Different critical geometries for half-symmetric laser resonators

Volume 7l, number 3,4

OPTICS COMMUNICATIONS

15 May 1989

DIFFERENT CRITICAL G E O M E T R I E S FOR HALF-SYMMETRIC LASER RESONATORS J.P. TACHI~, A. Le FLOCH and R. Le NAOUR Laboratoire de Spectroscopie du Solide et d'Electronique Quantique, Physique des Lasers, U.A. CNRS 1202, Universit~ ite Rennes, L F-35042 Rennes Cedex, France

Received 20 January 1989

It is shownthat three different critical geometriescan exist for a half-symmetricresonatorwith an internal lens of variable focal length. Indeed, in addition to the previouslydemonstrated critical geometryleadingto a mode size on the spherical mirror which is stationarywith respectto the focallength, two other geometriesgivea stationary mode size on the plane mirror. An experimental arrangement leads to a verificationof the theoretical predictions.

1. Introduction A laser oscillating in the fundamental (TEM0o) mode produces a beam with a gaussian profile. The beam contour in the resonator is determined by the curvature of the mirrors and the distance between them [ 1 ]. The transverse extension of the beam is sensitive to internal lens, or lenslike effects, and this can lead to large variations of the basic parameters of the laser itself. For example the asymmetry of a nonlinear resonance such as the Lamb dip, or the saturated absorption peak, is very dependent on the mode size variations when the laser frequency is scanned [2,3 ]. On the other hand in high power lasers, such as N d : Y A G lasers, the so-called dynamic stable resonators have been proposed to minimize the transverse variations of the beam size [4,5]. It is worthwhile to note that in this case the insensitivity of the mode size is found around a finite value of the focal power of the active medium whereas in the Lamb dip type resonance the insensitivity is found around the zero value of the focal power. Here we restrict our discussion to the latter case and we consider the widely used half-symmetric resonator. For this geometry where the diffraction losses are essentially located at the spherical mirror end, we have already shown the existence of the so-called Also with Universit~ de Nantes, F-44072 Nantes Cedex 03, France.

critical geometry which leads to a stationary mode on this spherical mirror. Nonlinear resonances, such as the Lamb dip, appear to be very sensitive to the frequency-dependent diffraction losses due to the mode size variations. However, with the critical geometry, a symmetric Lamb dip [ 6 ] and a symmetric CH4 saturated absorption peak [ 7 ] are obtained. One may ask then, as suggested by Giacomo [8 ], if it possible to find geometries leading to a stationary mode size on the p l a n e mirror. It is the aim of this paper to investigate such possibilities.

2. Theory Consider the half-symmetric resonator with an internal lens of focal length f s h o w n in fig. 1 (a). The radii, of curvature of the mirrors are R~ and R2 (here R~= oo), d is the distance between the mirrors and d~ characterizes the position of the lens. The spot sizes of the gaussian beam are w~ and WEon the mirrors M1 and MErespectively and WEis the spot size at the lens position. As shown by Kogelnik these quantities can be calculated by using the ABCD law [ 1,9 ]. First, let us recall the derivation of the critical geometry related to the insensitivity of w2 on the spherical mirror. Calculating w2 and W2o (with and without lens) and requiring w2=W2o, we obtain the quadratic equation

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pointing out that a stationary mode on the mirror M2 implies that the mode is stationary between the lens and the mirror M2 (fig. 1 ( b ) ) , the preceding condition for the insensitivity o f w2 may be derived from

d ~d~

: ~

d 2 . - -

(s) Fil c o

Ra

f

15 May 1989

dWL/dWl = O.

(5)

The search of the insensitivity of w] on the plane mirror may be done by the two above mentioned methods. For instance using the second approach, we write (fig. 2 ( a ) )

dwL/dW2 = O. Fig. 1. (a) The half-symmetric resonator with an internal lens of variable focal length. (b) The insensitivity of w2 to focal power variations leads to no change in the hatched portion of the gaussian beam.

(6)

Starting from the virtual waist at the midplane o f the M2 mirror equivalent lens (fig. 2 ( b ) ) and using the A B C D law, we obtain 2 2[A2+ (2B/zcw2) 2 ] WL=W2

(f-d,)d2-Rz(f-d,

)d+dZ(f+d~ ) = 0 .

( 1)

For vanishing focal power values ( Ifl >> Re), as those involved in L a m b dip experiments, this equation leads to the so-called critical length dc which is given by

d 2 - R 2 d c +d2=O .

(2)

For a given cavity (R2 fixed ), if the lens (or the lenslike effect) is located at the middle o f the cavity, which is the case in most c o m m o n lasers, we find dc = 4R2/5 [ 6 ]. However if the lens is m o v e d inside the cavity, eq. (2) describes the variations o f the critical length de. Since in the empty cavity, we have the usual relation

7t2W2o/,~2=dR2 - d 2 ,

~e =A2~2 +B2/~2 •

(8)

Eq. (6) can then be written as d~L/d~2 = 0 ,

(9)

and this leads to

~2 = ( B / A ) 2 .

