1 Introduction

Polarization phenomena take the main role in our understanding and application of electromagnetic radiation across the entire spectral range [1]. Among the polarization, circularly polarized light (CPL) gets more attention due to its interesting application in various fields including quantum-based optical computing and communication, holography [2], biometry [3], and nanofabrication [4]. Among the electromagnetic spectra, circularly polarized (CP) in terahertz (THz) region, the region between 0.1 and 10 THz, is highly desired due to its much application in science such as drug delivery and biological sensing [5, 6] because many biomolecules exhibit chiral structures with rotational or vibrational modes in the THz regime that can interact with right-handed circularly polarized (RCP) and left-handed circularly polarized (LCP) lights. Also, CPL is used in THz telecommunication processes [7]. However, a few works have investigated the circular-polarization-sensitive absorption, absorbing RCP or LCP component of incident light while reflecting the other component, which could find applications in fields of the CP wave detection and electromagnetic communications [2, 8].

In previous works, perfect absorption of light based on various structures based on metamaterial and graphene have been studied [9,10,11,12], but the CPL-sensitive absorbers have been investigated in a few structures. For example, in 2014, Wang and et al. have studied graphene-based absorbers under a magnetic bias. They have illustrated that by increasing the magnetic field strength up to 7 Tesla, the absorption peaks show quite different electromagnetic responses for the two kinds of CPLs [8]. In the other work, in 2014, Yang and et al. have designed CPL graphene-based absorber rely on magnetic circular dichroism. The simulation results have shown that the LCP incident light within the angle range of ± 80° has efficiently been absorbed by the graphene structure [13]. In 2017, Rashidi and co-workers considered a graphene-based one-dimensional photonic crystal under the external magnetic field. The results have represented that the magnetically tunable absorption of the structure depends on the CP state, and magnetic circular dichroism [14]. In 2018, Pan and et al. supposed a structure comprised of molybdenum zigzag arrays to overcome the lack of circular-polarization-sensitive absorbers working at high temperatures [2]. In the same year, Vepachedu and et al. experimentally assembled a chiral sculptured thin film that exhibits the circular Bragg phenomenon and can therefore be used as a CP filter in a spectral regime [15]. In 2021, Danka and co-workers have presented a CPL absorber based on chiral and demonstrated proposed chiral structure was a good candidate for designing a circular polarization selective absorber in the THz region [16]. One of the materials that show different electromagnetic responses for RCP and LCP light is Indium antimonite (InSb) [17]. InSb is an interesting III–V semiconductor used in long-wavelength optoelectronic applications. Its melting point is 809 K [18]. It shows two different refractive indices for RCP and LCP lights under magnetic field strength and highly depends on environmental temperature [19].

In this work, we proposed a new scheme to design a CPL-sensitive absorber containing InSb and Copper (Cu) layers that would control LCP and RCP absorption and frequency of the peak by the temperature and the magnetic field strength.

This manuscript is organized as follows: Sect. 2 covers the theoretical method. In this section, we have represented our proposed circular-polarization-sensitive absorber and used the 4 by 4 transfer matrix method to obtain the absorption of light by the structures. To this aim, we need the frequency-dependent optical properties of InSb. We have briefly represented the permittivity of the semiconductor InSb under the magnetic field in this section. In Sect. 3 we have studied the sensitivity of RCP and LCP absorption by the considered structure in the absence and presence of the Cu layer. To continue, by showing high sensitivity of the structure for RCP and LCP lights in the presence of Cu layer, we investigated the behavior of RCP and LCP absorption spectra versus varying the external magnetic field strength, environmental temperature, and InSb layer thickness. We concluded our results in Sect. 4.

2 Model and methods

Figure 1 schematically shows the proposed circular-polarization-sensitive absorber composed of an InSb and a Cu with the arrangement of InSb/Cu, where the InSb is a semiconductor with a thickness of dInSb, and Cu is the metal and non-magnetic (diamagnetic) [20] layer with a thickness of dCu. We suppose the whole structure is surrounded by air, and the electromagnetic wave propagates in the z-direction, and an external static magnetic field Bext is applied perpendicular to the structure along the propagation direction (z-axis). For the Cu layer, the permittivity is written as a function of the incident electromagnetic waves frequency (υ) in the THz region in the form of \(\varepsilon_{{{\text{Cu}}}} = - 1.7 \times 10^{5} + 1.1i \times 10^{6} \upsilon^{ - 1}\)[21]. We should note that the Cu melting point is mentioned at T = 1356 K (about 1082.85 °C) [22].

