As well as that have been discussed in last section, the features of the vector singularities have been experimentally observed in laser modes with the interrelated behavior of spatial structures and polarization states [Gil93, VKMR01, CLH06, LCH07, Erdo92, PTMA97, CHLL03b]. However, so far all experimental demonstrations were related to the regular lasing modes of integrable optical cavities;
no experiments have demonstrated explicitly the entanglement of polarization and spatial structures in chaotic laser resonators. In this section we use the VCSEL that is similar to that used in Sec. 3.4 to generate the 2D chaotic vector fields.
Figures 5.2-1(a) and (b) show the polarization-resolved near-field patterns with operating temperature of T=265 K, the threshold current of I=26.9 mA, and polarizations in 0° and 90° directions, respectively. The orthogonally polarized modes clearly exhibit to have remarkably distinct chaotic patterns. The measurement of the optical spectrum indicates that the whole experimental wave is phase synchronized to a single frequency at 806.45 nm. As a consequence, the orthogonally polarized components can mutually interfere to lead to a greatly different pattern in the polarization resolved near-field image, as shown in Fig. 5.2-1 (c) for 45°
polarization and Fig.5.2-1 (d) for -45° polarization. Explicitly, the entanglement of spatial structures and polarization states lead to the formation of an optical vector field.
We investigated the dependence of the 2D chaotic vector field on the operating parameters, and it turns out that the experimental vector field remains unchanged for 262.5 K < T < 267.5 K and for 26.9 mA < I < 27.6 mA. The width of these ranges indicates that generation of 2D chaotic vector fields is a robust phenomenon. To our best knowledge, the present result proffers the first experimental realization of 2D chaotic vector fields in a microcavity laser.
Since it is not feasible to measure polarization vector fields in a straightforward way, the reconstruction of the orthogonally polarized wave functions is practically
useful for analyzing the property of vector singularities. We use the same eigenfunctions expansion method as that described in Sec 3.4 to reconstruct the polarization resolved patterns. Figures 5.2-2(a) and 5.2-2(b) depict the patterns of
| ) , (
|p xi yj for two orthogonally polarized modes shown in Fig. 5.2-1(a) and 5.2-1(b), respectively. Figures 5.2-3(a) and 5.2-3(b) show the intensity plots of obtained from Eq. 3.4-5 for the experimental polarization-resolved modes at 0° and 90°, respectively. Figures 5.2-4(a) and (b) depict the wave patterns of the analytical wave functions corresponding to the experimental polarization-resolved modes at 0° and 90°, respectively. It can be clearly seen that the experimental polarization-resolved patterns are well-reconstructed with the analytical wave functions.
1,2
|Cn n |
Let and denote the polarization-resolved wave functions at 0° and 90°, respectively. In terms of and , the vector field distribution for the experimental pattern is given by
0( , )
With the vector field the polarization-resolved wave functions at 45° and -45° are given by
Figures 5.2-4(c) and 5.2-4(d) depict the numerical results for the intensity patterns of and
2 45( , )|
| x y |45( , ) |x y 2, respectively. The good agreement between the numerical and experimental patterns evidences the accuracy of the reconstructed wave function in representing the observed vector field.
To further validate the experimental observation to be a chaotic vector field, we use the reconstructed wave functions to calculate the amplitude and intensity distributions. For the chaotic wave function of Berry’s conjecture, the amplitude distribution is a Gaussian function (Eq. 2.3-2) and the intensity distribution is shown to be a Porter-Thomas distribution (Eq. 2.3-3). Figures 5.2-5 (a)-(b) show the amplitude distributions of the reconstructed wave functions with polarizations in ,
, , and , respectively. In addition, 5.2-6 (a)-(b) illustrate and intensity distributions corresponding to Fig. 5.2-5 (a)-(b), respectively. All amplitude and intensity distributions of the polarization-resolved wave functions are found to be fairly good agreement with the theoretical distributions.
