Optical Vortices Generated by VCSEL
In Sec. 4.1 we first review the free time evolution of a sine function with suddenly removal of the 1-D infinite potential well. Next, the problem is extended to the transient dynamics of various types of coherent waves released from square billiard in second section. In 1-D systems the current flow is monotonous since it is linear and can only flow in two direction, x or x axes. However, the 2-D probability current density becomes much complicated because it forms a vector field.
As indicated in Sec. 1.3, 2-D current field has three kinds of vector singularities, sink and source, saddle, and vortex, which correspond to the phase minima and maxima, saddle, and singularity, of the wave function. Moreover, angular momentum, which is an important physical quantity both in classical- [GPS02] and quantum-mechanical [BVD65] sys , will naturally arise due to the 2D current flow.
On the other hand, the analogy between transverse modes emitted from VCSEL and wave functions released from quantum billiards has been established and experimentally verified in Sec. 4.3. However, the analogies between paraxial optics and 2-D quantum system are not only restricted to the correspondence between amplitude distribution and wave function. For a non-stationary state, the probability current is defined by the continuity equation of probability density
tems
( , , ) Im[ *( , , ) ( , , )]
ve is related to the amplitude distribution of electric field
On the other hand, the transverse linear momentum density of a linearly polarized
quasi-TEM wa by
B07] (See Appendix B for a detailed discussion.)
( , , )
ilar behavior as the optical p x y z
. Moreover, the orbital angular momentum (OAM) density of the two systems expressed by
( , , ) ( 0) ( , , ) from VCSELs has never been investigated. In this section our aim is to analyze the linear and angular momentum densities of the light beam emitted from VCSELs by analogously calculating the pr
coherent waves released from quantum billiard.
have to first deal with
ectively, are also in the same mathematical form.
Recent years have been increased attention being given to optical OAM [ABSW92, FAAP08] for its wide applications in atom trapping [KTS+97], optical tweezers [MRS+99], and optical spanner [SADP97]. Furthermore, OAM of light beam can be encoded as qudit and has great potential applications in quantum information [MVWZ01]. However, the OAM carried by the cohe
obability current and angular momentum densities of
j
We in order to obtain . By definition, is written as
ˆ , )ˆ
Consider the eigenstate of square billiard Eq. (4.2.3), we have
1 2,
Hence, the probability current density of a eigenstate released from square billiard an be expressed as
y. In the vector plot, the arrows point to the directions of the flow on that
position and the length of the arrows is proportional to the strength of the flow.
re billiard, the OAM density is given by by the current flow
plots are given aside with unit in
, and 1.0T , respectively. (negative) value of angular m rotation with rota is point
t ome
s to
spatial range fo ndicates a counter
r calculations. The positive -clockwise (clockwise) ntum i
ˆz
a .
anti-symmetric and such that the net OAM computed by
y
is always zero for any . A fu
density depends on the choice of rotation axis, but the net value
We can see that the distribution of OAM
,
r analysis indicates that the distribution of OAM
1 2
does
not. The OAM has such an axis-independent net value was said to be intrinsic [CZDV06]. The OAM can be validated to be intrinsic by veri
OAM densit is defined as follow
fying the relation
|2
. The wave function should be normalized
information [MVWZ01]. Although with
such that ( ) 1
. The OAM spectrum is experimental measurable [GCP+04]
density depends on t (or z for a light beam), the OAM is an invariance of t (or z ) [MTTT02]. Fig. 4.4-3 shows the OAM spectrum of the eigenstate 15,15( , , )
P
and has great potential app the distribution of OAM
x y t
. Since the eigenstate 15,15( , )x y has a / 2 symmetry (i.
e. the wave-function distributions in the four quadrants are identical), the OAM
spectrum . The net OAM can be
alternatively calculated from OAM spectru
only has values as equals to the multiples of four
Obviously, we have 0 for the symmetric OAM spectrum and this result is consistent with the value calculated by Eq. (4.4.9). The OAM spectra of all eigenstates of square billiard are symmetrically distributed and have their peaks centered at . Hence, all eigenstates have their net OAM to be zero. As revealed by Zambrini and Barnett, it is more accurate to say that the OAM with
indicator rotation axis for calculating
OAM density is the dimensionless variance of OAM spectrum [ZB06], which is given by
0
axis-independent net value but axis-dependent density distribution is quasi-intrinsic [ZB06]. The of the relevance of the position of
( ) ( )2 0 our analysis may provide useful information for the chessboard-like patterns emitted from phase-coupled VCSEL arrays or photonic resonator crystals [PKM02].
