Optical Resonators
7.6 Mode Selection
7.6.2 Longitudinal Mode Selection
We have seen in Section7.2that the various longitudinal modes corresponding to a transverse mode are approximately separated by a frequency interval of c/2n0d.
As an example if we consider a 50 cm long laser cavity with n0 = 1, then the longitudinal mode spacing would be 300 MHz. If the gain bandwidth of the laser is 1500 MHz, then in this case even if the laser is oscillating in a single transverse mode it would still oscillate in about five longitudinal modes. Thus the output would consist of five different frequencies separated by 300 MHz. This would result in a much reduced coherence length of the laser (see Problem 7.3). Thus in applications such as holography and interferometry where a long coherence length is required or where a well-defined frequency is required (e.g., in spectroscopy) one would require the laser to oscillate in a single longitudinal mode in addition to its single transverse mode oscillation.
Referring to Fig.7.6we can have a simple method of obtaining single longitudi- nal mode oscillation by reducing the cavity length to a value such that the intermode
δν δν
Gain
Threshold
(a) (b)
c/2d c/2d
Modes with Gain > Loss
Gain
Threshold
Oscillating mode
ν ν
Fig. 7.6 (a) The longitudinal mode spacing of a resonator of length d is c/2d. Different modes having gain more than loss would oscillate simultaneously. (b) If the resonator length is reduced, the mode spacing can become less than the gain bandwidth and if there is a mode at the line center, then it would result in single longitudinal mode oscillation of the laser
spacing is larger than the spectral width over which gain exceeds loss in the cavity.
Thus if this bandwidth isνgthen for single longitudinal mode laser oscillation the cavity length must be such that
c
2n0d > νg
For a He–Ne laserνg∼1500 MHz and for single longitudinal mode oscillation one must have d < 10 cm. We should note here that if one can ensure that a resonant mode exists at the center of the gain profile, then single mode oscillation can be obtained even with a cavity length of c / n0νg.
One of the major drawbacks with the above method is that since the volume of the active medium gets very much reduced due to the restriction on the length of the cavity, the output power is small. In addition, in solid-state lasers where the gain bandwidth is large, the above technique becomes impractical. Hence other tech- niques have been developed which can lead to single longitudinal mode oscillation without any restriction in the length of the cavity and hence capable of high powers.
It is important to understand that even if the laser oscillates in a single longitudi- nal mode (single-frequency output), there could be a temporal drift in the frequency of the output. In many applications it is important to have single-frequency lasers in which the frequency of the laser should not deviate beyond a desired range. To achieve this the frequency of oscillation of the laser can be locked by using feedback mechanisms in which the frequency of the output of the laser is monitored contin- uously and any change in the frequency of the laser is fed back to the cavity as an error signal which is then used to control the mirror positions of the cavity to keep the frequency stable. There are many techniques to monitor the frequency of the laser output; this includes using very accurate wavelength meters capable of giving frequency accuracy in the range of 2 MHz.
Oscillation of a laser in a given resonant mode can be achieved by introducing frequency selective elements such as Fabry–Perot etalons (seeChapter 2) into the laser cavity. The element should be so chosen that it introduces losses at all but the desired frequency so that the losses of the unwanted frequencies are larger than gain resulting in a single-frequency oscillation. Figure7.7 shows a tilted Fabry–Perot etalon placed inside the resonator. The etalon consists of a pair of highly reflecting parallel surfaces which has a transmission versus frequency variation as shown in Fig.7.8. As discussed inChapter 2, such an etalon has transmission peaks centered at frequencies given by
νp=p c
2nt cosθ (7.37)
Fabry-Perot etalon Fig. 7.7 A laser resonator
with a Fabry–Perot etalon placed inside the cavity
7.6 Mode Selection 161
0 0.2 0.4 0.6 0.8 1
νp–1 νp νp+1
Fig. 7.8 Transmittance versus frequency of a Fabry–Perot etalon. The higher the reflectivity the sharper are the resonances
where t is the thickness of the etalon and n is the refractive index of the medium between the reflecting plates andθis the angle made by the wave inside the etalon.
