Suppression of Transverse Modes

Let us first consider the selection of transverse modes. In Sect. 5.2.3 it was shown that the higher transverse TEM , modes have radial field distributions that are less and less concentrated along the resonator axis with increasing transverse order n or m. This means that their diffraction losses are much higher than those of the fundamental modes TEMoo (Fig. 5.12). The field distribution of the modes and therefore their diffraction losses depend on the resonator parameters snch as the radii of curvature of the mirrors Ri, the mirror separation d, and, of conrse, the Fresnel number (Sect. 5.2.1). Only those resonators that fiilfiU the stabiUty condition [291, 314] 

In Fig. 5.34, the ratio yio/yoo of the diffraction losses for the TEMio and the TEMoo modes in a symmetric resonator with gi = gi = g is plotted for different values of g as a function of the Fresnel number Np. From this diagram one can obtain, for any given resonator, the diameter 2a of an aperture that suppresses the TEMio mode but still has sufficiently small losses for the fundamental TEMoo mode with beam radius w. In gas lasers, the diameter 2a of the discharge tube generally forms the limiting aperture. One has to choose the resonator parameters in such a way that a 3w/2 because this assures that the fundamental mode nearly fills the whole active medium, but still suffers less than 1 % diffraction losses (Sect. 5.2.6). 

Let us first consider the selection of transverse modes. In Sect. 5.2.3 it was shown that the higher transverse modes have radial field distributions 

In Fig. 5.32, the ratio Kio/koo of the diffraction losses for the TEMio and the TEMqo modes in a symmetric resonator with = g2 = g is plotted for different values of g as a function of the Fresnel number N. From this diagram one can obtain, for any given resonator, the diameter 2a of an aperture 

The laser can, however, still oscillate on several longitudinal modes, and for true single-mode operation, the next step is to suppress all but one of the longitudinal modes. 

Because the frequency separation of the transverse modes is small and the TEMjQq-mode frequency is separated from the TEMqq frequency by less than the homogeneous width of the gain profile, the fundamental mode can partly saturate the inversion at the distance r from the axis, where the TEM Qq mode has its field maximum. The resultant transverse-mode competition (Fig.5.34) reduces the gain for the higher transverse modes and may suppress their oscillation even if the unsaturated gain exceeds the 

---

All the phenomena described above are absent in a 2D-junction when the effects of transverse mode quantization are neglected [7]. We have considered the limiting case of a single (transverse) channel because this is the case when the effects induced by a dispersion asymmetry in the electron spectrum are most pronounced. The anomalous supercurrent Eq. (7) is a sign alternating function of the transverse channel index since for neighboring channels the spin projections of chiral states are opposite [4]. Besides, the absolute value of the dispersion asymmetry parameter decreases with transverse-channel number j. So, for a multichannel junction the effects related to a dispersion asymmetry phenomenon will be strongly suppressed and they completely disappear in the pure 2D case. 

In the previous sections we have seen that without specific manipulation a laser generally oscillates in many modes, for which the gain exceeds the total losses. In order to select a single wanted mode, one has to suppress all others by increasing their losses to such an amount that they do not reach the oscillation threshold. The suppression of higher-order transverse TEM modes demands actions other than the selection of a single longitudinal mode out of many other TEMqq modes. 

The main pecuharity here is the negative sign of y. This particular feature means that urdike polymorphic and transversal modes at the formation of the nuclei at the longitudinal mode, the concentration gradient stimulates nucleation. Therefore, at any concentration gradient, nucleation at longitudinal mode is always possible (in the thermodynamic sense). However, the kinetics may suppress it. 

HeNe Laser at A = 632.8 nm. The Doppler width of the Ne transition is about 1500 MHz, and the width of the gain profile above threshold, which depends on the pump power, may be 1200 MHz. With a resonator length of d = 100 cm, the spacing of the longitudinal modes is Su = c/2d = 150 MHz. If the higher transverse modes are suppressed by an aperture inside the resonator, 7 to 8 longitudinal modes reach threshold. The homogeneous width is determined by several factors the natural line width = 20... 

As mentioned before, the longitudinal excitation always promotes tunneling, A , =i > A =o- The effect of transversal excitation, however, depends on the behavior of the effective frequency 9 x). For instance, for monotonically growing (decreasing) 9 r) the excitation of the transversal mode suppresses (promotes) the tunneling splitting. We see below that the effective frequency is generally determined as a solution of auxiliary differential equation that describes the nodal structure of the semiclassical wave function. This corresponds to the discussion of Takada and Nakamura [30,31] (see Chapter 4). 

In the transverse direction, a nominal abutment stiffiiess equal to 50 % of the elastic transverse stiffiiess of the adjacent bent can be used this nominal stiffness has no direct correlation or relevance to the actual residual stiffness (if any) provided by the failed shear key but is meant to suppress unrealistic response modes associated with a completely released end conditiOTi. 

---