Microwave Open Resonator Techniques – Part I: Theory

This chapter is devoted to the theory of open resonators. It is well known that lasers uses open resonators as an oscillatory system. In the simplest case, this consists of two mirrors facing each other. This is the first but not the only application of open resonators, whose salient features consist in the fact that their dimensions are much larger than the wavelength and the spectrum of their eigenvalues is much less dense than that of close cavity.

For microwave applications a reliable description of the coupling between the cavity and the feeding waveguide is necessary.Several papers deal with the coupling through a small hole or a rectangular waveguide taking into account only for the fundamental cavity and waveguide mode [7,8,17].
In [4] a complete analysis of the coupling between a rectangular waveguide and an open cavity has been developed taking into account for all the relevant eigenfunctions in the waveguide and in the cavity.
In this paper we review the general theory of Open Resonators and propose to study the coupling between a feeding coupling aperture given by a rectangular or circular waveguide.
Starting from the paraxial approximation of the wave equation, we derive the modal expansion of the field into the cavity taking into account for the proper coordinate system.The computation of the modal coefficients takes into account for the characteristics of the mirrors, the ohmic and diffraction losses and coupling.

Parabolic approximation to wave equation
A parabolic equation was first introduced into the analysis of electromagnetic wave propagation in [13] and [14].Since then, it has been used in diffraction theory to obtain approximate (asymptotic) solutions when the wavelength is small compared to all characteristic dimensions.As open resonators usually satisfy this condition, the parabolic equation finds wide application in developing a theory of open resonators.
A rectangular field component of a coherent wave satisfies the scalar wave equation: where k = 2π/λ is the propagation constant in the medium.
For a wave traveling in the zeta forward direction, assuming an e jωt time dependance, we put: where ψ is a slowly varying function which represents the deviation from a plane wave.By inserting (2) into (1) and assuming that ψ varies so slowly with z that its second derivative can be neglected with respect to k ∂ψ ∂z , one obtains the well know parabolic approximation to the wave equation: The differential equation (3), similar to the Schrodinger equation, has solution of the type: where: The parameter P(z) represents a complex phase shift associated to the propagation of the beam along the z axis, q(z) is the complex parameter which describe the Gaussian beam intensity with the distance r from the z axis.
The insertion of ( 4) in (3) gives the relations: The integration of (6) yields: which relates the intensity in the plane z 2 with the intensity in the plane z 1 .
A wave with a Gaussian intensity profile, as (4), is one the most important solutions of equation ( 3) and is often called fundamental mode.
Introducing (9) in the solution (4), we obtain: Now the physical meaning of these two parameters becomes clear: • R(z) is the curvature radius of the wavefront that intersects the axis at z; • w(z) is the decrease of the field amplitude with the distance r from the axis.
Parameter w is called beam radius and the term 2w beam diameter.The Gaussian beam contracts to a minimum diameter 2w 0 at beam waist where the phase is plane.The beam parameter q at waist is given by: and, using (8), at distance z from the waist: Combining ( 12) and ( 10), we have: and where z R is the Rayleigh distance: The beam contour is an hyperbola with asymptotes inclined to the axis at an angle: In (14) w is the beam radius, w 0 is the minimum beam radius (called beam waist) where one has a plane phase front at z = 0 and R is the curvature radius of of the phase front at z.It should be noted that the phase front is not exactly spherical; therefore, its curvature radius is exactly equal to R only on the z-axis.The parameter of the Gaussian beam are illustrated in Fig. 2. Dividing ( 14) by ( 13), the useful relation is obtained: The expression ( 17) is used to express w 0 and z in terms of w and R: Inserting ( 11) in (7) we obtain the complex phase shift at distance z from the waist: Integration of (20) yields The real part of P represent the phase shift difference Φ between the Gaussian beam and an ideal plane wave, while the imaginary part produces an amplitude factor w 0 w which gives the decrease of intensity due to the expansion of the beam.Now we can write the fundamental Gaussian beam: where:

Stability of open resonator
A resonator with spherical mirrors of unequal curvature is representable as a periodic sequence of lens which can be stable or unstable.The stability condition assumes the form: The above expression was previously derived in [3] from geometrical optics approach based on equivalence of the resonator and a periodic sequence of parallel lens and independently in [5] solving the integral equation for the field distribution of the resonant modes in the limit of infinite Fresnel numbers.
To show graphically which type of resonator is stable and which is unstable, it is useful to plot a stability diagram on which each type of resonator type is represented by a point (Fig. 3).