Fig. 1
figure 1

Schematic structure of circular-polarization-sensitive absorber arrangement with InSb/Cu where the external magnetic field (Bext) is applied perpendicular to the structure along the z-direction

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In the presence of an external magnetic field perpendicular to the InSb layer, the InSb becomes gyrotropic with an asymmetric relative permittivity tensor is given as [17]:

$$\overline{\overline{\varepsilon }} = \left( {\begin{array}{*{20}c} {\varepsilon_{xx} } & {\varepsilon_{xy} } & 0 \\ {\varepsilon_{yx} } & {\varepsilon_{yy} } & 0 \\ 0 & 0 & {\varepsilon_{zz} } \\ \end{array} } \right).$$
(1)

here

$$\varepsilon_{xx} = \varepsilon_{yy} = \varepsilon_{b} - \frac{{\omega_{p}^{2} \left( {\omega^{2} + i\gamma \omega } \right)}}{{\left( {\omega^{2} + i\gamma \omega } \right)^{2} - \omega^{2} \omega_{{c,{\text{InSb}}}}^{2} }},$$
(2)
$$\varepsilon_{xy} = - \varepsilon_{yx} = \frac{{i\omega_{p}^{2} \omega \omega_{c} }}{{\left( {\omega^{2} + i\gamma \omega } \right)^{2} - \omega^{2} \omega_{{c,{\text{InSb}}}}^{2} }},$$
(3)
$$\varepsilon_{zz} = \varepsilon_{b} - \frac{{\omega_{p}^{2} \left( {\omega^{2} + i\gamma \omega } \right)}}{{\left( {\omega^{2} + i\gamma \omega } \right)^{2} }},$$
(4)

The \(\varepsilon_{b} = 15.68\) is the background high-frequency dielectric constant of InSb and \(\omega = 2\pi \upsilon\) is the angular frequency of incident electromagnetic waves.\(\omega_{p} = \sqrt {{\raise0.7ex\hbox{${N_{{{\text{InSb}}}} e^{2} }$} \!\mathord{\left/ {\vphantom {{N_{{{\text{InSb}}}} e^{2} } {\varepsilon_{0} m^{*} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\varepsilon_{0} m^{*} }$}}}\), and the \(\omega_{{c,{\text{InSb}}}} = {\raise0.7ex\hbox{${eB_{{{\text{ext}}}} }$} \!\mathord{\left/ {\vphantom {{eB_{{{\text{ext}}}} } {m^{*} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${m^{*} }$}}\) denote the plasma frequency, and cyclotron frequency, respectively [17]. \(N_{{{\text{InSb}}}} = 5.76 \times 10^{20} T^{1.5} e^{{{\raise0.7ex\hbox{${ - 0.26}$} \!\mathord{\left/ {\vphantom {{ - 0.26} {\left( {2 \times 8.625 \times 10^{ - 5} \times T} \right)}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\left( {2 \times 8.625 \times 10^{ - 5} \times T} \right)}$}}}} {\text{m}}^{ - 3}\) and \(m^{*} = 0.014m_{0}\) are the carrier density, and effective mass in the InSb layer, respectively [17]. The \(m_{o}\) indicates free electron mass, and it has a value of about \(9.1 \times 10^{ - 31} {\text{Kg}}\). \(\varepsilon_{0} = 8.85 \times 10^{ - 12}\) F.m−1 is permittivity of a vacuum [23]. \(\gamma = {\raise0.7ex\hbox{$e$} \!\mathord{\left/ {\vphantom {e {\left( {\delta m^{*} } \right)}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\left( {\delta m^{*} } \right)}$}}\) stands for the collision frequency of carriers in which δ is the carrier mobility and can be modelled as \(\delta = 7.7\,\left( {{\raise0.7ex\hbox{$T$} \!\mathord{\left/ {\vphantom {T {300}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${300}$}}} \right)^{ - 1.66} {\text{m}}^{2} .{\text{V}}^{ - 1} .{\text{s}}^{ - 1}\)[17]. T is environmental temperature. To calculate the reflectance, transmittance of the proposed structure, we used the 4 × 4 transfer matrix method [24]. In this method, by solving the Maxwell equations the optical field inside each layer is given as the sum of four normal modes: right and left circularly polarized waves propagating in forward and backward directions normal to the structure. Then determine the values of the optical fields on the layer boundaries; we can obtain the total transfer matrix of the whole structure as:

$$M = \left[ {D^{(0)} } \right]^{ - 1} \left[ {S_{{{\text{InSb}}}} S_{{{\text{Cu}}}} } \right]D^{(0)} ,$$
(5)

where D(0) and S are the dynamic matrices in air, and characteristics matrix in the n-th layer (n = InSb, and Cu), respectively, are given by:

$$S_{n} = \left[ {\begin{array}{*{20}c} {\cos \beta_{n}^{ + } } & {\frac{ - i}{{N_{n}^{ + } }}\sin \beta_{n}^{ + } } & 0 & 0 \\ { - iN_{n}^{ + } \sin \beta_{n}^{ + } } & {\cos \left( {\beta_{n}^{ + } } \right)} & 0 & 0 \\ 0 & 0 & {\cos \left( {\beta_{n}^{ - } } \right)} & {\frac{ - i}{{N_{n}^{ - } }}\sin \beta_{n}^{ - } } \\ 0 & 0 & { - iN_{n}^{ - } \sin \beta_{n}^{ - } } & {\cos \left( {\beta_{n}^{ - } } \right)} \\ \end{array} } \right],$$
(6)

and

$$D^{(0)} = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 \\ 1 & { - 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & { - 1} \\ \end{array} } \right],$$
(7)

here \(\beta_{n}^{ \pm } = \frac{\omega }{C}N_{n}^{ \pm } d_{n}\), \(N_{n}^{ \pm } = \sqrt {\varepsilon_{xx}^{n} \pm i\varepsilon_{xy}^{n} }\) are the complex refractive indices in the n-th layer in which the signs “+” and “−” stand for RCP and LCP lights, respectively. dn denotes the thickness of the n-th layer and C is the light speed in the vacuum, respectively. It should be noted that for the Cu layer \(\varepsilon_{xy} = 0\). The reflectance (R), and transmittance of RCP and LCP lights can be respectively derived by the total matrix elements as [24]:

$$R_{{{\text{RCP}}}} = \left| {\frac{{M_{21} }}{{M_{11} }}} \right|^{2} ,R_{{{\text{LCP}}}} = \left| {\frac{{M_{43} }}{{M_{33} }}} \right|^{2}$$
(8)
$$T_{{{\text{RCP}}}} = \left| {\frac{1}{{M_{11} }}} \right|^{2} ,T_{{{\text{LCP}}}} = \left| {\frac{1}{{M_{33} }}} \right|^{2}$$
(9)

Finally, by the aid of Eq. (8) and Eq. (9), the absorptance (A) can be obtained by:

$$A_{{\text{RCP(LCP)}}} = 1 - R_{{\text{RCP(LCP)}}} - T_{{\text{RCP(LCP)}}}$$
(10)