(10)

Since we look for solving eq. (10) around 1 / f = O, we take for w2 the value o f the empty cavity and we have

~2 =dR22/ ( R2 - d ) .

(11)

M

I

R1=co '

d

~

M2

(a)

R2

(4)

and variations o f de~R2 versus d~/R2 have already been represented graphically in ref. [ 7 ]. These results may also be obtained by an equivalent approach suggested by G i a c o m o [8 ]. Indeed, 180

where A = 1 - d2/R2 and B = d2 are the corresponding elements o f the A B C D matrix between the two considered planes. With ( = n w 2 / 2 , eq. (7) becomes

(3)

we deduce from eqs. (2) and (3), that for the critical geometry we can write d] = nW21o/~,i.e. d] is equal to the Rayleigh range associated with wm (see, e.g., ref. [ 10 ] ). In other words, for this critical geometry, the lens is located where the m i n i m u m radius of curvature occurs for the wavefront. Eq. (2) can also be written as

( dc/R2 -- 1 / 2 ) 2 + ( dl/R2)2= 1 / 4 ,

(7)

l

Fig. 2. (a) When w~ is insensitive to focal power variations, the gaussian beam does not vary in the hatched region. (b) The unfolded cavity.

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From eqs. (10) and ( 11 ), we find that the geometries leading to the insensitivity of wt are given by 2d~ - (3R2 +4dl )d 2

(12)

+(R2+2d2+4dlR2)dc-d2R2=O.

Solutions of eq. (12) are shown in fig. 3. They correspond to small values of dl, i.e., d~/R2 <. (dl/R2) m, where the maximum value of d~/R2 is

(d,/R2)m=(½x/~- l )[½( l + v/5) ] '/2 =0.15014...,

(13)

and is indicated by the arrow in fig. 3. Note that, for a fixed d value, variations of d~ allow us to observe only one solution. To have a better understanding of the changes in the spot size w~ at the plane mirror as the focal power of the internal lens varies, we have studied the relative variations Awl/W~o= ( Wm-- Wlo) /Wlo, Wlo being the spot size without lens in the resonator. These variations were calculated by the ABCD method for

,o]

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geometries around one critical geometry and plotted versus 1/f. The results are shown in fig. 4 for a given cavity length (R2 = 60 cm, d = 54 cm, i.e., d/RE = 0.9 ) and three different positions of the internal lens, the various parameters being chosen to be suited to the experimental setup described below. The three geometries considered here correspond respectively to the points L, M and N, in fig. 3. Point M, with an abscissa d~/R 2= 0.15 as obtained from eq. ( 11 ) for de~R2 = 0.9, corresponds to the critical geometry. Inspection of fig. 4(b) clearly shows that, as expected in this case, w~ is practically independent of 1 / f (more accurately, Aw~/wto< 1× 10 -4 for If[ i> 15 m). For the resonator associated with point L (fig. 4(a) ), note that a convergent lens reduces the spot size w~ at the plane mirror whereas a divergent lens increases this spot size. This usual behavior is not observed for the resonator associated with point N (fig. 4 ( c ) ) :

~( L

\\'x.

[/ I/

0.1

I -0.05

L \. t /

Awl/wlo('/.)

/

o.5~'

0.05

M

/J

"/

/

,

/////// O.5

(b)

0.1 o.os

-0.05

l

1/f(m-11

-0.2

, ,¢/ "

J."

/,///

(a)

N

dl

0

R"=

Fig. 3. In the d~/R2-d/R2 plane, the critical geometries yielding a spot size wt at the plane mirror which is insensitive to focal power variations around zero are represented by the solid line. Those leading to the insensitivity of ,,2 are represented by the dashed-dotted line. Note that for a given position of the lens three critical geometries exist for a half-symmetric resonator, provided the condition d~/R2~ 0.15 (see eq. ( 13 ) ) is satisfied. Only the points situated above the straight dashed line (slope unity) represent an actually realizable resonator.

, . 0 . 1 (c) 0,05

J Fig. 4. Relative variations Aw~/W,o versus the focal power 1/ffor three different positions of the internal lens in a given resonator (R2=60 cm, d = 5 4 cm), (a) dt = 14 cm, (b) d, = 9 cm and for this critical geometry w, is as expected practically independent of the focal power (in the considered range), (c) dt = 7 cm. These three geometries are represented by the points L, M and N, respectively in fig. 3.

