Interference pattern

In general, when two or more scalar waves with the same polarization interfere in space, complete destructive and constructive interference occurs on lines called nodal lines, and on points called phase singularity, wave dislocation, or optical vortices. The standard Mach–Zehnder interferometer can merge two beams at the output of the interferometer. If one beam be tilted from the optical axis, irradiance of interference pattern can be obtained.

When using circular approach for the synthesis of cylindrical vector beams, the opposite topological charge was adopted in the interferometer. Consideration of tilting one beam leads to the interference pattern by passing through a polarizer. From difference of interference pattern, we can easily distinguish between the radially-polarized beams and azimuthally-polarized beams and check the precision of this synthesis.

3.2.1 Analytical description of interference

The adopted method of synthesizing the inhomogeneous beams is circular polarization approach, so the two coherent beams of interference are with particular circular polarization. One of the incident beams, i.e. E , is a wave propagating from a direction in the x-z plane making a small angle with respect to the z axis. The mathematical properties of two coherent beams are described as following:

E , , E e e y (3-4)

E , , E e e y (3-5)

E e e e cos sin ̂ y .

When the two waves are e d, the field E The total field could be decomposed into three components depending on the state of polarization. The intensity profile as seen through a polarizer in the x-transmission

d as ollowing:

S bstit ting cos into (3-6) due to consistency of the coordinate, we obtain

I E 1 cos 2 cos cos cos sin cos 1

which can be abbreviated into

I 1 cos cos sin 2cos cos sin .

(3-7) In another case of using a polarizer with y transmission axis, the following formula was derived :

I 2sin cos sin . (3-8)

When combining the Eqs. (3-7) and (3-8), the total field irradiance could be observed by the CCD at exit of interferometer.

I I_x I cos^{2 1 2} In the experiment of synthesizing the cylindrical vector beams with titled angle, the irradiance of emergent beam is Ix or Iy depending on the transmission axis of a polarizer. The experiment result will be shown in the Chapter 3 where it is well match with theoretical derivation.

3.2.1 Simulation of interference pattern

In this section we will discuss three cases which are distinguished with different topological charge and tiled angle. For the case 1, m1=m2=1, it can be linked to our synthesis of cylindrical vector beams and the interference pattern could be obtained by experiment which is introduce in the chapter 3. From Eq. (3-9) the total irradiance of interference can be can be abbreviated into

I I_x I cos^{2 1}

2 cos sin sin cos sin . (3-10)

We can use mathematical software such as matlab to simulate intensity distribution of the emergent beam by using the formula derived in previous section. Fig. 3-2 (b - d) shows the interference pattern. If the tiled angle does not

equal to zero, the wires of the interference pattern become broader due to the decreasing tiled angle. The pattern of the first row is in the condition of x-transmission axis of polarizer. Similarly, the second row is in the condition of y-transmission axis of polarizer. We can find that they are spatially complementary between the first and second rows. From the relationship between pattern and tilted angle it can be applied to judge the precision of interferometer. As we decrease the titled angle carefully, the density of pipe wires decrease simultaneously. Till the tilted angle approaches zero, the fan-shaped pattern is observed as shown as Fig. 3-2 (a).

(a) (b) (c) (d) Fig. 3-2 Simulated interference pattern of 1 in different angle (a) 0°,(b) 0.02°, (c) 0.05° ,(d) 0.1°. The first row was represented in the x-transmission axis of polarizer. The second row was represented in the y-transmission axis of polarizer.

Note that m=m1 and m=-m2. The density of pipe wires decrease when the titled angle is decreased.

For the case 2, m1=m₂, we consider that the beam with the same unsigned topological charge was interfered when the tilted angle is 0.02°. The formula which describes the intensity of emergent beam from the polarizer with x-transmission axis is

I cos cos sin . (3-11)

Each of the interference patterns shown as Fig. 3-3 is with the same unsigned topological charge. The extra folk of interference pattern can be calculated by compare the wires of the upper plane of pattern with ones of the bottom plane. For instance, the mtotal equal to 2 in the Fig. 3-3(a). In general, mtotal can be expressed as

. (3-12) According to this equation, we can know how many topological charges the emergent beam has by calculate the extra forks. Moreover, it is observed that all patterns have good symmetry in spatial 2D plane. This phenomenon of symmetry will be discussed in the case 3 in comparison with the different topological charge. Another observed phenomenon is that the interference of the odd topological charge leads to the dark wire of centerline in y-axis. In contrast, the interference of the even topological charge leads to the bright wire of midline in y-axis.

Fig. 3-3 Simulated interference pattern of spiral phase with (a) 1, (b) 2, (c) 3, (d) 4. The tilted angle is 0.02° and the transmission axis of the polarizer is x-axis.

For the case 3, , the beam with the different unsigned topological charge was interfered when the tilted angle is 0.02°. The formula which describes the intensity of the emergent beam from the polarizer with x transmission axis is-

I cos cos sin .

The interference pattern with different topological charge was shown as Fig. 3-4. The well symmetry of interference pattern in x axis is obtained due to the same unsigned topological charge, i.e. m1=m₂. Interference of different unsigned topological charge leads to spatial asymmetry in x axis, as showed as Fig. 3-4(b)(d)(f)(h). The general rule to describe the symmetry of pattern is to check parity of the sum of topological charge. If the sum of m1 and m2 is even, the irradiance of good symmetry is observed.

Differently, the odd number of summing m1 and m2 result in the spatial asymmetry in the bottom plane. Note that the general rule to judge the extra wires of pattern is the same with Eq. (3-12) derived in the case 2.

During the synthesis procedures as creating cylindrical vector beam, the precision of optical path of our setup can be checked by these interference patterns.

Only the perfect supposition of two beams in the interferometer could lead to the perfect cylindrical vector beams. When one optical axis coincide with the other in the interferometer, we observe the fan shape pattern in the exit pupil plane due to the perfect overlap of high precision, as showed in the Fig. 3-5.

Fig. 3-4. Comparison of simulated interference pattern with different topological charge. The tilted angle is 0.02° and the transmission axis of the polarizer is x-axis.

Fig. 3-5. Simulated patterns of superposition of m1 + m2= (a) 2, (b) 3, (c) 4, (d) 5, (e)6 with no tilted angle.