Chapter 2. High-order lasing mode and free space propagation of large-aperture
2.2 Large-aperture equilateral-triangular VCSEL
2.2.1 Theoretical analysis
Compared to the square shape infinity potential well, the shape in equilateral-triangular is classically integrable but non-seperable system. Let three vertices of an equilateral-triangular to be set at (0, 0), (a/2, 3 / 2a ), and (-a/2, 3 / 2a ).
The eigenstates in an equilateral-triangular infinity potential wall have been derived by several groups [9-11] and the wave functions of the two degenerate stationary states can be expressed as
Hence, the condition of m≥2n is required to keep all eigenstates to be linearly independent to each other. Figures 2.10(a) and 2.10(b) show some of the Φ( )m nC, ( , )x y
and Φ( )m nS, ( , )x y with their quantum number labeled below each picture. As similar to the wave function in square shape potential well, the eigenstates do not manifest the localization on periodic orbits even if the quantum number approaches to infinity.
For the stationary coherent states in equilateral-triangular infinity potential well, first, the traveling wave states are represented from linear combination of eigenstates:
Next, the stationary coherent states associated with periodic orbits denoted by ( , , )p q
φ
in equilateral-triangular infinity potential well can be expressed as below [12,13].Fig. 2.10 (a) Some eigenstates of equilateral-triangular 2D infinity potential well
( )C, ( , )
m n x y
Φ .
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Fig. 2.10 (b) Some eigenstates of equilateral-triangular 2D infinity potential well
( )S, ( , )
m n x y
Φ . When m=2n, Φ( )m nS, ( , ) 0x y = .
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Fig. 2.11 Stationary coherent states of Ψ42,20+ ( , ;1,0, )x y φ with different
φ
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2.2.2 Experimental setup
The experimental setup is the same as shown in Fig. 2.4. The VCSEL device was placed in a cryogenic system with a temperature stability of 0.01 K in the range of 80–300 K. A DC power supplier (KEITHLEY 2400) with a precision of 0.005 mA is used to drive the VCSEL device. The near-field patterns were reimaged into a charge-coupled device (CCD) camera (Coherent, Beam Code) with an objective lens (Mitsutoyo, NA of 0.9). The far-field patterns are measured with a CCD placed behind a screen.
The schematic of the laser device structure with equilateral-triangular shape is shown in Fig. 2.12. The edge length of the oxide aperture is approximate 66.8
µ
m.Fig. 2.12 Schematic of the large-aperture equilateral-triangular laser device structure.
66.8μm
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2.2.3 Results and discussion
The experimental near-field morphologies are shown in Figs. 2.13(a)–(c) with different temperatures as labeled in each figure. Additionally, the corresponding far-field patterns for the honeycomb eigenmode, the cane-like superscar mode, and the superscar mode with ( , ) (1,1)p q = [14] are shown in Figs. 2.13(a’)–(c’). We can see that the near-field patterns for the honeycomb eigenmode and the superscar mode with ( , ) (1,1)p q = are conspicuously different to each other. However, their far-field patterns display fairly similar directional emission. The directional emission for a superscar mode can be easily traced from its localization in the near-field feature, as shown in section 2.1.3. Conversely, it is demanding to find the directional emission of a honeycomb lasing mode.
As described in section 2.2.1, the eigenstates for the standing waves are given by
( )
In terms of Φ( )m nS, ( , )x y , the experimental honeycomb pattern can be well reconstructed with n=60 and m=6, as depicted in Fig. 2.14(a). On the other hand, the Superscar modes associated with classical periodic orbits can be analytically expressed with the coherent states.
The experimental superscar pattern in Fig. 2.13(b) can be well reconstructed by
2 , ( , ; , , )
CN M+ x y p q φ with N=36, M=9, and ( , , ) (1,0,0.23 )p q
φ
=π
, as depicted in Fig.2.14(b). In the same way, the experimental superscar pattern in Fig. 2.13(c) is related to the theoretical solution CN M+, ( , ; , , )x y p q φ 2 with N=22, M=6, and
( , , ) (1,1,0.3 )p q
φ
=π
, as depicted in Fig. 2.14(c). The theoretical analysis of the far field can be implemented by combining the numerical near-field pattern and the equation of Fraunhofer diffraction. The calculation results of far field are shown in Fig. 2.14(a’) for honeycomb eigenmode, Fig. 2.14(b’) for superscar (1,0) mode, and Fig. 2.14(c’) for superscar (1,1) mode. The excellent agreements between the experimental and numerical patterns again confirm that the model of quantum system in infinity potential well is great important to analogically simulate the transverse lasing modes of the VCSELs, and this work was proposed in Ref. [15].40
The experimental and theoretical results show that the near-field patterns for the honeycomb eigenmode and the superscar mode (Figs. 2.13(a) and (c) and Figs. 2.14(a) and (c)) are apparently different to each other. One spreads over the triangular space while the other one is localized on the periodic orbit with trajectory parallel to the three edges of the triangle. However, their far-field patterns display fairly similar directional emission with the angles at integral multipliers of 60. As a consequence, we confirm that the far-field directional emission is just a necessary not sufficient condition for the emergence of a superscar mode [15].