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3. Experiments We propose now to experimentally demonstrate the existence o f critical geometries related to the insensitivity o f the spot size w~ at the plane mirror. To achieve this, we use the lenslike effects induced in the active m e d i u m o f a laser by the transverse inhomogeneity in saturation due to the gaussian profile of the mode. The focal power associated with these lenslike ~ffects varies around zero when the frequency is scanned across the laser line. In fact, we use the sensitivity o f the L a m b dip asymmetry to small diffraction losses variations due to the frequency dependent lenslike effects. In a laser oscillator small variations o f losses result indeed in rather large variations in the output power. The experiment is performed with a 3.39 ~m HeNe laser. The discharge tube has a short active length ( l 0 c m ) and is filled with a 4: 1 mixture o f 3He-Z°Ne gases at a total pressure o f 0.4 Torr. The resonator consists o f one plane mirror and one spherical mirror ( R 2 - - 6 0 c m ). The cavity length is d = 54 cm and the position o f the active m e d i u m between the mirrors can be changed. To restrict the oscillation in the

Y

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fundamental mode, a 1.2 m m diam aperture is located in front o f the plane mirror ( 1.2 m m ~ 2.4 Wl ). The inner tube diameter ( ~ 7 m m ) is chosen greater than the mode size in order to avoid lenslike effects associated with the transverse inhomogeneity in the unsaturated population inversion profile (see, e.g., ref. [ 11 ] and references therein). As the laser frequency is scanned, the output power o f the laser is measured for three different positions o f the active m e d i u m which correspond to working points L, M and N, in fig. 3. The experimental results are given in fig. 5 (a). They clearly show that for the critical geometry a symmetrical Lamb dip is obtained and that the L a m b dip asymmetry changes sign around the critical geometry. For a laser exhibiting a Lamb dip, the output power versus the frequency may be written as [ 2 ] P(X) =

A [ 1 - B ( X ) e x p ( X 2) ] 1 +DC2/(CEd-X2)

(14)

'

where A is a scale factor and X = ( v - / / o ) / A P D is a normalized frequency, A/2D being the Doppler linewidth (half-linewidth at the 1/e point). The factor C = ~ / A P o characterizes the inhomogeneous broad-

v

u

(b)

Fig. 5. (a) Experimental and (b) theoretical laser output power versus frequency ~ for three different positions of the internal lens corresponding to the points L, M and N, in fig. 3. In fact, for the experimental curves the distance between the middle of the active medium and the plane mirror is successively around 16, 11 and 9 cm instead of 14, 9 and 7 cm. The theoretical curves are plotted for AZ,D= 175 MHz, B (mean value)=0.72, C=0.14 and D=0.5. 182

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ening o f the laser line and ? is the homogeneous linewidth. D is a factor which reduces the strength of the Lamb dip and is due to cross relaxation (velocitychanging collisions and radiation trapping [ 1 2 ] ) . The quantity B(X) is the ratio of the overall losses, including diffraction losses, to the unsaturated gain and its variations with frequency are essentially related to the dispersion associated with the homogeneous (lorentzian) contribution to the laser line [ 2, l 1 ]. For the different positions o f the active medium considered in our experiment, the magnitude o f the diffraction losses variations is, in first approximation, indicated by the slope o f the curves given in fig. 4. So, using eq. (14), we can theoretically plot the output power versus the frequency and the results are shown in fig. 5 ( b ) . They compare favourably with experimental data and this good agreement confirms the validity o f our analysis.

4. Conclusion For a half-symmetric resonator with an internal lens of weak and variable focal power, we have shown that three different critical geometries are available for some given values o f d~, provided the cavity length satisfies the condition d > RE/2. Although these critical geometries do not have the same convenience, knowledge o f their existence is needed so as, for example, to avoid spurious asymmetries, due to /enslike effects, in nonlinear intracavity resonances.

15 May 1989

Acknowledgement

The authors wish to thank P. Giacomo for his helpful suggestions. This work was supported in part by the "Centre National de la Recherche Scientifique" and by the "Direction des Recherches Etudes et Techniques".

References

[ 1] H. Kogelnik and T. Li, Proc. IEEE 54 (1966) 1312. [ 2 ] A. Le Floch, R. Le Naour, J.M. Lenorrnand and J.P. Tachd, Phys. Rev. Lett. 45 (1980) 544. [3] A. Le Floch, J.M. Lenormand, R. Le Naour and P. Brun, IEEE J. Quantum Electron. QE-19 ( 1983) 1474. [4] J. Steffen, J.P.I./~rtscher and G. Herziger, IEEE J. Quantum Electron. QE-8 ( 1972) 239. [5] J.P. I35rtscher, J. Steffen and G. Herziger, Opt. Quantum Electron. 7 (1975) 505. [ 6 ] A. Le FIoch, J.M. Lenormand, R. Le Naour and J.P. Tach6, J. Physique-Lenres 43 (1982) L-493. [ 7 ] A. Le Floch, J.M. Lenormand, G. Ropars and R. Le Naour, Optics Len. 9 (1984) 496. [ 8 ] P. Giacomo, rapport BIPM-86/4 (1986). [9] H. Kogelnik, Bell Syst. Tech. J. 44 (1965) 455. [10]A.E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986)chap. 17. [ 11 ] J.P. Tach6, A. Le Floch and R. Le Naour, Appl. Optics 25 (1986) 2934. [ 12 ] P.W. Smith, IEEE J. Quantum Electron. QE-8 ( 1972) 704.

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