1 Introduction

Quantum secure communication technology is one of the effective means to overcoming the threat of quantum computation at this stage, and free space quantum secure communication technology is an important part of the air-ground integrated quantum secure communication network [1]. Free space quantum secure communication technology often applies polarization coding or polarization multiplexing regimes, and thus requires an optical system with good polarization maintaining capability [2, 3]. Establishing a method to analyze the polarization aberration of optical systems for free space quantum secure communication, and instruct the designing of optical systems is an urgent problem for researchers.

Zhang Qing et al. established a mathematical model of polarization signal in free space quantum communication system, theoretically deduced the mathematical relationship between polarization contrast ratio and phase delay, and shows some guidance suggestions for the design of optical system. Finally, the dynamic characteristics of the polarization base deflection angle are analysed [4]. Wu Jincai et al. analyzed the phase delay mechanism of the polarization optics system, pointed out that the phase delays of the optics in the system are linearly superimposed, and a method to improve the system polarization holding ability by rotating the optics is proposed [5]. Lin Shuo et al. studied the polarization characteristics of the Cassegrain optical system, and analyzed the light incidence angle and the polarization state thoery, and numerically analyzed the polarization state of the signal by using the modeling of the optical system, which plays a certain reference to the polarization control [6]. Yang Yufei et al. investigated the effect of periscope turntable on the polarization characteristics, and derived the functional relationship between the rotation angle of two mirrors and the attenuation and phase delay of the system through geometrical modeling, matrix calculation and numerical analysis. And further studies were conducted to compare the differences in the polarization aberration of the system using different metal coatings [7]. Lei Jingwen designed a free space coherent laser communication optical system, and analyzed the polarization characteristics of the orthogonal reflector set. Finally, the polarization error of the system is suppressed by using waveplate set polarization compensation scheme [8]. Chen et al. designed a laser communication optical system based on polarization spectroscopy, and designed the polarization-preserving film system. Polarization loss simulations and experimental measurements of the coated optical telescope, the communication transmitter mirror group and the communication receiver mirror group are researched, and the polarization loss of the optical system was less than 0.1% [9].

In this paper, the polarization signal transmission model of the free space quantum secure communication optical system is established, and the polarization aberration analysis method of the optical system is obtained based on the vector light tracing theory. Taking a collimation and beam reduction optical system containing two lenses as an example, the relationship between polarization maintaining, polarization attenuation and the lens membrane layer, polarization state of the incident light, field of view of the incident signal parameters are analyzed. Also, the polarization aberration is optimized by adjusting the design parameters of optical system, and the polarization purity and amplitude transmittance of exiting polarized light are 0.98 and 0.95 respectively.

2 Methods

In order to obtain the effect of the optical system on the incident polarized light, the polarization component of the signal at each lens surfaces of the optical system can be obtained step by step using the ray tracing method. For the free space quantum secure communication system, the separation of the two orthogonal polarization components of the signal is the most important, so it is essential for optical system maintaining the polarization states of linearly polarized light. In order to simplify the complexity of the polarization aberration analysis process, a device coordinate system is established for each lens surface during polarization light transmission parameters calculations, and the origin of the coordinate system is made to lie in the curvature spherical center of the lens surface. The transmission process of the incident signal on the surface of the first lens is talked in this section, and the transmission of the polarized light in the optical system can be analyzed with a similar process.

2.1 Transmission Direction of Polarized Light

In this paper, the rectangular coordinate system is used for calculating the light propagation, while the spherical coordinate system is used in representing the light propagation direction and the oscillate direction of the polarized component, and the relationship between the two coordinate systems is shown in Fig. 1.

Fig. 1.
figure 1

Relationship between rectangular and spherical coordinate systems

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The coordinate transformation relationship between the two coordinate systems is

$$ \begin{array}{*{20}l} {r = \sqrt {x^{2} + y^{2} + z^{2} } } \hfill \\ {\theta = \arccos \left( {\frac{z}{{\sqrt {x^{2} + y^{2} + z^{2} } }}} \right)} \hfill \\ {\varphi = \arctan \left( \frac{y}{x} \right)} \hfill \\ \end{array} $$
(1)

