Unidirectional diode-laser-pumped Nd:YAG ring laser with asmall magnetic field
W. R. Trutna, Jr., D. K. Donald, and Moshe Nazarathy
Hewlett-Packard Laboratories, 1651 Page Mill Road, Palo Alto, California 94304
Received December 2, 1986; accepted January 9, 1987
Single-mode operation of a monolithic semiconductor-laser-pumped unidirectional ring Nd:YAG laser has beendemonstrated at 1319 and 1338 nm. The ring design optimizes the polarization eigenvalue difference betweencounterpropagating modes of the ring. The laser threshold is 4.5 mW, and the magnetic-field requirement forunidirectional operation is approximately 100 G.
Multilongitudinal-mode oscillation occurs in Nd:YAGlasers because standing waves in the YAG crystal de-plete the gain in a spatially inhomogeneous way.' Tosuppress multilongitudinal-mode oscillation inNd:YAG lasers, discrete element versions of unidirec-tional ring lasers have been constructed,2A as has themonolithic isolated single-mode end-pumped ring(MISER) design of Kane and Byer.5 We have built anend-pumped single-mode unidirectional ring laserbased on a modified MISER design. The laser oscil-lates on the 1319- and 1338-nm Nd:YAG transitionswith a 4.5-mW pump threshold and a magnetic-fieldrequirement for unidirectional lasing of only 100 G.The main advantage of our new quasi-planar design isthe low magnetic-field requirement; in the MISER therequired magnetic field is impractically large in small,low-threshold devices.
This Letter contains a description of the quasi-pla-nar ring design, a comparison of the design with theMISER, a summary of the Jones calculus polarization-mode analysis, a pump-laser threshold calculation,and experimental results.
Unidirectional ring lasers contain three essential el-ements: a polarizer, a half-wave plate (or equivalent),and a Faraday rotator. For one direction of propaga-tion around the ring, the polarization rotations causedby the Faraday rotator and the half-wave plate cancel,yielding a low-loss linear polarization eigenmode. Forthe opposite direction of propagation the polarizationrotations add, yielding a higher-loss elliptical polariza-tion eigenmode. Unidirectional lasing occurs whenthe difference between the loss eigenvalues is suffi-ciently large.
The polarizer, the half-wave-plate equivalent, andthe Faraday rotator are all embodied in our quasi-planar ring laser sketched in Fig. 1. With a magneticfield H present in the direction shown, the YAG crys-tal itself acts as the Faraday rotator, the out-of-planetotal internal reflection (TIR) bounces labeled B, C,and D act as the half-wave plate, and the output cou-pler (mirror A) acts as a partial polarizer. In addition,mirror A is spherical for cavity stability. The essen-tial difference between the MISER design5 and thecurrent one is that in the current design the ray pathABCD is quasi-planar with the angle a of the order of1°, whereas in the MISER design a = 90°.5
The principle of operation for both designs consistsof emulating the ideal equivalent discrete element de-sign of a half-wave plate with a fast axis rotation anglethat is half of the Faraday rotation angle. Since theFaraday rotation is small, the equivalent wave-platerotation angle should also be made small. We showbelow that almost flattening the ray path leads to theoptimal value of the small-wave-plate rotation angleas well as to 1800 relative retardation of the equivalentwave plate, whereas the MISER design amounts to alarge rotation angle and a relative retardation thatdeviates from 1800.
The polarization state and loss for the clockwise(CW) and counterclockwise (CCW) oscillation direc-tions can be computed from the eigenvectors and ei-genvalues of the Jones matrix describing one roundtrip in the ring. Before analyzing the entire ring, weanalyze three TIR's and compared them for both theMISER and the quasi-planar ring geometries.
An analysis of the differential phase shift between sand p polarizations for a TIR mirror as a function ofangle of incidence is presented by Born and Wolf.6Conceptually, one can represent each TIR reflector asa wave plate whose axes are defined by planes parallel(p polarization) and perpendicular (s polarization) tothe plane of incidence. If the three bounces labeled B,
[TTT1IBA-�-DILi�JJFig. 1. The quasi-planar ring design.
Fig. 2. The three TIR's of Fig. 1 unfolded into a linear pathwith equivalent wave plates. The coordinates are refer-enced to the ADB plane. S represents the wave-plate axisequivalent to the axis perpendicular to the plane of inci-dence of the TIR mirror.