0 90 45 45
With the reconstructed vector field we can do the similar process as what has been done in last section to analyze the properties of vector singularities in the chaotic case. The angle function is again employed to describe the vector singularities:
0 90
( , ) ( , ), ( , )
c x y angle E x y E x y
. (5.2.4)
Figure 5.2-7(a) depicts the numerical pattern of the angle function c( , )x y for the experimental vector field. Different from the lattice structure of regular vector field, the chaotic polarization vector field is clearly seen to reveal a highly sophisticated interlace pattern. Fig. 5.2-7(b) is the zoom-in views of the central region with edge length equal to . Although the singularities are randomly distributed, the sign rule of the nearest neighbor singularities is still obeyed. The polarization vectors of the chaotic field as show in Fig. 5.3-7(c) become very intricate. However, one can still find that the singularities with
/10 a
topological charge are all vortices. 1
As mentioned by Freund [Freu95], the phase of a chaotic wave with real and imaginary parts to be and is identical to the orientation phase shown in Fig. 5.3-7(a). In fact such a complex chaotic wave function does exist.
The vector field expressed as eq. (5.2.1) can be decomposed into a linear combination of orthogonal circularly-polarized helical modes
0( , )
E x y E90( , )x y
( , ) R( , ) ( , ) ˆR L E x y E x y a E x y a
ˆL (5.2.5)
, where
0 90
( , ) [ ( , ( , )] / 2
ER x y E x yiE x y (5.2.6) , and
0 90
( , ) [ ( , ( , )] / 2
E x yL E x y iE x y . (5.2.7) ˆR (ˆx ˆy) /
a a ia 2 and ˆaL (aˆxiaˆy) / 2 are the helical basis unit vectors for the right- and left-handed circular polarizations, respectively. Hence, the phase function of E x yR( , ) is completely the same as c( , )x y . In addition to singularities, it is also meaningful to analyze the critical points in c( , )x y . Based on the thorough numerical analysis, it is found that all saddle points are manifestly found to be open saddles with no joined arms. In other words, no phase extrema are observed in the experimentally generated random phase filed. This result is consistent with the theoretical analysis that the phase extrema are really rare because there is little room left in the phase field to accommodate them [Freud95]. Since the circular polarization of light corresponds to the spin of quantum wave, the analyses of
( , )
c x y
can provide important information for chaotic quantum-billiard systems (such as ballistic quantum dots) with consideration on electronic spin.
Fig. 5.2-1. Experimental polarization-resolved near-field patterns observed at the operating temperature of T=265 K with polarization in (a) 0°(perpendicular) (b) 90° (horizontal) (c)45° (d)135°.
(c) (d) (b) (a)
(a) (b)
Fig. 5.2-2. (a) and (b) Intensity plots of the positive wave functions
| ) , (
| p xi yj for experimental results shown in Figs. 5.2-1(a) and 5.2-1(b), respectively.
(a) (b)
n1 n1
n2 n2
Fig. 5.2-3. (a) and (b) Distribution of the coefficients obtained by Eq.
(3.4.6) for experimental results shown in Figs. 5.2-1(a) and (b), respectively.
|Cm n, |
Fig. 5.2-4. (a)-(d): Reconstructed patterns with the eigenfunction expansion method for experimental results shown in Fig. 5.2-1(a)-(d), respectively.
(c) (d) (b) (a)
(a)
Fig. 5.2-5. Amplitude distributions of the polarization-resolved wave functions (blue step lines) for experimental results shown in Fig. 5.2-1(a)-(d), respectively. Red lines: Gaussian distributions (Eq. (2.3.2)).
-5 -4 -3 -2 -1 0 1 2 3 4 5
Fig. 5.2-6. Intensity distributions of the polarization-resolved wave functions (blue step lines) for experimental results shown in Fig. 5.2-1(a)-(d), respectively. Red lines: Porter-Thomas distributions (Eq. (2.3.3)).
(a)
(a)
(b) (c)
2π
0 a
a/10
Fig. 5.3-7. (a) The contour plot of the angle function C( , )x y . (b)-(c) Zoom-in view of the two small regions with the hollow circles on the singularities.