In the spherical coordinate system, a point on the unit sphere can be determined by a pair of parameters \(\left( {\begin{array}{*{20}c} \theta & \varphi \\ \end{array} } \right)\), which we use to denote the direction of the unit vector from the origin to the point. Assuming that the incident direction of light is \(\left( {\begin{array}{*{20}c} {\theta_{i} } & {\varphi_{i} } \\ \end{array} } \right)\), the oscillate direction of the electric field of the incident light is \(\left( {\begin{array}{*{20}c} {\theta_{Ei} } & {\varphi_{Ei} } \\ \end{array} } \right)\), and the direction of the normal at the point of incidence is \(\left( {\begin{array}{*{20}c} {\theta_{F} } & {\varphi_{F} } \\ \end{array} } \right)\). The refractive index of the lens is \(n_{o}\), the refractive index of the membrane is \(n_{if}\), the incident angle of polarized light is \(\theta_{in}\), the refraction angle in the membrane and in the lens are \(\theta_{if}\) and \(\theta_{o}\) respectively.

According to the Snell's formula,

$$ \sin \theta_{in} = n_{o} \cdot \sin \theta_{o} = n_{if} \sin \theta_{if} $$
(2)

Here the refractive index of air is assumed to be 1, and the thickness of the film layer is neglected. The incident angle of light at any point on the lens surface can be obtained from the direction vector of the incident light and the lens surface normal

$$ \theta_{in} = \arccos \left( {\left| {\sin \theta_{F} \cos \varphi_{F} \cdot \sin \theta_{i} \cos \varphi_{i} + \sin \theta_{F} \sin \varphi_{F} \cdot \sin \theta_{i} \sin \varphi_{i} { + }\cos \theta_{F} \cos \theta_{i} } \right|} \right) $$
(3)

Based on the co-planarity of incident light, refracted light and surface normal, the propagation direction of the refracted light in the membrane layer and in the lens can be obtained,

$$ \frac{{\sin \theta_{o} }}{{\sin \theta_{df} }}\left[ {\begin{array}{*{20}c} {\sin \theta_{i} \cos \varphi_{i} } & {\sin \theta_{i} \sin \varphi_{i} } & {\cos \theta_{i} } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} {\sin \theta_{F} \cos \varphi_{F} } & {\sin \theta_{F} \sin \varphi_{F} } & {\cos \theta_{F} } \\ \end{array} } \right] $$
(4)
$$ \frac{{\sin \theta_{o} }}{{\sin \theta_{dg} }}\left[ {\begin{array}{*{20}c} {\sin \theta_{i} \cos \varphi_{i} } & {\sin \theta_{i} \sin \varphi_{i} } & {\cos \theta_{i} } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} {\sin \theta_{F} \cos \varphi_{F} } & {\sin \theta_{F} \sin \varphi_{F} } & {\cos \theta_{F} } \\ \end{array} } \right] $$
(5)

where \(\theta_{df} = \theta_{in} - \theta_{if}\) is the difference between the incident angle and the refraction angle in the film layer, and \(\theta_{dg} = \theta_{in} - \theta_{o}\) is the difference between the incident angle and the refraction angle in the lens. It is should be noted that the above two direction vectors should be mode normalized when used for calculating.

2.2 Calculation of the Orthogonal Polarization Component

Assuming that the polarized light is incident to the optical system from air, and the orthogonal components of the light in the lens film layer can be obtained from Fresnel's theorem,

$$ \begin{array}{*{20}l} {E_{fs} = \frac{{2\cos \theta_{in} \sin \theta_{if} }}{{\sin \left( {\theta_{in} + \theta_{if} } \right)}}E_{s} } \hfill \\ {E_{fp} = \frac{{2\cos \theta_{in} \sin \theta_{if} }}{{\sin \left( {\theta_{in} + \theta_{if} } \right)\cos \left( {\theta_{in} - \theta_{if} } \right)}}E_{p} } \hfill \\ \end{array} $$
(6)

where \(E_{{\text{s}}}\) and \(E_{p}\) are the s-component and p-component of the incident light, respectively. The oscillating direction of the s-component is perpendicular to the incident plane, but the oscillating direction of the p-component is in the incident plane. The change of the signal propagation direction leads to the change of the oscillating direction of the light polarization component. The oscillating direction of the \(E_{fs}\) in the film layer is

$$ norm\left( {\left[ {\begin{array}{*{20}c} {\sin \theta_{i} \sin \varphi_{i} \cos \theta_{F} - \sin \theta_{F} \sin \varphi_{F} \cos \theta_{i} } \\ {\sin \theta_{F} \cos \varphi_{F} \cos \theta_{i} - \sin \theta_{i} \cos \varphi_{i} \cos \theta_{F} } \\ {\sin \theta_{i} \cos \varphi_{i} \sin \theta_{F} \sin \varphi_{F} - \sin \theta_{F} \cos \varphi_{F} \sin \theta_{i} \sin \varphi_{i} } \\ \end{array} } \right]} \right) $$
(7)

