Resonators confocal

The analysis has been extended by Boyd, Gordon, and Kogelnik to resonators with confocally-spaced spherical mirrors [5.20,5.21] and later by others to general laser resonators [5.22-5.30]. For the symmetric confocal case (the two foci of the two mirrors with equal radii R =R2 = R coincide, i.e., the mir-rorseparation d is equal to the radius of curvature R). 

For this case (5.28) can be separated into two one-dimensional homogeneous Fredholm equations that can be solved analytically [5.20,5.24]. The solutions show that the stationary amplitude distributions for the confocal resonator can be represented by the product of Hermitian polynomials, a Gaussian fxmction, and a phase factor  

C is a normalization factor. The function is the Hermitian polynomial of ffJth order. The last factor gives the phase (p zo, r) in the plane z = zo at a distance r = x + from the resonator axis. The arguments x and y depend on the mirror separation d and are related to the coordinates x, y 

The fundamental modes have a Gaussian profile. For r = w z) the intensity decreases to 1/e of its maximum value io = C on the axis r = 0). The value r = w z) is called the beam radius or mode radius. The smallest beam radius wo within the confocal resonator is the beam waist, which is located at the center z = 0. From (5.31) we obtain with d = R 

Note that wo and w do not depend on the mirror size. Increasing the mirror width 2a reduces, however, the diffraction losses as long as no other limiting apertme exists inside the resonator. 

The smallest beam radius wq within the confocal resonator is the beam waist, which is located at the center z = 0. From (5.31) we obtain with d = R 

The fundamental modes have a Gaussian profile. For r — w(z) the intensity decreases to 1/e of its maximum value Iq = C on the axis (r = 0). The 

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We have chosen to develop the quasioptical theory needed for understanding the spectrometer by considering first the properties of lenses and reflectors. In the analysis of resonators, a very fruitful approach is to unfold the multiple reflections of the resonator into a series of lenses in circular apertures spaced by the mirror separation for a confocal resonator (Kogelnik and Li, 1966). The semiconfocal resonator is a special case of the confocal resonator. We use a flat mirror, which images the curved mirror at minus the mirror separation. In such a resonator, it is impossible... 

Fig. 5.10. Phase fronts and intensity profiles of the fundamental TEMqo rnode at several locations z in a confocal resonator with the mirrors at z = J/2...

The phase fronts of the fundamental modes inside a confocal resonator close to the resonator axis can be described as spherical surfaces with a z-dependent radius of curvature. For za) = R/2 R = R. This means that at the... 

At the center of the resonator z = 0-> o = 0 R = oc. The radius R becomes infinite. At the beam waist the constant phase surface becomes a plane z = 0. This is illustrated by Fig. 5.10, which depicts the phase fronts and intensity profiles of the fundamental mode at different locations inside a confocal resonator. 

It can be shown [5.1,5.24] that in nonfocal resonators with large Fresnel numbers N the field distribution of the fundamental mode can also be described by the Gaussian profile (5.32). The confocal resonator with d = R can be replaced by other mirror configurations without changing the field configurations if the radius Rf of each mirror at the position zo equals the radius R of the wavefront in (5.37) at this position. This means that any two surfaces of constant phase can be replaced by reflectors, which have the same radius of curvature as the wave front - in the approximation outlined above. 

These resonators are equivalent, with respeet to the field distribution, to the confocal resonator with the mirror radii R and mirror separation d= R. 

The second factor is a function of d and becomes maximum for d= R. This reveals that of all symmetrical resonators with a given mirror separation d the confocal resonator with d = R has the smallest spot sizes at the mirrors. 

The confocal resonator with the smallest spot sizes at a given mirror separation d according to (5.39) also has the lowest diffraction losses per round-trip, which can be approximated for circular mirrors and Fresnel numbers N > hy 15.1]... 

This reveals that for gi = 0 or g2 = 0 and for gigi = E the spot sizes become infinite at one or at both mirror surfaces, which implies that the Gaussian beam diverges the resonator becomes unstable. An exception is the confocal resonator with = g2= 0, which is metastable , because it is only stable if both parameters gi are exactly zero. For gig2 > 1 or g g2 < 0, the right-hand sides of (5.42) become imaginary, which means that the resonator is unstable. The condition for a stable resonator is therefore... 

Lasers as Spectroscopic Light Sources b) confocal resonator... 

Equation (5.49) reveals that the frequency spectrum of the confocal resonator is degenerate because the transverse modes with q = q and m- -n = 2p have the same frequency as the axial mode with m = n = 0 and q = q - -p. Between two axial modes there is always another transverse mode with m- -n -f 1 = odd. The free spectral range of a confocal resonator is therefore... 

Fig.5.19a-c. Degenerate mode frequency spectrum of a confocal resonator (d = R) (a), degeneracy lifting in a near-confocal resonator (d = 1.1/ ) (b), and the spectrum of fundamental modes in a plane-mirror resonator (c)... 

The output beam from an HeNe laser with a confocal resonator (R = L = 30 cm) is focused by a lens of / = 30 cm, 50 cm away from the output mirror. Calculate the location of the focus, the Rayleigh length, and the beam waist in the focal plane. 

J.R. Johnson A method for producing precisely confocal resonators for scanning interferometers. Appl. Opt. 6, 1930 (1967)... 

G.D. Boyd, H. Kogelnik Generalized confocal resonator theory. Bell Syst. Techn. J. 41, 1347 (1962)... 

Lebedev N, Strycharz-Glaven SM, Tender LM. Spatially resolved confocal resonant raman microscopic analysis of anode-grown Geobacter sulfurreducens biofilms. ChemPhysChem 2014 15(2) 320-327. 

Figure 5.9 a Stationary one-dimensional amplitude distributions Am x) in a confocal resonator b two-dimensional presentation of linearly polarized resonator modes y) for square... 

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