C. A. Hill is with the Royal Signals and Radar Establishment, Malvern, ... HILL AND HALL: WAVEGUIDE LASER RESONATORS WITH TILTED MIRROR. I079.
1078
IEEE JOURNAL OF Q U A N T U M ELECTROKICS. VOL. QE-22. NO. 7, JULY 1986
Waveguide Laser Resonators with a Tilted Mirror CHRISTOPHERA.HILL
AND
DENIS R. HALL
very near the guide. In the vocabulary familiar in waveguide laser literature, this may be called a dual-Case I design [lo]. It minimizes both “wasted” resonator space and coupling losses. Our main deliberate simplification is that we consider only resonators of this type. Because the waveguide reflector coupling losses for the guide modes arevery low, and there is little mixing among these modes upon recoupling, a well-aligned dual-Case I laser has transverse modes very similar to those of the I. INTRODUCTION guide itself. During operation, the resonating mode is deHIS paper treats some fundamental problems of gas termined mainly by guide losses, which usually favor the waveguide lasers. These devices have become firmly EHI mode. Quasi-TEM,, output is therefore a natural reestablished over the past 15 years and many of their fea- sult [9], [ lo]. It is well known that tilting one of the mirrors causes tures have been well reviewed 111-[3]. There is continuhere RF the output mode to change considerably. We propose ing research onseveralfronts,includingtransverse excitation [ 4 ] , [ 5 ] ,the material and construction needs for a simple modelof tilt-dependent behavior. The problemis sealed-off lasers [6], and high-pressure pulsed [7] or CW interesting in itself,but especially becausea near-field Littrow diffraction grating appears roughly as a tilted Case [8] tuning. line. We hope Also, a dozen years after theinitial work on waveguide I mirror to near neighbors of the chosen specifically to gain insight into the problems of line-hoplaser resonators [9], [lo], there is a growing literature on their transverse mode behavior. This is the subject of the ping in tunable waveguide CO, lasers. In Section I1 we derive the coupling coefficients which present paper. The desirability of a Gaussian-like (quasidescribe reflection from a tilted Case I mirror. These are TEMoo)outputbeam profile underalloperatingconditions ‘is widely accepted, but the practical influences on used in Section I11 to model the resonator loss, power outmode quality have remained a little obscure. Thisis partly put, and resonator transverse mode content of a dual-Case I laser as functions of mirror tilt. The predictions of our because real lasers have so many complicating features well withtheexperimental results thataccuratemodeling is very difficult. It may include modelaccordfairly misalignments, longitudinal and transverse variations in (power outputs and beamprofiles) reported in Section IV. Sections V and VI discuss these results and the implicagainandsaturation,waveguideimperfections(bends, roughness,andimpuritieswhich all contribute to the tions for grating-tuned devices. In an Appendix we conmode-dependent waveguiding loss), and the recoupling of sider the problems of modeling propagation behavior in radiation at each end of the guide from a diffraction grat- typical nonideal waveguides. Our present results refer to waveguides of square cross ing or a plane or curved mirror. section. Wehave tried to illuminatethreeofthesefeatures: principally the dependence of resonator transverse mode 11. COUPLING COEFFICIENTS FOR A TILTEDCASEI on mirror misalignment, but also waveguiding losses and MIRROR the line selectivity of grating-tuned devices. We believe our simple model offers a reasonable explanation of sevThewaveguide reflector weconsider consists ofthe eral hitherto puzzling observations. 2a and a near-field square waveguide aperture of width A waveguide laser usually has a long straight thin dis- tilted plane mirror (Fig. 1). Henderson [11] treated the charge tube with areflector at each end. Probably the sim- E H I I coupling losses caused by small tilts or displaceplest and most common design has plane mirrors placed ments of intracavity waveguide modulators. His results apply equally to radiation recoupled into the guide from Manuscript received September 27, 1985; revised March 11, 1986. This amisalignedplanemirror,and are readily extended to work was supported in part by an SERC CASE studentship with Culham arbitrary EH,, modes (for waveguide mode notation see Laboratory. we will consider only C. A . Hill is with the Royal Signals and Radar Establishment, Malvern, the Appendix). For convenience, England. tilts around the axes (mirror diameters) lying along the x D. R. Hall is with the Department of Applied Physics, University of and y axes: a y-tilt or an x-tilt, respectively. Hull, Hull HU6 7RX, England. In the extreme near field it is unnecessary to follow IEEE Log Number 8608789. Abstract-We present a multimode theory of square-bore waveguide laser resonators with one well-aligned and one tilted plane mirror. Our theoretical results for mode content, resonator loss, and power output agree reasonably well with experimental results from an RF-excited C 0 2 laser with a 376 X 1.5 X 1.5 mm AI/AIZO3waveguide. The mode propagation coefficients of this structure are discussed. We model a diffraction grating as aplane mirror with a wavelength-dependent tilt, and conclude that previous single-mode theory has underestimated the severity of the line-hopping constraint on tunable waveguide CO, laser performance.
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0018-9197/86/0700-1078$01.00 O 1986 IEEE
HILL AND HALL: WAVEGUIDE LASER RESONATORS WITH TILTED MIRROR
I079
Square Waveguide
rn’rror
Fig. I . Case I waveguide reflector with tilt.
Henderson’s general method of expanding the guide field in Gaussian-Hermite modes. The coupling loss is
r
= 1 -
l c rl
TILT ANGLE
(
MRAD
)
(a)
2
(1) where the amplitude couplingcoefficient c, is given by the overlap integral
EH11-EHP1 \p=l
where the radiated waveguide field E, and the reflected field E, are expressed in the misaligned coordinate system (x’, y ’ , z’) of the “tilted” guide. Following Henderson further, we have x =x’
+ + z’ sin + z z’ cos + - y’ sin + (3) for they-tilt shown in Fig. 1, and the corresponding transformation for an x-tilt. For z’ = 0 and +
