1. Introduction

The principle of operation of most devices dedicated to non-reciprocal light transmission is based on the Faraday effect [1], where the polarization orientation of the light is changed by an external magnetic field as the light propagates through a transparent material. Due to the Faraday effect, the polarization orientation of the light is rotated in the same direction regardless of the propagation direction. The polarization rotation angles of the forward and backward propagating light do not compensate as usual but add up instead. This is because the change of polarization direction is defined only by the direction of the magnetic field and the Verdet constant. Therefore, by implementing specifically aligned polarizers, a device for non-reciprocal light transmission can be constructed, in which the forward propagating light is transmitted but backward propagating light is blocked. In addition, by placing two Faraday rotators in a Sagnac interferometer, a polarization-independent optical isolator can be designed [2]. Alternatively, non-reciprocal light transmission can be realized by simple rotating mechanical components. In one of the proposed methods, the non-reciproca $L$l transmission of light is formed by rotating half-wave plates [3,4]. Such optical isolator consists of two half-wave plates separated by a distance and rotating at an angular velocity $\Omega $ . In addition, the fast axis of the second half-wave plate is rotated by a fixed angle of ${\pi / 8}$ radians with respect to the first half-wave plate. The polarization of the light propagating in the forward direction through such optical isolator does not change, and for light propagating in the backward direction the polarization rotates perpendicularly to the initial orientation and it can be blocked by external polarizer. However, the practical implementation of this method is impossible due to the extremely high rotation velocity of the wave plates and/or the distance between them, because the following condition must be satisfied: $\Omega \,L = {{\pi \,c} / 8}$, where c is the speed of light between the wave plates. A practical improvement of this device may be realized by replacing half-wave plates with electro-optical crystals which can be electrically controlled. Nonlinear optical effects can also be exploited to generate non-reciprocity [5]. Another known device for the non-reciprocal light transmission consists of a rotating spherical resonator made from silica and a flying coupler placed over it [6]. In this case, the non-reciprocal transmission of light is realized mainly due to the Fizeau effect in the rotating sphere [7–9]. Similarly, non-reciprocal light propagation can be realized using only the Sagnac effect in its purest form [7,10]. Inspired by this idea, we proposed and investigated a four-port circulator based on a rotating fiber ring interferometer, where non-reciprocal light transmission is achieved due to the Sagnac effect. To the best of our knowledge, this is the first experimental demonstration of non-reciprocal light transmission based solely on the pure Sagnac effect.

2. Design and operation of the Sagnac circulator

The proposed Sagnac circulator is based on a ring interferometer rotating at an angular velocity $\Omega $. The light beam is inputted into the ring interferometer and divided into two perpendicularly polarized and counter-propagating light beams of equal power. Due to the Sagnac effect, a phase difference ${\pm} {\pi / 2}$ rad is obtained between the counter-propagating light beams. The sign of the phase difference depends on the direction of propagation of the light beams with respect to the direction of rotation of the ring interferometer. In addition, a fixed phase difference ${\pi / 2}$ rad is added between the counter-propagating and perpendicularly polarized light beams. The fixed phase difference is independent on the velocity and direction of rotation of the ring interferometer. Thus, the sum phase difference between the counter-propagating light beams is either $\pi $ rad or 0. Then counter-propagating light beams are combined into a single light beam which is outputted from the rotating ring interferometer by using a different path than the one through which the light beam is inputted into it [10].

The ring interferometer of the Sagnac circulator investigated in this work consists of two polarization maintaining (PM) fiber segments of similar length ${L_1}$ and ${L_2}$ and a polarizing beam splitter (PBS) (Fig. 1). The fiber segments are crosswise spliced to form an angle of 90 deg between their slow (or fast) axes. Collimators are attached to the ends of the fibers. The collimators are oriented so that the slow axis of the fiber with a length ${L_1}$ is parallel to the plane of incidence on the PBS and the slow axis of the fiber with a length ${L_2}$ is perpendicular to the plane of the incidence on the PBS. The PBS reflects the s-polarized light beam (s polarization orientation is perpendicular to the plane of incidence on the PBS) and transmits the p-polarized light beam (p polarization orientation is parallel to the plane of incidence on the PBS). The slow axes of the input/output fibers F1 and F2 are mirror-symmetrically oriented at an angle of +45 degrees with respect to the plane of incidence on the PBS. Thus, light polarized along the slow axis (slow-axis mode) of the fiber F1 or F2 is a superposition of s and p-polarized light beams of equal power and equal phase. Similarly, light polarized along the fast axis (fast-axis mode) of the fiber F1 or F2 is a superposition of the s and p-polarized light beams with equal power, but with a phase difference $\pi $rad.

