Centerline Laser Radiation Intensity in an Unstable Cavity H. M'irels and S. B. Batdorf

Aerospace Corporation, El Segundo, California 90245. Received 16 June 1972.

The variation of laser radiation intensity along the axis of an unstable optical cavity is investigated theoretically. The present study is an extension of the work of Rigrod1 to situations in which geometric divergence of the radiation must be taken into account. Diffraction effects are neglected.

Consider an unstable cavity as illustrated in Fig. 1. Cylindri­cal or spherical mirrors are located at stations a and d. The opti­cal gain region is located between stations b and c. We confine our attention to conditions along the axis. According to the theory of Siegman,2 the radiation reflected from the mirrors at

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stations a and d appears to originate from centers P and P, re­spectively. The location of these centers is noted in the Appen­dix. Let r, r denote axial distance as measured from the reflection centers P and P, respectively. Radiation intensity I is normal­ized by the saturation value Is (i.e., I/Is ≡ I). The intensity in the r, f direction is denoted by I+, I–. Assume tha t the local gain g is related to the local intensity I = I+ + I–by

where go is the zero power gain and m = 1, 2 for homogeneous and inhomogeneous broadening, respectively.

Due to the divergence of the reflected radiation, the increase in intensity along the r and r axis is

where β+ = I+rσ, β– = I–(r)σ, and σ = 0,1,2 for plane, cylindri­cal, and spherical mirror systems, respectively. (The value σ = 0 corresponds to the case where both mirrors are plane and thus represents the Rigrod1 solution. Note tha t β as defined here is a generalization of the corresponding quanti ty in Ref. 1.) Since dr = –dr, an integral of Eq. (2) is

where K is a constant. Equation (2) also indicates tha t β+ and β– are constant in regions where g0 = 0. Thus

In the low-power limit g — g0, the intensity distribution in the gain region (rb ≤ r ≤ rc) is

I t follows tha t the laser threshold condition is given by

where l = re – rb = rb – rc, Ra = Ia+/Ia

– and Rd ≡ Id–/Id

+ (i.e., Ra and Rd are mirror reflection coefficients).

The intensity distribution along the axis of an unstable. cavity can be found when go,m,Ra,Rd, and cavity geometry are specified. A numerical integration of Eq. (2a) in the interval rb, ≤ r ≤ rc is generally required. To avoid iteration, it is convenient to con­sider the indirect problem, i.e., given Ra, βa

+, find the correspond­ing value of Rd. In the latter case,

where we find βc and βc+ – by noting βb+ = βa

+ and integrating Eq. (2a) from rb to rc The numerical integration is facilitated by Eq. (2) expressed in the form

Fig. 2. Variation of transmitted intensity Td with mirror trans-missivity td at centerline for unstable cavity in which pa < 0, |pd| = ∞. Conditions correspond to configuration in Ref 3; σ = 2, L = 91.4 cm, l = 17.8 cm, gr0 = 0.10 cm– 1 , Ra = 0.99, Rd

= 0.99 – td: (a) m = 1; (b) m = 2.

where ξ ≡ (r – rb)/(rc – rb) is a normalized independent variable and r = ra + ra – r. In the direct problem, iteration is used to find the value of βa

+ corresponding to a specified value of Rd.

When βa+, Ra, and Rb are known, β+ and β– are defined

everywhere exterior to the lasing region rb < r < rc. If mirror d has a transmissivity td, the intensity of the transmitted radia­tion Td is found from

A plot of Td vs td is shown in Fig. 2 for conditions corresponding to an experimental cw chemical laser cavity.3

If m = 1, an analytical solution can be obtained tha t is correct to order l/rb, l/rc. The procedure is as follows. The integral of Eq. (2a) between the limits rb and rc can be expressed in the form

In

where r* and r* are mean values of r and in the integration in­

terval and are defined by

Fig. 1. Unstable cavity nomenclature

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It can then be shown that the incident intensity at mirror a and d equals

where

Equation (12) can be used to obtain the absorbed and trans­mitted power at the mirrors, provided suitable estimates of r* and r* are available. The use of an arithmetric mean

is sufficiently accurate for most purposes and, by use of a mean value theorem in Eq. (11), is correct to order l/rb, l/rc. Im­proved estimates for r* and r* are being investigated.

Analytic results from Eqs. (12) and (13) are included in Fig. 2(a). It is seen that these results are in exact agreement with the numerical integration of Eq. (8) to the accuracy of the figure scale. For the case |pa| = 635, l/rb approximately equals 0.1. Hence, the error is considerably less than l/rb.

The unstable cavity may be viewed as an oscillator-amplifier in which the region near the optical axis behaves like an oscillator. Radiation from this region diverges outward and is amplified in the outer (amplifier) portion of the cavity. The present results are therefore useful for establishing the conditions under which the region near the axis is saturated. The results are directly applicable for estimating transmitted intensity, near the axis, for cavities employing partially transmitting mirrors or gratings.3

The present results also provide the leading term in an expansion for intensity distribution off the axis.

Appendix—Determination of Reflection Centers. The lo­cation of the virtual reflection centers, P, P (Fig. 1), in terms of mirror radii and separation distance L is noted herein following the method of Ref. 2. Denote mirror radii at stations a and d by pa and pd, respectively. The sign convention is such that p is positive for concave mirrors and negative for convex mirrors (when viewed from the cavity). Define

The location of the virtual centers is found from2

When mirror d is plane (pd → ∞), these expressions reduce to

Note that pa is negative for the configuration in Fig. 1.

This work reflects research supported under the U. S Air Force Space and Missile Systems Organization contract F04701-71-C-0172.

References 1. W. W. Rigrod, J. Appl. Phys. 36, 2487 (1965). 2. A. E. Siegman, Proc. IEEE 53, 277 (1965). 3. R. A. Chodzko, H. Mirels, and F. Roehrs, "Application of

Single Frequency Unstable Cavity to a cw HF Laser," in preparation.

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