3 Results

For numerical calculations of light absorption from the proposed structure, we choose \(d_{{{\text{Cu}}}} = 0.6{\mu m}\) and the investigation is performed in the 1–14 THz frequency region. At first, we investigate the proposed circular-polarization-sensitive in absence of the Cu layer under different magnetic fields. The RCP and LCP light absorptions versus frequency for \(d_{{{\text{InSb}}}} = 2.2{\mu m}\) at T = 300 K under different magnetic field strengths have shown in Fig. 2. As can be seen in Fig. 2, the sensitivity of RCP and LCP light for InSb under different magnetic fields is weak and the absorption is small. For the RCP light absorption increases by increasing the magnetic field. In contrast, the LCP light absorption decreases by increasing the magnetic field. Figure 2 illustrates that by increasing the magnetic field strength, light can be sensitive for RCP and LCP in this structure but show a small absorption value. Figure 3 displays the influence of the Cu layer on the sensitivity of the proposed structure for RCP and LCP lights within 3–8 THz. As the result shows, the perfect absorption of the RCP light appears at 5.2 THz with absorption over 99% (Fig. 3a). However, as can be seen in Fig. 3b, the LCP light mostly reflected at the whole frequency. Transmittance is near zero because of using Cu as a reflector layer. So, the different absorptions for RCP and LCP lights around 5.2 THz indicate that the proposed structure can be utilized as a circular-polarization-sensitive absorber.

Fig. 2
figure 2

RCP and LCP light absorptions versus frequency in the InSb layer at T = 300 K under different magnetic field strengths

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Fig. 3
figure 3

RCP and LCP lights a absorption, and b reflection spectra versus frequency in the InSb/Cu at T = 300 K under Bext = 3 Tesla

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Figure 4 displays the sensitivity of RCP and LCP light absorptions to the magnetic field strength for \(d_{{{\text{InSb}}}} = 2.2{\mu m}\) at T = 300 K. Figure 4a displays RCP light absorption spectra of our proposed structure at different magnetic field strengths around Bext = 3 Tesla. ALCP illustrates that maximum absorption of about 99% occurred for Bext = 2.8–3 Tesla, and values of absorption peak decrease by getting away from Bext = 3 Tesla.

Fig. 4
figure 4

a RCP absorption of the InSb/Cu structure under around Bext = 3 Tesla, and RCP and LCP absorptions at T = 300 K under the different magnetic field strength along b z, and c − z-direction

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In general, by increasing the magnetic field the absorption peak shifts to higher frequencies, and its absorption value at lower magnetic fields (Bext < 3 Tesla) is increased by increasing magnetic field strengths. But as can be seen, for the higher value than 3 Tesla, its absorption decreases.

The influence of magnetic field strength on RCP and LCP absorptions for both directions of magnetic fields (b) z and (c) − z is represented in Fig. 4b, c, respectively. It is seen from Fig. 4b that the absorption peak of RCP light increases by increasing magnetic field strengths and the highest values of the peak occurs around the 3 Tesla at frequency 5.2 THz. For the magnetic field strength above 3 Tesla, the absorption peak values are decreased. In the whole frequency range, the absorption peak experiences a shift to higher frequencies, and the width of the peak increases by increasing magnetic field strength. Moreover, the LCP light absorption is too weak to be seen in conventional optical absorption. The results of RCP and LCP absorptions show the proposed structure is circular-polarization-sensitive for different magnetic field strength between 0 and 5 Tesla at the whole frequency range. Also, interestingly by reversing the direction of the external magnetic field, the absorption behaviors of RCP and LCP light, as indicated in Fig. 4c are interchanged. In this case, as seen in Fig. 4c perfect absorption peak for LCP light in the − z-direction occurs around Bext = − 3 Tesla. By increasing magnetic field strength along − z-direction, the absorption peak shows the same behavior as mentioned in Fig. 4(b) in the z-direction. Therefore, the proposed circular-polarization-sensitive absorber is polarization-selective through magnetic adjustment, and direction of the external applied magnetic field. Moreover, the frequency of absorption peaks can be tuned by magnetic field strength. The tunability of the RCP light for \(d_{{{\text{InSb}}}} = 2.2{\mu m}\) under Bext = 3 Tesla at different environmental temperatures is represented in Fig. 5a. The RCP light absorption spectra of our proposed structure at T > 270 K show perfect absorption of about 99% and almost its value maintains unchanged by increasing environmental temperature. In addition, the positions of the absorption peak shift toward lower frequencies by increasing the environmental temperature.

Fig. 5
figure 5

a RCP absorption of the InSb/Cu structure, and RCP and LCP absorption spectra under Bext = 3 Tesla at different environmental temperatures for magnetic field direction along b z, and c − z

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Figure 5b, c illustrate the effect of environmental temperature on RCP and LCP absorption peak under Bext = 3 Tesla and Bext = − 3 Tesla, respectively. Figure 5b shows by increasing environmental temperature RCP absorption peak value increases and shifts toward higher frequencies.