The width of each peak depends on the reflectivity of the surfaces, the higher the reflectivity, the shaper are the resonance peaks. The frequency separation between two adjacent peaks of transmission is
ν = c
2nt cosθ (7.38)
which is also referred to as free spectral range (FSR).
If the etalon is so chosen that its free spectral range is greater than the spectral width of the gain profile, then the Fabry–Perot etalon can be tilted inside the res- onator so that one of the longitudinal modes of the resonator cavity coincides with the peak transmittance of the etalon (see Fig.7.9) and other modes are reflected away from the cavity. If the finesse of the etalon is high enough so as to introduce sufficiently high losses for the modes adjacent to the mode selected, then one can have single longitudinal mode oscillation (see Fig.7.9).
Example 7.9 Consider an argon ion laser for which the FWHM of the gain profile is about 8 GHz. Thus for near normal incidence (θ∼0) the free spectral range of the etalon must be greater than about 10 GHz.
Thus
c
2nt >1010Hz
Taking fused quartz as the medium of the etalon, we have n∼1.462 (atλ∼510 nm) and thus t≤1 cm
Another very important method used to obtain single-frequency oscillation is to replace one of the mirrors of the resonator with a Fox–Smith interferometer as shown in Fig.7.10. Waves incident on the beam splitter BS from M1 will suffer multiple reflections as follows:
Reflection 1: M1→BS→M2→BS→M1
Reflection 2: M1→BS→M2→BS→M3→BS→M2→BS→M1, etc.
Only oscillating mode Gain spectrum
(a)
(b)
(c)
Cavity modes
Fig. 7.9 The figure shows how by inserting a Fabry–Perot etalon in the laser cavity one can achieve single longitudinal mode oscillation
BS
M2 M3 M1
d3
d2 d1
Fig. 7.10 The Fox–Smith interferometer arrangement for selection of a single longitudinal mode Thus the structure still behaves much like a Fabry–Perot etalon if the beam split- ter BS has a high reflectivity.1For constructive interference among waves reflected toward M1from the interferometer, the path difference between two consecutively reflected waves must be mλ, i.e.,
1It is interesting to note that if mirror M3is put above BS (in Fig.7.10), then it would correspond to a Michelson interferometer arrangement and the transmittivity would not be sharply peaked.
7.6 Mode Selection 163 (2d2+2d3+2d2−2d2)=mλ
or
ν=m c
2(d2+d3), m=any integer (7.39) Thus the frequencies separated byν=c / 2(d2+d3)will have a low loss. Hence ifνis greater than the bandwidth of oscillation of the laser, then one can achieve single mode oscillation. Since the frequencies of the resonator formed by mirrors M1and M2are
ν=q c
2(d1+d2), q=any integer (7.40) for the oscillation of a mode one must have
m
(d2+d3) = q
(d1+d2) (7.41)
which can be adjusted by varying d3 by placing the mirror M3on a piezoelectric movement.
Example 7.10 If one wishes to choose a particular oscillating mode out of the possible resonator modes which are separated by 300 MHz, what is the approximate change in d3required to change oscillation from one mode to another? Assumeλ0=500 nm, d2+ d3=5 cm.
Solution Differentiating Eq. (7.39) we have δν= mc
2(d2+d3)2δd3= ν (d2+d3)δd3 which gives usδd3=25 nm.
Problem 7.3 Consider a laser which is oscillating simultaneously at two adjacent frequenciesν1and ν2. If this laser is used in an interference experiment, what is the minimum path difference between the interfering beams for which the interference pattern disappears?
Solution The interference pattern disappears when the interference maxima produced by one wavelength fall on the interference minima produced by the other wavelength. This will happen when
l=mc ν1 =
m+1
2 c ν2
, m=0, 1, 2...
Eliminating m from the two equations we get
l= c
2(ν2−ν1)=d
where d is the length of the laser resonator. Thus the laser can be considered coherent only for path differences of l=d, the length of the laser. On the other hand, if the laser was oscillating in a single mode the coherence length would have been much larger and would be determined by the frequency width of the oscillating mode only.