C, and D in Fig. 1 are unfolded into a linear path, thechange in polarization state can be described by threewave plates with axes at angles relative to the ABDplane, as is shown in Fig. 2. Let us denote the retarda-tions (phase-shift differences between the s and p po-larizations) of each of the mirrors by Rb, Rc, and Rd,respectively, and let 01 be the angle between the ADBand ADC planes of Fig. 1 and 02 the angle between theADC and DCB planes. In the MISER design point Cis taken on the upper face of the crystal; thus 02 isapproximately 900, resulting in a single equivalentwave plate with overall phase shift Rb - RC + Rd in-clined at angle 01 to the reference ADB plane. In thepublished MISER design, 01 is 200 and Rb - Re + Rd =
960. In other words, the net effect of the out-of-planebounce is to insert a quarter-wave plate inclined at 200into an equivalent planar ring resonator. On the oth-er hand, in the case of the quasi-planar ring, Oi - 02 -10. In this case, the retardations add rather thansubtract, leading to an overall phase shift of Rb + RC +Rd, which by careful choice of incident angles can bemade precisely 1800. In YAG, with refractive index1.82, the incident angle on mirrors B and D is 480, andthe incident angle on mirror C is 54°. This impliesthat the incident angle on the output coupler A is 300.Thus the TIR reflections of the quasi-planar ringamount to an equivalent half-wave plate rotated by asmall angle in an equivalent planar ring cavity.
The Verdet constant for YAG at 1064 nm is 1.80 X10-7 rad/mm-G.5 For a 6-mm-long sample, in a mag-netic field of 1000 G, the Faraday rotation is less than0.10. Based on the previous argument that the half-wave plate should cause a polarization rotation that iscanceled by the Faraday rotation, it follows that thering should be nearly planar, so the geometric rotationangle is small. The small geometric rotation angleand the 1800 phase shift are features of the quasi-planar ring design but not of the MISER design.
The output-coupling mirror reflectivities are chosento maximize the differential loss between the CW andCCW directions with the additional constraint thatthe output coupling be less than 0.2% in order to have alow pump threshold. Our device has s- and p-polar-ization power reflectivities of 99.9 and 85% with arelative phase shift of 180°. A i800 phase shift ispreferred because low-loss linear polarization remainsan eigenmode of the ring when the Faraday rotationcancels rotations caused by the other cavity elements.
A Jones matrix analysis was carried out on the qua-si-planar ring design. The dimensions L and Wshown in Fig. 1 are 6.0 and 4.88 mm. The radius ofcurvature of the output coupler is 20 mm. Some re-sults are plotted in Fig. 3. There are four polarizationeigenvalues, two CW and two CCW. Figure 3(a)shows the magnitude squared of the largest polariza-tion eigenvalue, y, as a function of the out-of-planerotation angle a (Fig. 1) for several magnetic-fieldmagnitudes. Since 1 - ,y12 gives the power loss perround trip, the polarization mode with the largesteigenvalue is the one that oscillates. Note that forlarger magnetic fields, larger rotation angles are re-quired to maximize the eigenvalue. This is consistentwith the argument that larger rotation of the effectivehalf-wave plate is required to compensate for the larg-er Faraday rotation. The differential loss betweenthe larger CW and CCW eigenvalues is shown in Fig.3(b). The largest differential loss occurs at about 50,but in order to keep the output coupling below 0.2% arotation angle of 1.25° was chosen for the design. Em-pirical evidenice25 indicates that a conservative esti-mate of the differential loss required for unidirection-al operation is 0.01%. This is easily achieved with thedesign represented by Fig. 3.
For a 1319-nm YAG ring laser that is semiconduc-tor-diode-laser pumped, the lasing threshold should
100.0
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W99.4(.
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8
z 6
OW4
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0 2 4 6 8 10OU DEGREES
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0 2 4 6
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Fig. 3. (a) The magnitude squared of the lowest-loss polar-ization eigenvalue, plotted against the out-of-plane rotationangle a. The secondary variable is the magnetic-field mag-nitude. (b) The differential loss plotted versus a for thesame magnetic fields as in (a).
250 OPTICS LETTERS / Vol. 12, No. 4 / April 1987
E
00.