 

Fig. 1. Fiber ring interferometer with input/output fibers F1 and F2 and quarter-wave plates QWP1 and QWP2.

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We will determine the fixed phase difference between the s and p-polarized light beams in a non-rotating ring interferometer, i.e., in a stationary case. Suppose the light beam enters the ring interferometer through fiber F1 and is polarized along one of the axes of the fiber. The s-polarized light beam reflects from the PBS and enters the fiber segment of length ${L_1}$, the fast axis of the fiber is oriented perpendicularly to the plane of incidence on the PBS and matches the polarization orientation of the beam reflecting from the PBS. The light propagates through the fiber segment of length ${L_1}$, then it enters another fiber segment of length ${L_2}$, slow axes of which are rotated by 90 degrees with respect the fiber segment of length ${L_1}$, so from this point the light continues propagating in the slow-axis mode until it reaches the PBS through the collimator of fiber segment of length ${L_2}$. The slow axis of the fiber segment of length ${L_2}$ is oriented perpendicularly to the plane of incidence on the PBS and the outgoing beam is s-polarized. The s-polarized light beam reflected from the PBS is coupled into the fiber F2. Thus, at the input and output of the ring interferometer, the light beam is s-polarized; the polarization orientation does not change. Meanwhile, the p-polarized light beam initially propagates through PBS, then it is coupled into the fiber segment of length ${L_2}$. In this fiber segment the light propagates in fast-axis mode, and in the fiber segment of length ${L_1}$ – in slow-axis mode. The polarization of the output light of the fiber segment of length ${L_1}$ is parallel to the plane of incidence on the PBS, so the light beam is p-polarized. The p-polarized light beam is transmitted through the PBS and coupled into fiber F2. In this case, the light beam at the input and at the output of the ring interferometer is p-polarized. Based on the examples described above, it can be easily shown that the fixed phase difference acquired between the contra-propagating s and p-polarized light beams is the same independently from which side the light beams are inputted into the ring interferometer:

Here $B = {n_{slow}} - {n_{fast}}$ is the birefringence of the fiber, that is, the difference between the effective refractive indices of the slow and fast-axis modes of the fiber, $\lambda $ is the wavelength in vacuum, ${\phi _R}$ is the phase change due to the reflection of the s-polarized light beam from the first and second sides of the PBS, ${\phi _T}$ is the phase change resulting from the transmission of the p-polarized beam through the PBS from both sides. As we can see, the smaller is a difference between the fiber lengths ${L_2}$ and ${L_1}$, the smaller is the fixed phase difference between the s and p-polarized light beams. The fixed phase difference between the s and p-polarized light beams is constant regardless of the rotational velocity and rotation direction of the ring interferometer. Since in practice it is very difficult to make the fiber lengths ${L_2}$ and ${L_1}$ equal with a high accuracy, temperature adjustment is used for fine adjustment of the phase difference. The part of the fiber segment of length ${L_1}$ denoted as ${L_{h1}}$ is placed on a heater (see Fig. 1). This way the birefringence of the fiber can be controlled with temperature. The decrease in fiber birefringence is approximately proportional to the increase in temperature [11]:

Here γ is the proportionality factor. Typical γ-factor values reported in the literature for PANDA PM fibers are $- 5.93 \times {10^{ - 7}}\;^\circ {\textrm{C}^{ - 1}}$ and $- 5.29 \times {10^{ - 7}}\;^\circ {\textrm{C}^{ - 1}}$ [11]. For comparison, our measured γ-factor value for the Fujikura PM980 fiber is $- 5.57 \times {10^{ - 7}}\;^\circ {\textrm{C}^{ - 1}}$ (see Supplement 1). The change in phase difference due to the change in birefringence is several orders of magnitude larger than the change due to the thermal expansion of the fiber [12], therefore the thermal expansion of the fiber is not taken into account. Thus, the change in phase difference during heating of the fiber is:

If the fiber segment of length ${L_2}$ is heated, then a plus sign should be written in the expression (3).