The probabil ty curr nt of coherent state i e N Mp q, ,, ( , , )x y t can be expressed as
Since the partial differential operator is linear, we
presented in Fig. 4.4-4 (a)-(f), respectively. As expected, the current flux displays high directionality indicated by the motion classical particle. The OAM density distribution of a coherent state is much interesting and can be easily expected from the distribution of current density. The OAM density of superscar can be expressed as
, , , ,
has only negative OAM component. Besides, the peak does not center at 0 but shifts to 62.
1,1,0.6
The net OAM calculated from OAM spectrum has the same value as L35,13( , , )x y t .
It can be seen that there are two small peaks embedded in a big peak. To further understand this phenomenon, we show the OAM spectra of the coherent states
The OAM variance is evaluated to be 255.7.
1,1,
35,13 ( , , )x y t with equals to 0, 0.25 , and 0.5
in Fig. 4.4-7(a’)-(c’),
We find that two opposite segments of the perscar will result in one h the O M spectrum only ha lues as even
Since the PO has two pairs of opposite segments, the OAM spectru of
stationary coherent state becomes a standing wave and has its OAM spectrum
, the two peaks completely overlap and the OAM spectrum only has values as equals to multiples of four due to the / 2 symmetry. Hence, there are actually only two partially overlapped peaks in the OAM spectrum shown in Fig. 4.4-6.
The predominant lasing modes in the broad-area square-shaped oxide-confined VCSEL are the superscar modes tha
standing-wave representation of stationary coherent state t Although
real, it becom mplex oved
t are analogously interpreted by the
, ,
The probability current of CN Mp q, ,, ( , ,x y t) is given by
CN M x y t is a standing wave that is composed by two completely overlapped traveling waves, one (N Mp q, ,, ( , , )x y t ) rotates clockwise and the other (N Mp q, ,, ( ,x y t, )) counter-clockwise. Although the two components begin to split as the coherent state
, , , ( , ,
p q
CN M x y t ased, they still partially overlap in some regions. In these overlapped regions, the currents of the two traveling-wave ponents interfere and destroy each other. As the time tma2/ , the two components start to merge.
ically distributed and has a zero net value.
, , ( , , )
y t However, such a OAM spectrum had a variance
as large as .
The zero net value and large variance of OAM make the
by t have less applications. However, Zou and Mathis recently light beams with differen onent [ZM05]. If such a device
decom ing-wave t into two traveling waves
3718
( , , )y
p
pose the stand
lasing modes interpreted
, , , p q
CN M x
OAM com
proposed a scheme for OAM beam splitter to separate t can be realized, we can employ it to
. The light beam with their lasing mode interpreted by
directions. For the chaotic wave function chaos( , , )
has its OAM as shown in Figs. 4.4-5 and 4.4-6 to be m for
e three regular cases, the probability currents all flow in definite ore convenient
x y t
, the current flux becomes much complicated. We first write down the expression of probability current
( , , ) Im[ ( , , ) ( , , )]
By the principle of superposition, we have
( , ) 0.4T , 0.55T , 0.8T respectively. Unlike the regular current flows,
ectors mly distributed flux. In order to make a more explicit vis tion o
-in views of small regions of j ( , , 0.1 )x y T are shown in Fig. 4.4-12 (b)-(d). Strikingly, several pronounced vortices are induced in the current flux as the chaotic wave function is released.
Such discrete vortices have been widely observed in Bose-Einstein condensate [MAH+99], superfluid [MFDM03] and Type-II superconductor films [MFDM03].
The length of the vectors has been modified by setting j' j/ | j|0.9 to enhanced the vortex structures. As discussed in Sec. 1.3, the vortices in probability current essentially correspond to the phase singularities of the com In order to verify the vortices, we draw the contour plots of the phase
plex scalar field.
( , , )x y t arg[ chaos( , , )]x y t
(4.4.24)
and show them as backgrounds of the vector plots for a convenient comparison. The contour plot is color-coded with red and purple corresponding to 2 and , respectively. The singularities are at the points wh
white and black curves in the contour pl
imaginary parts of wave function, respectively. e ection
. i
singularities more quickly and accurately.
out the singularities with topological charges equal to
0
s nd the ere all colors get together. The ots stand for the nodal lines for real and
By the definition, th This result can help us
inters to f of white and black curves are singularities
The red squares and pink triangles point
and 11 , respectively. It can be easily checked that the clockwise and count -clockwise vortices are
ities. analysis wel demo trates t sign at th
labeled by “1, 2
er
Besides, our
coincident with the positive and negative singular l
ns he rule th e nearest neighbor singularities on any contour of constant phase are required to have opposite signs [Freu95].