0:
:3
8
6
4
2
00 10 20 30 40 50
INPUT POWER (mW)
Fig. 4. The dependence of the unidirectional ring-laseroutput power on (0) the 590-nm dye-laser pump power and(X) the 795-nm semiconductor laser pump power.
not exceed about 10 mW because of the power limita-tions of available semiconductor-diode lasers. Thisvalue is much lower than the achievable thresholdbased on the published version of the MISER, whichhas a length L = 38 mm and a threshold pump power of150 mW for 1064-nm oscillation. The difficulty oflowering the threshold to below 10 mW is compoundedby the fact that the inherent gain of the 1319-nm line isonly one fourth that of the 1064-nm line.7
Threshold reduction is accomplished by reducingthe output coupling from 1 to 0.2% and by reducing themode volume, which is accomplished by reducing thesize of the ring. A smaller mode implies that thepump beam can be focused more tightly, thereby in-creasing the gain, which is proportional to pump pow-er density. This result can be derived from a simplerate-equation analysis and is expressed mathematical-ly in the following equation 6:
Pthreshold =AhcL
Xpra(l - e-aPD)
where A is the mode cross-sectional area, L is thefractional round-trip loss, a- is the stimulated emissioncross section (= 0.42 X 10-19 cm2,6,7 r is the upper laserlevel lifetime (= 230,gsec),8 Xp is the pump-laser wave-length, cp is the pump-laser absorption coefficient, his Planck's constant, c is the velocity of light, and D isthe ring perimeter.
The estimated threshold for 590-nm dye-laserpumping of the quasi-planar ring is 5.5 mW, assuminga 0.2% round-trip loss, apD >> 1, and a 50-Mm moderadius. The predicted threshold for 795-nm diode-laser pumping is 4.1 mW.
There is a trade-off between the goals of reducingthreshold and maximizing the differential loss toachieve unidirectional oscillation. Reducing the ringsize to reduce threshold decreases the Faraday rota-tion, which in turn reduces the differential loss. Forthe quasi-planar ring both low-threshold and unidi-rectional operation are readily achievable. However,this does not apply to the MISER design, as was re-vealed by a comparative study of the differential lossof the MISER and the quasi-planar designs. For ringsof approximately the same dimensions and output
coupling, the differential loss of the MISER design is300 times smaller, leading to impractically large mag-netic fields for a unidirectional semiconductor-laser-pumped 1319-nm-wavelength MISER.
The quasi-planar ring-laser design described abovehas been fabricated and tested. Testing was donewith both a 590-nm dye laser and a 795-nm diode laseras the pump source. Some of the key experimentalresults are presented here.
The pump threshold for the ring laser is 6 mW forthe dye-laser pump and 4.5 mW for the semiconduc-tor-laser pump, in close agreement with the predictedvalue. The magnetic field required to induce unidi-rectional operation is only about 100 G. The unidirec-tional output power versus pump power is measured inthe presence of a 600-G magnetic field and is shown inFig. 4.
Unidirectional lasing is observed up to the maxi-mum available pump power of 60 mW. Above thresh-old, the slope efficiency is about 17%.
The laser spectrum was measured with a scanningmonochromator. Laser output was found at both1319 and 1338 nm. However, when the dye pumplaser was detuned slightly from the wavelength ofmaximum absorption, the 1338-nm line disappeared.We offer no explanation for this behavior. The mono-chromator has sufficient resolution, 3.5 GHz, to re-solve the longitudinal and transverse modes of thelaser, which are spaced by about 11 and 4 GHz, respec-tively. Single-longitudinal-mode lasing was alwaysobserved for both laser lines, although mode hoppingcould be induced by tilting the crystal. When thepump beam was properly aligned and focused, single-transverse-mode lasing was observed as well. The1319- and 1338-nm lines are sufficiently far apart thatthey can be easily separated with a filter or grating toensure a single-frequency output.
A single-mode unidirectional 1319- and 1338-nmNd:YAG quasi-planar ring laser has been demonstrat-ed. The quasi-planar design is an improvement overthe MISER design in that both a low pump thresholdand a low magnetic-field requirement for unidirec-tional operation can easily be achieved. A diode-la-ser-pumped quasi-planar ring is a potential candidatefor a compact, efficient 1338- or 1319-nm single-fre-quency source for coherent optical communications.
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