3.1

Polarization wave and Poincare sphere

Polarization is an abstract concept for many student. Student know the electric field has the vector form [3], as shown in (1) eq(1).

Electric field E_x and E_y and their amplitude A_x and A_y. There are phase retardation between E_x and E_y and the retardation value is δ. The value of E_x, E_y and δ can categorize these waves into linear polarization, circular polarization or elliptical polarization. The total electric wave is the combination of x and y direction field, and it is very difficult to draw a circular polarization in a blackboard. Some animation is helpful for construction this geometric relationship between the electric field in x, y and the combination.

Another important concept of polarization is Poincare sphere. Different polarization state can also describe on the point of the sphere. The sphere is in Cartesian coordinates with three axes s₁, s₂, and s₃. The positive s₁ axis direction is horizontal polarization, while vertical polarization is in the opposite direction of s₁ axis. The positive s₂ axis direction is linear polarization in 45 degree, while the opposite direction is linear polarization in 135 degree. The positive s₃ axis direction and negative s₃ axis direction are north pole and south pole, which are left hand and right hand circular polarization. In eq.(2), the inverse tangent of the amplitude ratio A_y over A_x makes ψ.

The azimuthal angle is 2ψ which describes the latitude where the point lies on the sphere and the retardation δ describe the angle between the point and S₃ axis. With 2ψ and retardation δ, user can point out the polarizaiton state in Poincare sphere as eq.(3). Without computer visualization, it is not easy for student to understand.

Linearly polarized wave is the electric field oscillate in a constant plane (in xy-plane). The phase retardation between x and y polarized light is δ = δ_y – δ_x = 0 or π.

Figure 1

(a) linear polarized wave (b) linear polarization state on Poincare sphere (c) circular polarized wave (d) circular polarized wave on Poincare sphere (e) elliptical polarized wave (f) elliptical polarization state on Poincare sphere

Circularly polarized wave is the electric field vector propagate with uniform rotation electric field in the xy-plane. This occurs when A_x = A_y and . The relationship between E_x and E_y is a circle, as shown in eq.(4).

Through derivation calculations, students can understand that this is a circle equation. Circular polarization wave is hard to imagine how it works. I tried to use interactive software in teaching. You can use the mouse to change the angle of view angle. For example, observe the wave on xy plane from z direction. It is much easier to understand the left-handed and right-handed circular polarizations. The polarization state on Poincare sphere at the right side of program changes according to the amplitude and relative retardation, as shown in Figure 2. Moreover, it is easier to observe the polarization state change from circular polarization to the linear polarization. Also, the electric field shown in eq.(1) can be written as an elliptic function, such as eq.(5). Through the continuous changing processor parameter, we can let students operate various polarization states.

Figure 2

Polarization Wave and Poincare Sphere

Therefore, the polarization state is not only a mathematical form, but also an object, which can be manipulate. Student use the simulator in the classroom, as shown in Figure 3(a). In the classroom, you can also do simple experiment. Many students have mobile phones. Nowadays, mobile phone screen is polarized light. Teachers can prepare polarized sunglasses or circular polarizer lenses for cameras to show this phenomenon. The brightness of the screen laptop changes as shown in Figure 3(b).

Figure 3

(a) student use laptop to manipulate the polarization state and Poincare sphere (b) student observe the brightness of laptop screen when the sunglass and circular polarizer plate with different rotating angle

How do we generate polarization? Sometimes I ask student this question. We can use a polarizer to absorb specific polarization direction, or we can generate polarization state by reflection. For example, we can use Fresnel equation to describe the light transmitted and reflected from surface. The s-polarization and –polarization states have different reflection coefficient as shown in eq. (6) (7).

Students can observe the reflection, relation coefficient and the Brewster Angle by actually operating the angle of incidence or changing the refractive index of the medium, as shown in Figure 4

Figure 4

Fresnel equation for reflection and transmittance intensity

In daily life, the LCD screen is the application of polarization light. Changing the brightness by change the polarization after pass through liquid crystal. Optical structure of LCD is two orthogonal polarizers, with a transmission axis at 0 and 90 degrees. A liquid crystal act as a wave plate sandwiched between polarizers. By changing the optical axis of the liquid crystal, while the phase retardation Γ = 2π(n_e – n_o)d/λ. The parameters n_e, n_o and λ represent the extraordinary, ordinary refractive index and wavelength, respectively. The transmittance can be explain Jones calculus as shown in eq. (8) [4]. The transmittance intensity is shown in eq. (9).

The students actually change the value of the transmittance axis angle of polarize and analyzer, retardation and optics axis angle of wave plate in simulator as shown in Figure 5. According to Fresnel equation, we explained that most of the ground reflections are s-polarization that is why sunglasses has horizontal absorption axis. Students rotated polarized sunglasses and observed the brightness change of laptop, as shown in Figure 6.