where \(norm\left( \cdot \right)\) denotes the mode normalization of the vector. The oscillating direction of \(E_{fp}\) is

$$ norm\left( {\left[ {\begin{array}{*{20}c} {\sin \theta_{F} \cos \varphi_{F} + \frac{{\sin \theta_{if} }}{{\sin \theta_{df} }}\sin \theta_{i} \cos \varphi_{i} } \\ {\sin \theta_{F} \sin \varphi_{F} + \frac{{\sin \theta_{if} }}{{\sin \theta_{df} }}\sin \theta_{i} \sin \varphi_{i} } \\ {\cos \theta_{F} + \frac{{\sin \theta_{if} }}{{\sin \theta_{df} }}\cos \theta_{i} } \\ \end{array} } \right]} \right) $$
(8)

Similarly, the polarized components refracted through the membrane to the lens are

$$ \begin{array}{*{20}l} {E_{os} = \frac{{2\cos \theta_{if} \sin \theta_{il} }}{{\sin \left( {\theta_{if} + \theta_{il} } \right)}}E_{fs} } \hfill \\ {E_{op} = \frac{{2\cos \theta_{if} \sin \theta_{il} }}{{\sin \left( {\theta_{if} + \theta_{il} } \right)\cos \left( {\theta_{if} - \theta_{il} } \right)}}E_{fp} } \hfill \\ \end{array} $$
(9)

where \(\theta_{if}\) and \(\theta_{il}\) are the incident angle and refraction angle when light refracts from film into lens, \(\theta_{il} = \theta_{o}\). The oscillating direction of the \(E_{os}\) component is always perpendicular to the incident plane, so it is the same as the oscillating direction of \(E_{fs}\). The oscillating direction of the \(E_{op}\) component is

$$ norm\left( {\left[ {\begin{array}{*{20}c} {\sin \theta_{F} \cos \varphi_{F} + \frac{{\sin \theta_{o} }}{{\sin \theta_{dl} }}\sin \theta_{i} \cos \varphi_{i} } \\ {\sin \theta_{F} \sin \varphi_{F} + \frac{{\sin \theta_{o} }}{{\sin \theta_{dl} }}\sin \theta_{i} \sin \varphi_{i} } \\ {\cos \theta_{F} + \frac{{\sin \theta_{o} }}{{\sin \theta_{dl} }}\cos \theta_{i} } \\ \end{array} } \right]} \right) $$
(10)

So far the propagation direction, oscillating direction and amplitude of the two orthogonally polarized components in the membrane layer and in the lens are obtained, and the light transmission inside the lens is analyzed hereinafter.

In the same way, a coordinate system \(\left( {O^{\prime}:X^{\prime},Y^{\prime},Z^{\prime}} \right)\) is established taking the back surface of the lens as a reference, and the relationship \(\left( {O^{\prime}:X^{\prime},Y^{\prime},Z^{\prime}} \right)\) and \(\left( {{\text{O}}:X,Y,Z} \right)\) is shown in Fig. 2. Assume that the curvature radius of the front and back lens surfaces are \(r_{1}\) and \(r_{2}\), respectively, and the center thickness of the lens is \(d\). In \(\left( {O^{\prime}:X^{\prime},Y^{\prime},Z^{\prime}} \right)\), the coordinates of the point on the front surface where normal direction is \(\left( {\begin{array}{*{20}c} {\theta_{F} } & {\varphi_{F} } \\ \end{array} } \right)\) are \(\left( {\begin{array}{*{20}c} {r_{1} \sin \theta_{F} \cos \varphi_{F} } & {r_{1} \sin \theta_{F} \sin \varphi_{F} } & {r_{1} \cos \theta_{F} } \\ \end{array} {\text{ + d}} - r_{1} - r_{2} } \right)\).

