504
19 Dielectric Structures
19.2 Microscopic Resonators
19.2.1 Microdiscs
A microdisc is a cylindrical resonator with a thickness d that is small compared to the radius R. It can
be fabricated from semiconductors and semiconductor heterostructures using patterning and materialselective etching. With underetching a mostly free-standing disc can be made that resides on a post
(Fig. 19.22).
The coordinate system is (ρ, φ, z) with the z direction being perpendicular to the disc area. Typically,
the disc is so thin that there is only one node along z. Solving the wave equation in this geometry
[1645], the modes are characterized by two numbers (m, l). m describes the number of zeros along
the azimuthal direction φ with the field amplitude being proportional to exp(±imφ). Thus, except
for m = 0, the modes are simply degenerate. Modes with E z = 0 are called TE modes. This is the
preferred polarization of emission. The number l denotes the number of zeros in the radial direction.
Only for modes with |m| = 1, is the intensity nonzero on the axis, i.e. for ρ = 0. All other modes have
vanishing intensity in the disc center.
The light intensity in whispering gallery modes is preferentially concentrated along the circumference of the disc as shown in Fig. 19.23a. Since the light can only escape via evanescent waves, the light
is well ‘captured’ in such a mode. The Q-factor (19.36) is extremely high and takes values of several
500nm
1µm
(a)
(b)
Fig. 19.22 a Side view of a 3-µm diameter disc containing one 10-nm InGaAs quantum well between 20-nm
(In,Ga)(As,P) barriers standing on an InP pillar that has been selectively underetched using HCl. b Top view SEM
image of a 5-µm diameter (In,Ga) (As,P) microdisc. The pedestal shape is a rhombus due to anisotropic etching of the
HCl. Adapted from [1644], reprinted with permission, ©1992 AIP
(a)
(b)
6
5
4
3
2
1
0
4
0.0
0.05
0.1
Fig. 19.23 a Field intensity for whispering gallery mode (10, 0) (TM-polarized) for a circle with 1 µm radius (shown as
white line) and n = 1.5. The image size is 4 × 4 µm 2 . b Theoretical quality factor of a 2-µm InP microdisc as a function
of the deformation parameter (19.56). The insets show (8,0) whispering gallery modes at a wavelength of 1.55 µm for
n = 3.4. Part (b) adapted from [1647]
19 Dielectric Structures
19.2 Microscopic Resonators
19.2.1 Microdiscs
A microdisc is a cylindrical resonator with a thickness d that is small compared to the radius R. It can
be fabricated from semiconductors and semiconductor heterostructures using patterning and materialselective etching. With underetching a mostly free-standing disc can be made that resides on a post
(Fig. 19.22).
The coordinate system is (ρ, φ, z) with the z direction being perpendicular to the disc area. Typically,
the disc is so thin that there is only one node along z. Solving the wave equation in this geometry
[1645], the modes are characterized by two numbers (m, l). m describes the number of zeros along
the azimuthal direction φ with the field amplitude being proportional to exp(±imφ). Thus, except
for m = 0, the modes are simply degenerate. Modes with E z = 0 are called TE modes. This is the
preferred polarization of emission. The number l denotes the number of zeros in the radial direction.
Only for modes with |m| = 1, is the intensity nonzero on the axis, i.e. for ρ = 0. All other modes have
vanishing intensity in the disc center.
The light intensity in whispering gallery modes is preferentially concentrated along the circumference of the disc as shown in Fig. 19.23a. Since the light can only escape via evanescent waves, the light
is well ‘captured’ in such a mode. The Q-factor (19.36) is extremely high and takes values of several
500nm
1µm
(a)
(b)
Fig. 19.22 a Side view of a 3-µm diameter disc containing one 10-nm InGaAs quantum well between 20-nm
(In,Ga)(As,P) barriers standing on an InP pillar that has been selectively underetched using HCl. b Top view SEM
image of a 5-µm diameter (In,Ga) (As,P) microdisc. The pedestal shape is a rhombus due to anisotropic etching of the
HCl. Adapted from [1644], reprinted with permission, ©1992 AIP
(a)
(b)
6
5
4
3
2
1
0
4
0.0
0.05
0.1
Fig. 19.23 a Field intensity for whispering gallery mode (10, 0) (TM-polarized) for a circle with 1 µm radius (shown as
white line) and n = 1.5. The image size is 4 × 4 µm 2 . b Theoretical quality factor of a 2-µm InP microdisc as a function
of the deformation parameter (19.56). The insets show (8,0) whispering gallery modes at a wavelength of 1.55 µm for
n = 3.4. Part (b) adapted from [1647]