For detailed exploring the far fields of the honeycomb eigenmode and the superscar mode, the experimental results are redraw in Fig. 2.15 and the numerical patterns are calibrated by changing the index of square in CN M+, ( , ; , , )x y p q φ 2 to
0.8 , ( , ; , , )
CN M+ x y p q φ for diminishing the contrast of the image and emphasizing the delicate morphology. There are two apparent differences between the far-fields in Fig.
2.15: the structures of the six points in each directional emission and the lines connected between the six points. Interestingly, these lines form two triangles with a shape in hexagram which is the well-known Magen David (Star of David).
In the far field of honeycomb eigenmode, there are two dots in each six directions of directional emission. As a result, there are total 12 dots in the far field.
These dots can be explained by dividing the eigenstate Φ( )m nS, ( , )x y into three parts, each part is familiar to the eigenstates of a rectangle-shape infinity potential well. The three parts are as listed:
As shown in the section 2.1.3, the far field emitted from a rectangle-shape aperture VCSEL has 4 points on each directional emission. Based on this result, the 12 dots and its locations in the far field of a honeycomb near-field pattern are apparently reasonable.
Despite the 12 dots, the lines in Magen David can not get well explain from the results of the far fields emitted from a rectangle-shape aperture VCSEL. Actually, the lines are not real line but the interference patterns, and amazingly, the lines used to connect the six dots form one of the triangular in Magen David with a close loop (one touch drawing), as shown in Fig. 2.16. The detailed formation about this line is related to the superposition of the wave function for constructing the near-field transverse modes. For example, a summation of a group of sin waves to create a sinc function, because the number of the sin waves in the group is finite, there will exist some fluctuations between the peaks of sinc function. As a result, in the far field of VCSELs, it usually exist some interference-line patterns connected between each dots [6,7,16].
Based on the explanation of the Magen David emitted from honeycomb near-field pattern, the Magen David in the far field of a superscar mode (1,1) is explained as follows. The locations of the 12 dots are related to the quantum numbers of the eigenstates as confirmed by the far field of a rectangle-shape aperture VCSEL. When a group of eigenstates with different eigenvalues is combined to generate a coherent state CN M+, ( , ; , , )x y p q φ 2, several groups of 12 dots will overlap to form six elliptical dots on each directional emission and the lines in Magen David will be plaited to a ribbon, as shown in Fig. 2.15.
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Fig. 2.13 Experimental near-field morphologies: (a) honeycomb eigenmode, (b) cane-like superscar (1,0) mode, (c) superscar (1,1) mode. The far-field patterns (a’), (b’), and (c’) correspond to (a), (b), and (c), respectively.
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Fig. 2.14 Theoretical near-field morphologies: (a) honeycomb eigenmode, (b) cane-like superscar (1,0) mode, (c) superscar (1,1) mode. The far-field patterns (a’), (b’), and (c’) correspond to (a), (b), and (c), respectively.
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Fig. 2.15 The experimental and Theoretical far-field patterns from honeycomb pattern (above) and superscar (1,1) mode (bottom) in near field. The morphology of the numerical patterns is enhanced.
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Fig. 2.16 Six points on one of the triangular of Magen David is connected with only one touch.
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Furthermore, we explore the free space propagations of the transverse modes emitted from VCSELs for understanding the directional emissions from some typical lasing modes in equilateral-triangular VCSEL. The light evolution emitted from VCSEL with a honeycomb lasing mode is shown in Fig. 2.17. The results are conspicuously different to the eigenmodes of square boundary in Fig. 2.8. The honeycomb lasing mode analogously splits into three honeycomb patterns and each pattern propagate out from the three edges of triangular boundary with direction vertical to each edge. During the evolution, an inverse triangular window is gradually opening on the center of the diffraction pattern. This window cuts the three divided honeycomb patterns to six dots with interference structure.
The free-space propagations of the superscar (1,0) and (1,1) modes are shown in Figs. 2.18 and 2.19, respectively. The second and fourth rows in the Figs. 2.18 and 2.19 depict the numerical patterns calculated from combining the theoretical near-field patterns and the Fresnel diffraction integral (eq. A.9). In Fig. 2.18, the characteristic of the diffraction patterns show a distorted trajectory followed the direction parallel to the left bevel edge as same as the diamond-like superscar mode in square-aperture VCSEL. For the diffraction of superscar (1,1) mode, there are another six points in the diffraction patterns (six lines in diffraction space) and such six points are explained in Fig 2.20. There are three short dash lines in the three corners which are hardly observed in the experimental measurement (Fig. 2.19), and these short dash lines will form the six points.
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Fig. 2.17 Experimental and theoretical results of free-space propagation of the honeycomb eigenmode. The experimental results are shown in first and third rows and the theoretical results are shown in the second and forth rows.
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Fig. 2.18 Experimental and theoretical results of free-space propagation of the cane-like superscar (1,0) mode. The experimental results are shown in first and third rows and the theoretical results are shown in the second and forth rows.
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Fig. 2.19 Experimental and theoretical results of free-space propagation of the superscar (1,1) mode. The experimental results are shown in first and third rows and the theoretical results are shown in the second and forth rows.
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Fig. 2.20 The formation about six points in the diffraction pattern of the experimental results. The schematic picture about these propagations of six points is also shown.
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