The minute feature of vortices and phase singularities in the small regions , and 3” in Fig. 4.4-13(a) of jchaos( , , 0.2 )x y T
are displayed in Fig.
4.4-14(b)-(d), respectively. From the figures, we can find that both the spatial density and size of vortices decreases as the distance from the origin increases. This result arises from the fact that all the currents flow out of originally-confined region in a radial way. Therefore, we suggest that the vortices y interference of the many randomly-oriented currents that is still inside the originally-confined region.
This phenomenon becomes more obvious as t increases. Fig. 4.4-14(b)-(d) depict the vector plots of chaos( , , 0.4 )
are formed b
j x y T
in the small regions marked in Fig. 4.4-14(a).
It can be seen that there is no vortex in the region shown in Fig. 4.4-14(d). As the wave completely leaves the originally-confined region, all currents are radially
flowing and the vortices become trivial.
For the chaotic wave funct calculated by
) ion released from the billiard, the OAM density is
( , , ) ( 0) ( , , )
chaos chaos
l x y t m r r j x y t
. (4.3.21
The density plots of |lchaos( , , ) |x y t
at t0.1T , 0.2T , 0.4T , 0.55T , 0.8T , and 1.5T are illustrated in Fig. 4.4-15 (a)-(f), respectively. We can see that the chaotic wave function has very complicated OAM density. Such a complex OAM density is validated to be intrinsic and has a zero net value. Since the chaotic wave function ( , , )chaos x y t is composed by eigenstates with ficient, the OAM spectrum of chaos( , , )
real expansion coef x y t
is still symmetrically distributed as displayed in Fig. 4.4-16. Due to the intricacy of the OAM spectrum, the OAM variance of
( , , )
chaos x y t
has an extremely large value of 2576. As revealed by the Zambrini and Barnett, this large variance of OAM is probably resulted from the formation of off-axis vortices as those shown in Figs. 4.4-12, 4.4-13, and 4.4-14.
(a)
(c) (f)
(d)
(b) (e)
Fig. 4.4-1. (a)-(f) The vector plot of j15,15( , , )x y t
at t0.1T, 0.2T , 0.3T , 0.4T , 0.5T , and 1.0T , respectively.
Fig. 4.4-2. (a)-(f) show the density plots of l15,15( , , )x y t
Fig. 4.4-3. The OAM spectrum of 15,15( , , )x y t .
( )
P
-50 0 50
0.00 0.03 0.06 0.09 0.12 0.15
( )
P
-50 0 50
0.00 0.03 0.06 0.09 0.12 0.15
(a)
(b)
(d)
(c)
(e)
(f)
Fig. 4.4-4. (a)-(f) The vector plot of J35,131,1,0.6( , , )x y t
at t0.1T, 0.2T , 0.3T , 0.4T , 0.5T , and 1.0T , respectively.
Fig. 4.4-5. (a)-(f) The density plots of L1,1,0.635,13( , , )x y t
Fig. 4.4-6. The OAM spectrum of 1,1,0.635,13( , , )x y t .
-100 -50 0
0.00 0.03 0.06 0.09 0.12 0.15
( )
P
-100 -50 0
0.00 0.03 0.06 0.09 0.12 0.15
( )
P
Fig. 4.4-7. (a)-(c) The intensity patterns of 1,1,35,13 ( , , )x y t with , 0 0.25, and 0.5 , respectively; (a’)-(c’) The OAM spectra of the coherent states shown in (a)-(c), respectively.
-150 -100 -50 0 50 100 150
(a) (d)
(c)
(e)
(f) (b)
Fig. 4.4-8. (a)-(f)The vector plot of Jc 1,1,0.635,13 ( , , )x y t
at t0.1T, 0.2T , 0.3T , 0.4T , 0.7T , and 1.0T , respectively.
(a)
Fig. 4.4-10. The OAM spectrum of C35,131,1,0.6( , , )x y t .
-100 -50 0 50 100
0.00 0.03 0.06 0.09 0.12 0.15
( )
P
-100 -50 0 50 100
0.00 0.03 0.06 0.09 0.12 0.15
( )
P
(a)
(b)
(d)
(c)
(e)
(f)
Fig. 4.4-11. (a)-(f) The vector plot of jchaos( , , )
at t 0.1T
x y t , 0.2T , 0.4T , 0.55T , 0.8T , and 1.55T , respectively.