The width and value of the absorption peak maintain unchanged for higher environmental temperatures. In contrast, the LCP light absorption is too weak and negligible. On the other hand, Fig. 5c shows the RCP and LCP light behaviors under Bext = − 3 Tesla. Results show the behaviors of RCP and LCP are interchanged by comparing with Bext = 3 Tesla. As seen in Fig. 5c, the absorption peak reveals for LCP light while absorption is negligible for RCP light when Bext = − 3 Tesla. By increasing the environmental temperature, LCP light peak absorption shifts to lower frequencies, its width and value maintain unchanged, particularly for higher temperatures (T > 270 K).

So, it can be concluded that not only proposed circular-polarization-sensitive absorber is tunable through environmental temperature but show a perfect absorption of about 99% at \(\upsilon = 5.2{\text{THz}}\) and Bext = 3 Tesla. Moreover, the frequency of absorption peaks can be tuned by environmental temperature with no change in peak absorption values.

Figure 6 shows the absorption of RCP and LCP lights for the structure with various thicknesses of InSb under Bext = 3 Tesla and T = 300 K. It can be seen from Fig. 6 that when dInSb is more increased the new peak in absorption spectra has been revealed for LCP light. This proves that the thickness of InSb has a significant impact on the values and frequency position of the absorption spectra.

Fig. 6
figure 6

a RCP absorption of the InSb/Cu structure, and RCP and LCP absorptions spectra under Bext = 3 Tesla at T = 300 K for different InSb layer thickness for magnetic field direction along b z, and c − z

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Figure 6b, c illustrate the effect of dInSb on the RCP and LCP lights of the structure under Bext = 3 Tesla and − 3 Tesla, respectively. Figure 6b shows the absorption of RCP light occurs for dInSb > 1 μm, and its perfect absorption happened for dInSb between 2.5 and 3.5 μm. When the dInSb reached above 3.5 μm the values of the absorption peak at lower frequencies decrease and new peaks in absorption spectra have been revealed. Moreover, as the dInSb gradually increases, the perfect absorption peak occurs in higher frequencies.

In other words, when dInSb becomes larger, the value of peak absorption becomes smaller, and the frequency point at the peak value moves to a lower frequency, as shown in Fig. 6a, b.

4 Conclusion

In summary, based on the different electromagnetic responses for right-handed circularly polarized (RCP) and left-handed circularly polarized (LCP) lights on InSb, we have designed a new type of circular-polarization-sensitive absorber in the THz frequency region with an arrangement of InSb/Cu.

We theoretically have studied the characteristics of the RCP and LCP lights absorption by varying the environmental temperature, magnetic field strength, and the InSb layer thickness. Theoretical calculation results have shown that the absorption peak of about 99% has occurred for RCP light at around 5.2 THz under Bext = 3 Tesla at T = 300 K. The LCP light has reflected at the whole frequency region. The influence of magnetic field strengths and their direction on The RCP and LCP light absorption have been investigated. Results have illustrated that maximum absorption under applying a magnetic field in z-direction for RCP light has been achieved only around Bext = 3 Tesla, and LCP light absorption was too weak to be seen in conventional optical absorption. For applying a magnetic field in − z-direction, LCP light maximum absorption has occurred only around Bext = − 3 Tesla, and RCP light absorption was too weak. When the magnetic field strength was increasing, the absorption peak shifted toward higher frequencies. By changing the environmental temperature, the absorption peaks have produced significant results. For RCP and LCP light above T > 270 K the perfect absorption has occurred around Bext = 3 Tesla and Bext = − 3 Tesla, respectively. As the temperature is increased, the peak absorption value almost maintains the same and experiences a shift toward lower frequencies.

Finally, the influence of the InSb layer thickness on the absorption peak of RCP and LCP lights has been researched. The achieved results show that by increasing the thickness of InSb the new peak in absorption spectra has been revealed, and perfect absorption occurs in higher frequencies for the same magnetic field strength.