The fixed phase difference acquired between the s and p-polarized light beams in the ring interferometer, including the temperature adjustment, is the sum of expressions (1) and (3):

When the ring interferometer rotates at an angular velocity ${\mathbf \Omega }$, the phase difference between the contra-propagating light beams due to the Sagnac effect is [13]:

Here ${\mathbf A}$ is the area enclosed by the fiber loop, $\lambda $ is the wavelength of light in vacuum and c the speed of light in vacuum. It is remarkable that the Sagnac effect is independent of the refractive index of the medium [7,14–16], suggesting a universal geometric origin for the effect. Hence the area law holds for ring interferometers made with optical fibers. The phase difference is proportional to the dot product of angular velocity ${\mathbf \Omega }$ and area ${\mathbf A}$. The sign of the phase difference depends on which side the light beam enters the ring interferometer with respect to its rotation direction. When the light beam enters the ring interferometer from the fiber F1, then the s-polarized light beam propagates in the clockwise (CW) direction and the p-polarized light beam propagates counter-clockwise (CCW). If the interferometer rotates in the CW direction (see Fig. 1), then the phase difference between the s and p-polarized light beams has a negative sign. The opposite situation is when the light beam enters the ring interferometer from fiber F2. In this case, the s-polarized light beam propagates in the CCW direction, and the p-polarized light beam propagates in the CW direction. In this case the phase difference has a positive sign. As we can see, in both cases the phase difference between the s and p-polarized light beams acquired due to the Sagnac effect has the same absolute value, but opposite signs.

Thus, the total-sum phase difference at the output of the rotating ring interferometer between the s and p-polarized light beams is $\Delta \varphi + \Delta \Phi + \delta $. Here δ is the initial phase difference between the s and p-polarized light beams at the input of the ring interferometer, which can be 0 or $\pi $ rad, depending on the polarization of the light in the input fiber, which can be aligned to slow or fast axis of the fiber, as already mentioned.

We will find out which part of the power of the s and p-polarized light beams at the output of the ring interferometer is coupled into slow and fast-axis modes of the output fiber, respectively. For this purpose, we will use the Jones calculus for polarized light [17]. The normalized Jones vector for the electric field at the ring interferometer output is:

Here ${E_s}$ and ${E_p}$ are the complex amplitudes of the electric fields of the s and p polarizations, respectively.

Assuming that the slow axes of the input/output fibers F1 and F2 are mirror-symmetrically oriented at an angle of +45 degrees to the plane of incidence on the PBS, the projection of the electric field (6) on the slow or fast axis of the output fiber is:

Here, the plus sign corresponds to the projection of the electric field on the slow axis of the output fiber, and the minus sign corresponds to the projection on the fast axis of the output fiber. The normalized power of the electric field along the slow or fast axes of the output fiber is:

In the latter expression, the plus sign corresponds to the light transmission function between modes of the same polarization in the input and output fibers, that is, when light propagates in slow or fast-axis mode in the both fibers simultaneously. In other cases, a minus sign must be chosen. When $\Delta \varphi + \Delta \Phi = 0$, then the polarizations of the light at the input and output are the same, and when $\Delta \varphi + \Delta \Phi = \pi $ rad, the polarizations are mutually perpendicular.