Fig. 4.4-12. (a) The vector plot of jchaos( , , 0.1 )x y T
. (b)-(d) Zoom-in views of small regions marked by the hollow squares in (a). Backgrounds are the
unctions.
corresponding contour plots of phase f 1
3 2
(a) (b)
(c) (d)
1
3 2 1
3 2
(a) (b)
(c) (d)
Fig. 4.4-13. (a) The vector plot of jchaos( , , 0.2 )x y T
. (b)-(d) Zoom-in vi small regions marked by the hollow squares
ews of in (a). Backgrounds are the corresponding contour plots of phase functions.
1
3 2
(a) (b)
(c) (d)
1
3 2 1
3 2
(a) (b)
(c) (d)
Fig. 4.4-14. (a) The vector plot of jchaos( , , 0.4 )x y T
. (b)-(d) Zoom-in views of squares in (a). B
small regions marked by the hollow ackgrounds are the corresponding contour plots of phase functions.
1
3 2
(a) (b)
(c) (d)
1
3 2 1
3 2
(a) (b)
(c) (d)
(a)
Fig. 4.4-16. The OAM spectrum of chaos( , , )x y t .
-100 -80 -60 -40 -20 0 20 40 60 80 100 0.00
0.03 0.06 0.09 0.12 0.15
( )
P
-100 -80 -60 -40 -20 0 20 40 60 80 100 0.00
0.03 0.06 0.09 0.12 0.15
( )
P
Chapter 5
Vector Fields and Vector
Singularities in VCSELs
In chapter3 we have shown many interesting near-field patterns observed at threshold currents. However, the presented experimental results are restricted to linear polarization. Unlike EEL has unipolarization, VCSEL has a more intriguing polarization state to the birefringence and isotropic gain region. VCSEL typically emits linearly polarized light field in one direction at near-threshold current. As the injection current increases, one common condition is that two orthogonal linear polarization states independently coexist. In this case the wavelengths of two polarization states are different. Besides, this condition is easy to operate in multi-mode lasing that will result in cloudy pattern. Another interesting phenomenon is the polarization switching, the lasing polarization state switches to the perpendicular one [AS01, MFM95, vEWW98] as the injection current increases.
Here a third circumstance that has the transverse patterns to be polarization-entangled, i. e. it has different morphology at different polarization angles, is concerned. In fact, this phenomenon corresponds to the formation of vector field which has been widely studied in various laser systems [[Gil93, VKMR01, CLH06, LCH07], as well as in VCSELs [Erdo92, PTMA97, CHLL03b]. Since the near-field pattern that is analogous to quantum-billiard wave function is purely real, VCSELs can be employed to manifest vector singularities. Vector singularities are isolated, stationary points in a plane at which the orientation of the electric field of a real vector field becomes undefined. Vector singularities as well as phase singularities play a vital role in singular optics.
This chapter is organized as follows. In first section we present a polarization-entangled pattern associated with two superscars modes in a square shaped VCSEL. We reconstruct the patterns in two orthogonal polarization states by SU(2) coherent states to manifest the vector field and vector singularities. Similar experimental method as that in Sec. 5.1 is applied to originally generate a chaotic vector. By using the eigenfunction expansion technique, the vector field is reconstructed to unambiguously analyze the vector singularities embedded in a chaotic vector field. Since the polarization of light corresponds to the spin of
quantum wave, the analyses of the vector fields in VCSELs can provide important information for quantum-billiard systems (such as ballistic quantum dots) with consideration on electronic spin.
5.1 Vector Fields in Square VCSEL
As revealed in the introduction of this chapter, the near-threshold lasing modes of VCSELs are usually linearly-polarized and VCSELs can simultaneously lase in two polarizations when injection current increase. However, the increase of injection current tends to lead to multi-mode lasing and result in cloudy pattern. Since the threshold current of VCSEL varies with device temperature, we can alternatively make the lasing thresholds of two orthogonally polarized modes to be nearly the same by means of adjusting the operating temperature. The temperature dependence shown in Fig. 3.3-3 (a) of one square VCSEL has neglected the polarization of the lasing modes. Fig. 5.1-1(b) shows the polarization-resolved temperature dependence of threshold currents of another VCSEL. It can be found that the two polarizations simultaneously lase at temperatures around 295 and . The 0 of the polarization is along the [110] direction of the (001)-GaAs crystal, as illustrated in Fig.
5.2-1(a). Fig. 5.1-2 (a)-(d) present the lasing patterns at temperature of in , , , and , respectively. It can be seen that the patterns in and
have different morphology and the patterns in and K intensity pattern that can be observed by removing the polarizer, as shown Fig.