Spherical cavity in cartesian coordinates
In cartesian (x, y, z) coordinates, the separate solutions for (3) are [12]: where 1 H m is a Hermite polynomial of order m (Appendix A).
Note that both the φ mp and ψ mp functions are orthonormal on the transverse planes z=cost.
When we assume that the mirrors are sufficiently large to permit the total reflection of the gaussian beams of any relevant order, we can put: 1 To be consistent with the parabolic approximation the condition |m + n + 1| << (kw 0 ) 2 must be satisfied.Resonance occurs when the phase shift from one mirror to the other is a multiple of π.Using (2), ( 4) and ( 9) this condition leads to: where q is the number of nodes of the axial standing wave pattern and 2l >> z R the distance between the mirrors (Fig. 4).
The fundamental beat frequency Δ f 0 , i.e. the frequency spacing between successive longitudinal resonances, is given by: where c is the velocity of light.From (10) the resonant frequency f of a mode can be expressed as: The combined use of eqs.( 2),( 4) and ( 9) yields: (32)

Coupling to feeding waveguide
Because the expressions for the solution of scalar wave equation ( 3) with the boundary condition Φ = 0 on the mirrors is given by (31) the electromagnetic field inside the cavity can be expressed in terms of (quasi) transverse electromagnetic modes (TEM): According to the results reported in sect.2, the expressions for the ŷ polarized modes are: and similarly for the x polarized ones.
In (33) the index n summarizes the indexes (mpq).
From Maxwell equations we get for the mode vectors [15]: and for the coefficients V n , I n where S is the cavity surface, * denotes the complex conjugate and ζ 0 is the free space impedance.Note explicitly that the tangential electric field appearing in expression (36) and (37) is the actual field over S.This is not given by expression (33), which, at variance with expression (34), does not provide a representation for the tangential components uniformly valid up to the cavity boundaries.Let us divide the surface S into three parts: the coupling aperture A, the mirrors M and the (ideal) cavity boundary external to the the mirrors, M (see Fig. 4).Hence: Strictly speaking, in one of the integrals over the mirrors surfaces the coupling aperture should be deleted.However, the error that we make in extending the integration to the whole mirror is negligible provided that the waveguide dimension is much smaller than that of the mirrors.
The Leontovic boundary condition: wherein σ is the electric conductivity of the mirrors and δ = 2 ωσμ is the penetration depth, can be applied to express the (tangential) electric field over the mirrors in terms of magnetic one, given by (16).Hence: being: Outside the mirrors we can assume that the field is an outgoing locally plane wave (Fresnel-Kirchhoff approximation), so that on M: Accordingly, we have for the last integral in (38): The last equality in (43) follows from the fact that: according to the orthogonality condition satisfied by the φ functions.
The cavity quality factor Q n for the n-th mode is defined as: where W n is the mean electromagnetic energy stored in the cavity and P n is the power lost when only the nth mode is excited at the resonance pulsation ω n .The power is lost due to diffraction and ohmic losses.
By taking (32) into account, we can express the diagonal term in (40) as a function of the quality factor for the ohmic losses, Q rn ,: being δ n the skin depth at the resonant frequency.
The diffraction losses of a cavity can be calculated by taking into account for the diffraction effects produced by the finite size of the mirrors.Under the simplifying assumption of quasi-optic nature of the problem (dimensions of the resonator large compared to wavelength and quasi-transverse electromagnetic fields) the Fresnel-Kirchhoff formulation can be invoked for the diffracted field from the mirrors.Hence we have for the diffraction loss of the nth mode: and for the corresponding quality factor for the diffraction losses, Q d ,: By using (40, 41,44, 45) and taking into account that σδζ 0 >> 1 and δ n /δ 1 for all relevant modes, equation (38) becomes: and: A metallic waveguide is assumed to feed the cavity.The waveguide field on the coupling aperture A is represented as: where e g and h g are TE electromagnetic modes of the waveguide: Assuming that the mirror curvature can be neglected over the extension of the coupling aperture,fields (51, 52) verify the following orthonormality relation: Expressing the field over the coupling aperture A by means of expression (52) we obtain from (3): where: and: By introducing the matrices A and B, whose elements are: and β nm respectively2 , and the vectors I ≡ {I n } and V g ≡ {V g n }, relation (3) can be written in a compact form as: wherein V + and V − are the vectors of the incident and reflected waveguide mode amplitudes respectively.By enforcing the continuity of the magnetic field tangential component over the coupling aperture, we get: wherein B + is the adjoint (i.e., the transpose, being B a real matrix) of B and ζ is the diagonal matrix whose elements are the modes characteristic impedances, normalized to ζ 0 .From ( 48) and ( 49) we immediately obtain: wherein I is the unit matrix and A −1 the inverse of the matrix A.
Solution of eq. ( 60) and ( 61) provides the answer to our problem.In particular, from (61) we get the (formal) expression for the feeding waveguide scattering matrix S:

Field on the coupling aperture
In order to solve the system (60-61) a suitable description of the field on the aperture A is necessary.Any device, able to support electromagnetic field matching the cavity field on the mirror (33-34), can be used to feed the cavity.In the following, the case of metallic and circular waveguide will be treated in detail.

Modes in rectangular waveguide
A rectangular metallic waveguide, with transverse dimensions a × b, is assumed to feed the cavity.The waveguide TE electromagnetic modes of the metallic rectangular waveguide, on the coupling aperture A, is represented as: (65) and the TM electromagnetic modes: Note explicitly that in expressions (63-68) the index n summarizes the indexes (pq).

Modes in circular waveguide
When a circular waveguide , with radius a, is assumed to feed the cavity, the TE electromagnetic modes are: and the TM electromagnetic modes: where: q or is the r-mo zero of Bessel function of order p and q or r-mo zero of the derivative of Bessel function of order p.
Note explicitly that in expressions (69, 71) the index n summarizes the indexes (pr).

Equivalent circuit
Let us consider systems (60) and (61) under the following assumptions: 1. Negligible intercoupling between cavity modes, i.e. : 2. Single cavity mode approximation, i.e., -see (3)-: 3. Beam diameter at the mirror much larger than the waveguide dimension, i.e. : and taking (75-76) into account, equations (60-61) became From (79,80) we immediately get: From (82) we get for the equivalent terminal impedance relative to the fundamental mode: Taking into account for the expression (55) for F 0 , we get the equivalent circuit representation of Fig. 5. where the explicit expression for its elements are collected under Table 1.
Table 1.Expressions for the circuit elements of Fig. 5 The value of L e depends on the feeding waveguide and is reported in the subsections for rectangular and circular waveguides.

Rectangular waveguide
The cavity is assumed to be excited by the incident fundamental TE 10 mode.From expressions (51-52) and (63) follows that in the cavity are excited only the (0,0,q) mode, in the feeding waveguide are excited the modes TE n0 .
When the cavity is fed by a rectangular waveguide under the approximation 3), the expression (56) for β 0k can be explicitly evaluated, leading to: Hence, for the sum at the right hand side of (83) we have:

Circular waveguide
The cavity is assumed to be excited by the incident fundamental TE 11 mode.From expressions (51, 52) and (69) follows that in the cavity are excited the (0,0,q) mode, in the feeding waveguide are excited the modes TE 1r .
When the cavity is fed by a circular waveguide under the approximation 3), the expression (56) for β 0k can be evaluated (Appendix A), leading to: ) allowing the computation of L e .

B. Coupling cavity -circular waveguide
The x component of the magnetic field, for TE modes, in the waveguide is: q pr a J p (k tpr ρ) cos φ cos (pφ) − p ρ J p (k tpr ρ) sin φ sin (pφ) The x component of the magnetic field in the cavity is: where According to the position of sect.4 we consider the mode (0,0,q) in the cavity, so the equation (95) reduces to: and the equation (94) on the mirror (for z=l):

Figure 1 .
Figure 1.Amplitude distribution of cavity fundamental mode Two real beam parameters, R and w, are introduced in relation to the above complex parameters q by1 q = 1 R − j λ πw 2(9)

Figure 2 .
Figure 2. Parameters of Gaussian beam mpq represent Hermite Gauss beams propagating from left to right and from right to left,respectively.

Figure 4 .
Figure 4. Spherical Open Cavity In the paraxial approximation the eigenfunctions Ψ spq satisfy the normalization relation