Figure 5

Transmittance simulation of a wave plate between polarizer and analyzer

Figure 6

A simple experiment of transmittance of a polarized light pass through a polarizer-sunglass. The green arrow is the polarization direction of laptop while the sunglass transmission axis is (a) 90 degrees (b) 135 degrees (c) 180 degrees (d) 225 degrees

3.2

Diffraction and laser dots projector System

Diffraction is also a common topic in the photonics. When I teaching this topic, I started from Fourier series, and then introduced the Fourier transform. Finally explained an application of diffraction, the 3D sensing structure light in the mobile phone. Series expansion of the periodic square wave as shown in eq. (10), can be expanded into the sum of many different frequency sine functions. The coefficients of each sinusoidal wave have different coefficients as shown in eq. (11). Superposition of those different frequency sinusoidal waves can reconstruct the original square wave, as shown in Figure 7 (a).

Figure 7

(a) Series expansion of periodic step function (b) The Fourier transform of a rectangular function rect(x/A), and the amplitude in frequency domain is |Sinc(Af)|

Similarly, we can also use any function to analyze the amplitude of each of its frequencies, and plot the frequency of each component against the intensity. It is the Fourier transform of the original function. For example, the rectangular function, whose relationship is shown in eq. (12),

Through the Fourier transform, you can transform the rect function to a sinc function on the frequency domain, as shown in eq.(12). Only a few specific functions have analytic Fourier transforms pairs. For example, rect function and sinc function are a pair of quite important Fourier transform pairs as shown eq.(13). This function is very important for optics, because sinc function is the electric field distribution from the single slit diffraction. The intensity distribution of diffraction is the square of the sinc function. In Figure 7 (b), it shows the original function g(x) and its amplitude |G(f)| in the frequency domain. In this case, it is |G(f)|= |Asinc(Af)|.

In the integral transform, the convolution integral is also very often used. It is defined as the integral of the product of the two functions after one is reversed and shifted. As shown in eq. (14). This is a very dynamic concept. It has not been easy for students to understand such concepts. Teachers often explain this process to students by constantly changing τ. I think this is also a mathematical concept that is quite suitable for introduction with interactive software. Through the change of τ parameter, h(t-τ) will be in different positions, we can observe this change phenomenon in time. Such as Figure 8 (a).

Figure 8

(a) convolution of two rectangular function (b)

The Fourier transform can change the complex convolution integral into a simple multiplication, F{g * h] = F{g}F{h}. The function I want to use here is Comb function III_T(t). The definition of Comb function is shown in eq.(15), which is a pulse train of delta function.

The convolution of g(x) and III_T (t) will be, g(x) is copied to each pulse train. As shown in Figure 8 (b), the mathematical formula can be written as eq.(16).

The Fresnel diffraction formula, such as eq. [5], can also be written in the form of convolution. The electric field in the initial position convolute with the transfer function . Therefore, he Fresnel diffraction formula can be rewritten into the form of eq. (18). Another method is to take the x, y terms out of eq.(17) and make eq.(17) become form of a two-dimensional Fourier transform, as shown in eq.(19). The mathematical formula is concise and symmetry, but for students still feel obscure about the distribution. Therefore, students can use software and change the aperture size and then observed the diffracted image directly, as shown in Figure 9 (a) and (b). To deepen the students’ understanding of diffraction optics, it is relatively helpful.

Figure 9

(a) Diffraction of square aperture in 2D intensity plot (b) Diffraction of circular aperture in 3D intensity plot

After some lecture and simulator operation, we can use the laser pen with DOE to demonstrate the relationship between the diffraction and the convolution. A simple demonstration is shown in Figure 12 (a). Finally, I will briefly introduce how to generate structured light and its application 3D Sensing on the mobile phone, such as iPhone X Face ID. The laser dots projector can be divided into three parts [6]: patterned-VCSEL, collimated lens and impulse DOE, As shown in Figure 10.

Figure 10

Patterned-VCSEL structured light system

The patterned-VCSEL array in dot projector is shown in Figure 11 (a). The diffraction pattern of the DOE is fan-out pattern similar to 2D Comb function as shown in Figure 11 (b). The diffraction pattern of the VCSEL array pass through the DOE is like copy the VCSEL pattern to every pulse on the pulse train as shown in Figure 11 (c). For large angle diffraction, we can imagine that the pattern of Figure 11 (c) is imaged on a spherical surface. This, when the spherical image is turned to the plane image, some pincushion distortion occurred, and the final image is as shown in Figure 11 (d). This is how laser dots projector works in consumer electronics. Figure 12 (b) is the laser dot projector pattern of OPPO Find X on the market.

Figure 11

(a) patterned VCSEL array (b) impulse of DOE (c) image of spherical surface s₁ (d) image of planar surface s₂.

Figure 12

(a) Diffraction image of laser projector from daul DOE (b) Diffraction image from pattern VCSEL and fan-out DOE in OPPO mobile phone Find X.