Let’s examine the case where the fiber lengths ${L_2}$ and ${L_1}$ are chosen such that the fixed phase difference between the s and p-polarized light beams is $\Delta \varphi = {\pi / 2}$ rad, and the phase difference obtained due to the Sagnac effect is $\Delta \Phi ={\pm} {\pi / 2}$ rad. Depending from which side the light beam enters the ring interferometer (assuming that the ring interferometer rotates in the CW direction, see Fig. 1), the sum phase difference between the contra-propagating s and p-polarized light beams may be 0 or $\pi $ rad respectively. When the beam enters the ring interferometer from the fiber F1 side, then the sum phase difference between the s and p-polarized beams at the interferometer output is $\Delta \varphi + \Delta \Phi = 0$ and according to expression (9) we find that the resulting light beams at the input and output of the interferometer, i.e., the superposition of the s and p-polarized beams, are identically polarized. Otherwise, when the beam enters the ring interferometer from the fiber F2 side, the sum phase difference between the s and p polarization beams is $\Delta \varphi + \Delta \Phi = \pi $ rad. The resulting beams are then perpendicularly polarized at the input and output of the ring interferometer. For the combinations of fibers F1 and F2 and their polarization modes, we assign the corresponding port numbers from 1 to 4 and list them in Table 1. The first column of the table shows the input fibers and their polarization modes and the assigned port numbers; the second column shows the sum phase difference acquired in the ring interferometer between the s and p-polarized light beams; the third column shows the output fibers and their polarization modes and the corresponding port numbers; the last column shows the directionality of light transmission between the ports. As can be seen, the light from port 1 is transmitted to port 2, from port 2 to port 3, from port 3 to port 4, and from port 4 to port 1.

Table 1. Operation of the four-port Sagnac circulator

View Table

So we have a four-port optical circulator. If we change the direction of rotation of the ring interferometer, or change the sign of the fixed phase difference from positive to negative, then the direction of light transmission between the ports changes in the opposite direction, that is, 1→4→3→2→1.

3. Experimental demonstration

For the demonstration and characterization of the Sagnac circulator, a prototype of a rotating fiber ring interferometer was designed and an experimental scheme for the characterization of the circulator was assembled (Fig. 2 and Fig. 3). The optical scheme of a fiber ring interferometer shown in Fig. 1 was mounted on a 160 mm diameter aluminum disk rotated by a hollow shaft electrical motor.

 

Fig. 2. Schematic of an experimental setup of a polarization-dependent four-port fiber optic circulator. The dotted line defines a rotating ring interferometer, the optical scheme of which is shown in Fig. 1.

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Fig. 3. Photo of an experimental setup of a polarization-dependent four-port fiber optic circulator during the experiment. 1 – port 1 and a stationary mounted fiber collimator; 2 – port 2 and a stationary mounted fiber collimator; 3 – port 3; 4 – port 4; 5 – rotating disk; 6 – hollow shaft electric motor; 7 – rotating fiber collimator and QWP1 or QWP2; 8 – stationary mounted QWP3 or QWP4 and PC1 or PC2; 9 – stationary mounted fiber end of the DL976; 10 - non-contact optical tachometer.

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The loop of the ring interferometer was made of two segments of PM fiber (PANDA type, Fujikura PM980) of equal length, about 1.7 m. Since the polarization beat length specified by the manufacturer is in the range of 1.5 - 2.7 mm @ 980 nm, the fiber lengths were selected with an accuracy of at least 1 mm. For a more accurate adjustment of the fixed phase difference ($\Delta \varphi $), temperature adjustment was implemented by heating the corresponding fiber segment mounted in the heater. A variable power, 976 nm wavelength diode laser (DL976) with multimode fiber output (125/105 NA 0.15) was used for heating. A diode laser illuminated the absorbent heater layer, which was thermally insulated from the disk on the opposite side. $\Delta \varphi $ was precisely tuned by controlling the power of the diode laser. The fiber of the ring interferometer was coiled into 9 loops and attached to the disk. The diameter of the windings was in the range of 120–125 mm (average total area equal to 0.106 m²), the estimated disk rotation velocity required to achieve the ${\pm} {\pi / 2}$ rad phase difference at a wavelength of 1063 nm according to expression (5) is 188 rad/s or 1795RPM. The opposite ends of the input/output fibers F1 and F2 with the attached collimators were mounted on the hollow shaft of the prototype by aligning the optical axis of the collimator with the axis of rotation of the disk. Since the collimators rotate with the ring interferometer, the plane of linear polarization rotates accordingly. In order to prevent rotation of the linear polarization, circularly polarized light is transmitted between the rotating and stationary parts of the circulator. Quarter-wave plates are used for this purpose, which convert linearly polarized light into circularly polarized light and vice versa. The quarter-wave plates QWP1 and QWP2 rotate together with the ring interferometer, and QWP3 and QWP4 are fixed. Polarizing cubes PC1 and PC2 are used to separate and direct perpendicularly polarized light beams to/from the respective input/output ports. A pigtailed distributed Bragg reflector (DBR) single-frequency PM diode laser (DL1063) with an integrated isolator with a central wavelength of 1063 nm was used to characterize the Sagnac circulator. Using a fiber optical coupler (FOC 50:50), the diode laser output was divided into two fiber branches and directed to ports 1 and 2 on opposite sides of the circulator. Four power meters (PD1-PD4), one for each port were used to characterize the Sagnac circulator. The rotational velocity of the disk with a ring interferometer was measured with a non-contact optical tachometer. Since the required heating power of DL976 to produce a fixed phase difference of ${\pi / 2}$ rad was not known in advance, during the experiment, the heating power was selected so that the maximum light transmittance was reached in the forward direction from port 1 to port 2, and the light transmittance from port 1 to port 4 was minimal, respectively. The heating power was tuned for each rotational velocity setting so that the condition $\Delta \varphi + \Delta \Phi = 0$ was ensured in the forward propagation direction of the light beam, regardless of the rotational velocity of the ring interferometer. This condition is satisfied when $\Delta \varphi ={-} \Delta \Phi $. However, such temperature tuning does not ensure that $\Delta \varphi = {\pi / 2}$rad, so in the backward propagation direction of the light beam, the sum phase difference is not necessarily equal to $\pi $ rad and can vary from 0 to $\pi $ rad depending on the rotational velocity. In the backward propagation direction, the signs of the phase differences coincide, i.e., $\Delta \varphi = \Delta \Phi $ and the sum phase difference is equal to:

After inserting the expression (10) into the expression (9) and normalizing it to the maximum transmittance (${T_{\max }}$), we find the light transmittance function from port 2 to ports 1 and 3, depending on the angular velocity:

A plus sign corresponds to the direction from port 2 to port 1, and a minus sign corresponds to direction from port 2 to port 3. ${T_{\max }}$ is measured experimentally, and area A is the fitting parameter of the function $T(\Omega )$. Under the above conditions, the experimentally measured light transmittance between the different circulator ports is shown in Fig. 4. The continuous curves represent the function $T(\Omega )$, where the fitting parameter $\textrm{A} = 0.1068\;{\textrm{m}^2}$. The measured optimal rotational velocity of the ring interferometer corresponding to $\Delta \Phi ={\pm} {\pi / 2}$ rad is 187 rad/s or 1786RPM.

 

Fig. 4. Light transmittance between different circulator ports depending on the rotational velocity of the ring interferometer. The legend indicates the directions and percentage of light transmission. For each measurement point the heating power was tuned to maximize light transmission from port 1 to port 2. Symbols represent measured values, continuous curves represent fitting of experimental data using function (11).

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Initially, until the disk rotation velocity is close to zero, the light transmittance between ports 1 and 2 in the forward (Fig. 4 black squares) and backward (Fig. 4 red triangles) directions is equal. By further increasing the velocity of rotation of the disk and maintaining the condition that the sum phase difference in the forward propagation direction is always zero, the transmittance in the backward direction, i.e., from port 2 to port 1 begins to decrease and more and more light power is directed to port 3 (Fig. 4 dark cyan inverted triangles) in accordance with function (11). Finally, when the optimal disk rotation velocity is reached (187 rad/s), the transmittance from port 2 to port 1 is minimal. The light is no longer propagating in the same way in the backward direction as in the forward direction, but it is directed to port 3 instead. When the optimal disk rotation velocity was reached, the light transmission from port 1 to port 2 was 41.6%, and from port 2 to port 1 was only 0.4%. The transmittance of the circulator between ports 1 and 2 in the forward and backward directions differed by a ratio of at least 104:1. As the disk velocity was further increased, a slight drop in transmittance in the forward direction from port 1 to port 2 was observed. This may have been related to the misalignment of the ring interferometer due to centrifugal forces. The power of the diode laser (DL976) used to heat the fiber, while maintaining the condition $\Delta \varphi + \Delta \Phi = 0$, was adjusted to the rotational velocity of the disk for each measurement point (Fig. 5 upper curve). The power of the DL976 had to be increased slightly faster than the rotational velocity was increased, although according to expression (3) the phase change is proportional to the temperature. This deviation was most likely caused by the increase of heater cooling efficiency as the disk rotation velocity increased. By changing the direction of rotation of the disc to the opposite, the heating power of the DL976, in contrast to the previous cases, had to be steadily reduced by increasing the rotation velocity of the disc (Fig. 5 bottom curve). This is because the $\Delta \Phi $ sign has changed and $\Delta {\varphi _h}$ must change in the opposite direction in order to maintain the above-mentioned condition $\Delta \varphi + \Delta \Phi = 0$. When the disk rotates in the opposite direction, i.e., in the CCW direction, the heating power had to be reduced at a lower rate compared to the case where the disk rotated in the CW direction. This difference arises from the increase of the heater cooling efficiency as the disk velocity increases.