5.2-2(e). Notice that the wave lengths of the two polarizations at and are measured to be different.
0 90
To understand this result, we can express the wave amplitude at arbitrary polarization angle by the phasor amplitude at 0 and 90
1
The experimental patterns are actually time-averaging observations
2 2 2 2
, the time average of the oscillating term
results in zero such that the interference term vanishes. Therefore, the observed patterns in and are just the total intensity pattern. However, if
then the orthogonally polarized components can mutually interfere to lead to various patterns in other polarizations.
Fig. 5.1-3 (a) and (c) show the polarization resolved near-field patterns in and at operating temperature of . It can be seen that the patterns in and are no longer the total intensity pattern (Fig.5.1-3(e)) and have greatly different morphologies, as presented in Fig.5.1-3(b) and (d). In other words, the pattern is linearly polarized, but the polarization is not the same for different spatial points. In contrast to the case at , the measurement of the optical spectrum indicates that the orthogonal polarization modes have the same wavelength. As 0
mentioned in the previous discussion, the fact that orthogonal polarization modes are phase synchronized to a common frequency is a basic requirement for a polarization- entangled pattern. From Eq. (5.1.1) we can see that, for the two polarization state with the same frequency ( 1 2
ˆ 9
( ,
), the phasor amplitude of the total field can be written in form of vector field
E x y( , )E x y a0( , ) x 0 ) ˆay E x y
45
. (5.1.4)
To understand the vector field and manifest vector singularities, we have to find the wave functions of lasing modes in two orthogonal polarizations as basis. In this case the lasing modes in and 45 are easier to reconstruct. Based on can be expressed as
45 45 45 respectively. The patterns in 0 and can be obtained by projecting the vector field into
The similarities between Fig. 5.1-3 (a)-(d) and Fig. 5.1-4 (a)-(d) verify our theoretical
reconstruction of the experimental results. Note that the formations of and are critically depended on and . Only if
and really match the experimental results, one can well reconstruct
and .
The vector singularities are generally described by the orientation angle function:
0 90
( , )x y angle E x y E[ ( , ), ( , )]x y
(5.1.8)
The vortices of the angle function (x,y) correspond to the vector singularities at which the orientation of the electric field vector is undefined. Figure 5.1-5(a) depicts the numerical pattern of the angle function (x,y) for the experimental vector field.
Here the angle is color-coded by hue and the singularities are at the points where all colors get together. A small region highlighted by white square area with edge equal to of the vector field is depicted in Fig. 5.1-5(b) to demonstrate the novel lattice structure of the vector singularities. The white and black curves stand for the nodal lines of and , respectively. It can be validated that the crossings of white and black curves coincide with the singularities at which all color get together. Besides, it is of pedagogical importance to confirm the sign rule that the nearest neighbor singularities on any contour of constant phase are required to have opposite signs [Freu95]: The singularities with topological charge and
/10 a
0( , )
E x y E90( , )x y
1 1 are labeled by white squares and black triangles, respectively. Furthermore, the vector field distribution in this region is manifested in Fig. 5.1-5(c). It can be seen that the singularities with topological charge equal to correspond to saddle points 1 of the vector flow and those with topological charge equal to 1 are all vortices, no source or sink point are found in our thorough analysis.
0°
Fig. 5.1-1. (a) Reference of the polarization angle (b) The threshold currents of the two polarizations. Simultaneous lasings occur at temperatures around
295K and 255K.
210 220 230 240 250 260 270 280 290 300 310 20
25 30 35
Threshold Current (mA)
Temperature (K) 45°
90
-45 (a)
(b)
90°
0°
(a) (b)
(c) (d)
(e)
Fig. 5.1-2. (a)-(d) The lasing patterns in 0, 45, 90, and 45 and (e) The total intensity pattern observed at 295K.
(a) (b)
(c) (d)
(e)
Fig. 5.1-3. (a)-(d) The lasing patterns in 0, 45, 90, and 45 and (e) The total intensity pattern observed at 255K.
(c) (d)
Fig. 5.1-4. (a)-(d) The reconstructed patterns of Fig. 5.2-3(a)-(d), respectively.
(b) (a)
(a)
(b) (c)
0 2π
a
Fig. 5.1-5. (a) The contour plot of the angle function ( , )x y . (b) Zoom-in view of the small regions highlighted by the white square. (c) The vector plot of the polarization vector with vortices and saddles labeled by “＋” and “－”
signs, respectively.
5.2 Chaotic Vector Field in VCSEL
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
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