Fig. 2.
figure 2

Relationship between the two coordinate systems

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Assume that the coordinate of the intersection point of the light and the lens rear surface is \(\left( {\begin{array}{*{20}c} {r_{2} \sin \theta_{Fb} \cos \varphi_{Fb} } & {r_{2} \sin \theta_{Fb} \sin \varphi_{Fb} } & {r_{2} \cos \theta_{Fb} } \\ \end{array} } \right)\), then the following formula is established

$$ \frac{{r_{2} \sin \theta_{Fb} \cos \varphi_{Fb} - r_{1} \sin \theta_{F} \cos \varphi_{F} }}{{r_{2} \sin \theta_{Fb} \sin \varphi_{Fb} - r_{1} \sin \theta_{F} \sin \varphi_{F} }} = \frac{{{{\sin \theta_{o} } \mathord{\left/ {\vphantom {{\sin \theta_{o} } {\sin \theta_{dg} }}} \right. \kern-0pt} {\sin \theta_{dg} }}\sin \theta_{i} \cos \varphi_{i} - \sin \theta_{F} \cos \varphi_{F} }}{{{{\sin \theta_{o} } \mathord{\left/ {\vphantom {{\sin \theta_{o} } {\sin \theta_{dg} }}} \right. \kern-0pt} {\sin \theta_{dg} }}\sin \theta_{i} \sin \varphi_{i} - \sin \theta_{F} \sin \varphi_{F} }} $$
(11)
$$ \frac{{r_{1} \cos \theta_{F} - r_{2} - r_{2} \cos \theta_{Fb} }}{{r_{1} \sin \theta_{F} \cos \varphi_{F} - r_{2} \sin \theta_{Fb} \sin \varphi_{Fb} }} = \frac{{{{\sin \theta_{o} } \mathord{\left/ {\vphantom {{\sin \theta_{o} } {\sin \theta_{dg} }}} \right. \kern-0pt} {\sin \theta_{dg} }}\cos \theta_{i} - \cos \theta_{F} }}{{{{\sin \theta_{o} } \mathord{\left/ {\vphantom {{\sin \theta_{o} } {\sin \theta_{dg} }}} \right. \kern-0pt} {\sin \theta_{dg} }}\sin \theta_{i} \cos \varphi_{i} - \sin \theta_{F} \cos \varphi_{F} }} $$
(12)

Thus, the light incident angle and incident point on the lens rear surface can be obtained, and the polarized components of the exiting light from the rear surface can also be obtained. With the same process, we can gradually obtain the effect of each optical system lens surface on the incident light, and finally obtain the polarization aberration of the optical system.

Linearly polarized light and circularly polarized light can be regarded as special elliptically polarized light, so elliptically polarized light is used to represent the polarization states of the light, which can be expressed in terms of azimuth angle \(\alpha\) and ellipticity angle \(\beta\). Assuming that the two polarization components of the optical signal are

$$ \begin{gathered} E_{x} = A_{x} e^{i\delta x} \hfill \\ E_{y} = A_{y} e^{i\delta y} \hfill \\ \end{gathered} $$
(13)

From the two components, the maximum values of the X and Y components of trajectory ellipse can be obtained, and the azimuth angle and ellipticity angle of the trajectory ellipse can be expressed as,

$$ \begin{gathered} \tan \left( {2\alpha } \right) = \frac{{2A_{x} A_{y} }}{{A_{x}^{2} - A_{y}^{2} }}\cos \left( {\delta_{y} - \delta_{x} } \right) \hfill \\ \sin \left( {2\beta } \right) = \frac{{2A_{x} A_{y} }}{{A_{x}^{2} + A_{y}^{2} }}\sin \left( {\delta_{y} - \delta_{x} } \right) \hfill \\ \end{gathered} $$
(14)

The ellipticity angle is 0 for linearly polarized light, then the azimuthal angle characterizes the angle between the oscillating direction of the linearly polarized light and the X-axis.

3 Results

In order to verify the validity of the polarization characteristic analyzing method, the polarization characteristics of a collimation and beam reduction optical system containing two lenses in the 1550 nm band are analyzed. And the feasibility of improving polarization maintenance capacity by optimizing the parameters of the optical system is discussed. The main parameters of the optical system are shown in Table 1.

Table 1. Parameters of optical system
Full size table

Four types of linearly polarized light are commonly used in the quantum key distribution systems, namely, H(\(\beta = 0,\alpha = 0\)), V(\(\beta = 0,\alpha = 90\)), + (\(\beta = 0,\alpha = 45\)）, −(\(\beta = 0,\alpha = 135\)）.The distribution of polarization states of the exiting signals is shown in Fig. 3, and Silicon dioxide is used as the coating film on the front and back surfaces of both lenses.

Fig. 3.
figure 3

Polarization state distribution of the exiting signal, (left to right H, +, V, −).