 

Fig. 5. Required heating power of the diode laser (DL976) to ensure the maximum light transmission from port 1 to port 2 versus the rotational velocity of the ring interferometer.

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4. Experiment using a broadband spectrum light source

The light transmittance characteristics of the Sagnac circulator using a pigtailed broadband polarization-maintaining superluminescent diode were also examined. Central wavelength was approx. 1045 nm; optical bandwidth at -3 dB level was 40 nm. It was observed that the characteristics of the circulator, i.e., isolation, became significantly worse, when using a broadband radiation compared to the earlier described case with the narrowband radiation. However, when using adjustable band-pass filter with the bandwidth of about 1 nm @ FWHM, only 1 nm-broad part was cut from a broad spectrum and the circulator isolation became comparable to the previous case. The difference in transmittance between forward and backward propagating light was more than 100 times. Each time the central wavelength was changed, the diode laser (DL976, see Fig. 2) power had to be adjusted to achieve maximum non-reciprocity, i.e., to maintain $\Delta \varphi = {\pi / 2}$rad (Fig. 6). However, the rotational speed of the disc was changed insignificantly to maintain $\Delta \Phi ={\pm} {\pi / 2}$rad due to the weak dependence on the wavelength. Assuming that the change in heater temperature is approximately proportional to the heating power (${P_h}$) of DL976, and using relations (1), (3), (4) and assuming that ${L_2} - {L_1} \approx 0$, we find that ${P_h} \propto ({{\phi_R} - {\phi_T} \pm {\pi / 2}} )\lambda $.

 

Fig. 6. Required heating power of the diode laser (DL976) to ensure maximum light transmission from port 1 to port 2 and minimum transmission in backward direction depending on the wavelength of light.

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In other words, Fig. 6 shows the phase characteristics of a polarization beam splitter (PBS) as a function of wavelength. The difference in heating power between the two nearest curves (blue and red, or black and red) is 12 W and corresponds to a phase change of $2\pi $ rad (the disk rotates in the CW direction). Thus, the strong wavelength dependence of the phase characteristics of PBS impairs the transmittance characteristics of the Sagnac circulator in the presence of a sufficiently broad spectrum of radiation.

5. Conclusions

In summary, an optical fiber circulator based on the Sagnac effect was proposed and investigated. The experimentally demonstrated light transmittance from port 1 to port 2 was 41.6% and from port 2 to port 1 was less than 0.4%. In other words, the forward and backward light transmittance differed by a ratio of at least 104:1 between the same circulator ports and the optical isolation in the backward direction was 24 dB. It was found that for broadband radiation, the properties of the Sagnac circulator are significantly affected by the wavelength dependence of the phase characteristics of the polarizing beam splitter. The proposed circulator can be attractive for practical application as it can be easily designed to operate at any wavelength. The main optical elements of the Sagnac circulator are the beam splitter, the polarization maintaining fibers and the quarter-wave plates, and the device does not use magneto-optical optically transparent materials, the properties of which are often strongly dependent on the wavelength. The Sagnac circulator can be also realized in a free-space configuration, with a ring interferometer consisting of a beam splitter and several mirrors. A constant phase difference between counter-propagating and perpendicularly polarized light beams can be induced by using a quarter-wave plate with its optical axis oriented parallel to or perpendicular to the plane of polarization of the light beam. The Sagnac circulator in the free-space configuration could be adapted for particularly high optical powers because it does not contain optical components susceptible to thermal effects or optical damage, in contrast to conventionally used devices based on magneto-optical materials.