a function of the B-field. The experimental data and the theoretical fit using Eq. (11) is shown in Fig. 8 for the case of low intensity.

Fig. 8. Experimental data for the case of low intensity (f = 6.98 cm) and high intensity (f = 5.18 cm) for the case of P = 20-50 mW. Theoretical data for low intensity (black thin line). From the fit, the value of the Verdet constant is V = -0.07248 deg/cm-G.

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To corroborate the results using the TPA, we measured the angle of rotation at low and high intensities (I_L = 0.0125 GW/cm², and I_H = 1.832 GW/cm²) using a halfwave plate, for different values of the B-field (B = 0.0 T - 0.5 T), see Fig. 9. From the results shown in Fig. 8 it can be seen that there is change of the Faraday rotation due to the photoinduced rotation of the order of Δθ_NL = 12^o for I = 1.832 GW/cm² at 0.55 T. This Giant nonlinear Faraday rotation is 60 times larger than the best reported value with an intensity three times larger than what we used [12–13].

Fig. 9. Net change of the photoinduced angle of rotation as function of the B-field, the red dots are the values measure with halfwave plate, while the blue triangles correspond to the data from Fig. 8.

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Finally, there is one interesting aspect that we have also explored, the possibility to modulate information in the polarization as a function of intensity. The EFTL can be driven by a pulse width modulating signal (PWM) (the experimental setup is shown in Fig. 10). We have used a square wave signal with a frequency of 600 Hz. The minimum current of the signal is f = 6.98 cm (low intensity) and f = 5.18 cm (high intensity) at the sample as a function of the B-field. In this case, the intensity modulation due to the TPA and Faraday plus Photoinduced rotation for ϕ_o = π /2 show the same frequency modulation as the driven frequency. On the other hand, if the high-intensity point (f = 5.18 cm) is located in the middle between the low current and the high current of the driving signal we were able to double the Frequency of the output signal. Figure 11 shows the experimental results with red curves for B = 0.5 T and Blue curves for B = 0 T.

Fig. 10. Pulse width modulation experimental setup, the EFTL is driven by A square wave that can have up to 2 kHz in frequency.

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Fig. 11. Oscilloscope signal as function of magnetic field in increments of 0.1 T B = 0.0 T (blue), B = 0.1 T (Cyan), B = 0.2 T (olive) B = 0.3 T (Navy), B = 0.4 T (Pink), and B = 0.5 T (red). The last graph is the normalized transmittance with B = 0.0 (blue) and B = 0.5 T (red).

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5. Conclusions

In conclusion, we have measured the photoinduced Faraday rotation in Cd_0.85Mn_0.15Te:Fe at 790 nm, and found a giant Faraday rotation of the order of 12 deg in a magnetic field B = 0.55 T and an intensity of the order of 1.832 GW/cm². This rotation is 60 times larger than previously reported [12–13]. This large linear and nonlinear rotation may be attributed to the presence of the Fe-ions as was investigated in [18]. In addition, we have explored the use of an EFTL to experimentally study the nonlinear magneto-optical properties of system. We showed that with only three traces (one with B = 0 and the other two with different B values B₁ and B₂) and exploring several points in the two none zero B value traces, we were able to obtain all the relevant linear and nonlinear magneto-optics properties of the sample. We have corroborated this by performing alternate experiments where the system is subject to low intensity (linear regimen) and high intensity (nonlinear regime) as a function of the B field and found excellent agreement with the results obtained in the nonlinear traces.

We show that in the case of Cd_0.85Mn_0.15Te:Fe the photoinduced polarization rotation is on the order of 12.0^o which is about 13% of the total linear Faraday rotation, this can be improved with very large static magnetic fields. So, it is possible to rotate light around 20^o ∼ 22^o in a magnetic field of ∼1 T, or by increasing the intensity of the light in the nonlinear regime.

Further research will focus in understanding experimentally and theoretically the effect of the Fe concentration on the imaginary part of the four-order magneto-electric susceptibility $\chi _{11}^{\prime\prime}$ and the third order nonlinear optical susceptibility $\chi _5^{\prime\prime}$ in addition to the Verdet constant of the material. The system can be optimized such that one can imagine encoding information in the light polarization by modulating the light intensity.

Acknowledgments

H. Garcia thanks SIUE for its financial support.

Disclosures

The authors have no conflicts of interest to declare that are relevant to the content of this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Giant nonlinear Faraday rotation in iron doped CdMnTe

Abstract

1. Introduction

2. Theoretical background

3. Experimental setup

4. Results and discussion

5. Conclusions

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Tables (1)

Equations (15)

Optical Materials Express