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From the results in Fig. 3, it can be seen that the four ideal linearly polarized light basically maintains its original polarization characteristics after exiting from the optical system, but the optical system introduces a certain amount of polarization aberration to the polarized light. In addition to a small data error introduced by the data truncation in the calculation process, the polarization aberration is mainly due to the fact that the polarized signal is divided into s-component and p- component at the surface of each optical lens and membrane layer, and propagates according to Fresnel's law. Since the transmission and reflection coefficients of the two orthogonal components do not coincide, leading to changes of the polarization states of the signal.

Different from imaging optical systems, communication systems are only concerned with the overall average polarization maintaining level of the received signal, i.e. the polarization state of the signal coupled into the optical fiber. In order to analyze the polarization maintaining characteristics of the optical system more intuitively, the polarization maintaining characteristics of the system with different film layers (Au film, Ag film, Al film, Sio2 film and NIR film) are studied. The polarization purity and amplitude of the exiting light are given in Fig. 4, respectively. In this paper, the polarization purity of the signal refers to the proportion of the exiting signal projected in the polarization direction of the incident signal, and the polarization amplitude is the amplitude of exiting signal projected in the polarization direction of the incident signal.

Fig. 4.
figure 4

Polarization purity(left) and amplitude(right) of the exiting signal

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In Fig. 4, the horizontal axis is azimuth angle of the incident polarized light. The results in Fig. 4 shows that the polarization purity of the exiting signal presents an approximate trigonometric trend with azimuthal angle when linearly polarized light is incident at different azimuthal angles, and the peak appears near 0° and 90°. Since the optical system analyzed in this paper does not contain anisotropic materials, it should be rotationally symmetric. The results in Fig. 4 are mainly due to the different spectroscopic and refractive processes of the signals on the surface of the lens in different coordinate systems, thus leading to different synthesis results of the polarization components. In order to avoid the influence of the calculation process, the influence of coating films on the polarization purity and amplitude is analyzed.

From the results in Fig. 4, it can be seen that the change of polarization purity and amplitude of the exiting signal is consistent for different membrane layers. The polarization maintenance of optical system with Al, Sio2 and NIR film is almost same, but is better than the system with Ag and Au film. For polarization amplitude of exiting signal, the optical system with NIR film gives the best result, and the worst two candidates are systems with Au and Ag film. Considering the polarization purity and amplitude characteristics of the exiting signal, the NIR film is recommended. In the following sections, the polarization characteristic of optical system with NIR film is analyzed, and the incident light is X linearly polarized light.

Fig. 5.
figure 5

Polarization purity and amplitude with different incident angles

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Figure 5 gives the polarization purity and magnitude of the exiting signal when the incident angle of X ideal linearly polarized light changes. Considering that communication systems tend to have a small field of view, the signal incident angle used in this paper is ±1°. As can be seen from the results in Fig. 5, the polarization purity and amplitude of the exiting signal gradually become smaller with the increase of the incident angle. Besides, a bigger incident angle has a more significant impact on the polarization purity obviously.

For improving the polarization maintaining performance of the system, the polarization maintaining performance of the system optimized by changing the distance between the two lenses in this paper, but sacrificing a certain beam collimation effect. The results are shown in Fig. 6, where the horizontal coordinate is the distance adjustment factor, which is defined as the ratio of the distance change to the initial distance, with a negative value for distance shortening and a positive value for distance increasing. The results in Fig. 6 show that the polarization purity of the system increases when the distance between the two lenses increases, while the change trend of polarization signal amplitude is opposite. And an important distance value is observed, which separates polarization maintaining and amplitude into two parts with big gradient and small gradient. Naturally, the distance is defined as the optimal distance in this paper. It is found that beam divergence angle of more than 90% of the output light at the optimal distance can be controlled within 1°.

Fig. 6.
figure 6

Polarization characteristics under different lens distances

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4 Conclusions

In this paper, a theoretical model for analyzing the polarization characteristics of optical systems is established based on vector optics theory. In order to verify the effectiveness of the model, a collimation and beam reduction optical system containing two lenses is studied, and the influence of lens film, polarization states of incident light and field of view on polarization characteristics is analyzed. Also, the polarization characteristics of the system are optimized by adjusting the distance between the two lenses, and the polarization purity and amplitude transmittance of exiting signal are 0.98 and 0.95 respectively. It should be noted that the method proposed in this paper is still effective for practical optical systems with phase errors.