Maximising transmission through a Fabry-P´erot cavity requires control of a number of
factors including:
Chapter 4 Cavity Dynamics and Control
Figure 4.3: A Standing Wave Resonating in a Cavity. When the cavity length is an exact number of half wavelengths the light circulates and each reflection constructively reinforces new light entering the cavity. The energy builds to many times the incident energy with a small portion of the circulating power being transmitted each time. (Again the beams have been shown as slightly separated for clarity but in fact they lie on top of each other.) Beams of similar but slightly different wavelengths begin to destructively interfere with themselves as they circulate, preventing the build up of power of that wavelength in the cavity. This light is reflected back from the cavity rather than resonating within it.
2. The degree to which the cavity is impedance matched. (Cavities which are not impedance matched reflect more light leaving less for transmission.)
3. The match of the incident beam’s wavelength to the cavity’s length.
4. The alignment and offset of the beam with respect to the cavity’s optical axis. 5. The position and size of the waist of the beam (which uniquely determines its diver-
gence).
6. The quality of the beam itself, specifically it’s wavefront.
Other factors which may have an effect are the line width and polarisation of the beam.
4.3.1 Mechanical Attributes
The first three of the attributes discussed above are mechanical characteristics of the cavity and cannot be controlled by manipulating the shape of the wavefront. For the experiments reported in the following chapter, the cavity was accepted as delivered (it was a good quality cavity) and the frequency of the laser was locked to the cavity length using Pound Drever Hall Locking (see section 4.4).
Cavity Structure
Clearly the physical characteristics of a cavity affect the fraction of incident power it can transmit. In particular, if the mirrors are of low quality or are misaligned, losses will be large, transmission will be minimal or, in extreme case, non-existent.
4.3 Maximising Cavity Transmission
Degree of Impedance Matching
Equation 4.4 showed that the reflection coefficients of the two mirrors comprising the cavity place an upper limit of the amount of incident light it is able to transmit. If these are equal (and the cavity is lossless and its length is an integer number of half wavelengths) 100% can be transmitted. If not, this amount is reduced by an amount determined by equation 4.4. For the cavity used in the experiment the reflection coefficients close to equal
at r1 = r2 = 99% so the maximum power of a mode matched wavelength that the cavity
can transmit is close to 100%.
Locking Cavity Length to Laser Frequency
In the real world the frequency of the laser and the length of a cavity are continually changing, see page 254, Ref [67]. For example, the cavity used in the alignment optim- isation experiment (Chapter 6) was 30 cm long and made of aluminium. Aluminium has
a coefficient of expansion of 2.5.10−5 C−1 which meant that every change in temperature
of 0.1◦C resulted in a change of cavity length of 0.75 microns. For a round trip within
the cavity, this is approximated 1.5λ for 1064 laser light. Similarly the frequency emitted by the laser is dependent upon the incident current, the ambient temperature and the behaviour of the laser’s gain medium, Ref [67].
When transmitted power is maximised, reflected power is minimised. Figure 4.4 presents the results of a simulation which shows how the power reflected from a cavity depends upon how closely the cavity length is an exact number of half wavelengths. Here, because
the reflection coefficients are not equal (R1 = 98.5% and R2 = 95%), the cavity is not
impedance matched and there will always be some reflected power. The reflected power is used to generate an error signal to lock the laser frequency to the cavity length and this is explained in section 4.4 below.
4.3.2 Higher Order Modes
Figure 4.5 shows the power transmitted by a cavity. It is plotted in the frequency domain as the frequency of the incident light is swept across multiple resonances. The power transmitted when the cavity resonates in the fundamental modes are shown in red and are separated by an FSR.
Other modes lie between the fundamental modes and are also separated by an FSR, (see section 4.6 and Ref [68]). In general, the odd modes are due to misalignment while the even modes are due to errors in waist size and position. Mode matching a cavity involves minimising these higher order modes and coupling as much of the power as possible into
the fundamental or T EM00mode.
4.3.3 Mode Matching
Three of the requirements for maximising transmission are concerned with matching the position and shape of the beam to the cavity and are the factors optimised in the following experiments. This optimisation is termed ”mode matching” and involves:
1. Ensuring the beam is aligned to the optical axis of the cavity (tilt and offset). 2. Ensuring the beam’s waist is the correct size and in the correct position.
Chapter 4 Cavity Dynamics and Control
λ/2 = 532nm
A B
Figure 4.4: The power reflected from a cavity changes as the length changes, being minimal when the cavity length is an exact number of half wavelengths. For a lossless cavity, shown here, it falls to zero. The blue circle shows a magnified picture of the bottom of the trough.
Frequency
Pow
e
r
FSR
TEM00 TEM00 TEM00
TEM10 TEM 20 TEM30
Figure 4.5: An illustration of the power transmitted by a poorly mode matched cavity in the frequency domain as the frequency of the incident light is swept across multiple resonances. The fundamental modes are shown in red and are separated by an FSR. Other modes lie between the fundamental modes.
3. Ensuring the beam is Gaussian in shape.
4.3 Maximising Cavity Transmission
Tilt and Offset
Managing tilt and offset is a common experience for anyone constructing an optical layout and typically involves adjusting two mirrors in the optical path, one far from the target (in this case a cavity) to manage offset and one close to the target to manage tilt, figure 4.6. Off course, as neither mirror is on top of the target or infinitely far from it, adjustments couple into each other but, in general, they are relatively distinct. The distance to the far mirror can be reduced by inserting a lens. Essentially it is a question of leverage. If a long stick is held at one end, (A), and pivoted the other end, (B), which is pointing at a target, B moves around considerably, producing a large offset from the original position. If the same stick is held near B and pivoted, the end B hardly moves at all and only the angle it presents to the target changes.
Far Mirror Close Mirror Diagram 1 Far Mirror Close Mirror Diagram 2 - Offset Far Mirror Close Mirror Diagram 3 - Tilt
Figure 4.6: Tilt and Offset can be managed reasonably independently using two adjustable mirrors (Diagram 1), one far from the target and one close to it. When the far mirror is adjusted the beam moves as shown by the blue trace (Diagram 2), affecting the offset of the beam. The mirror close to the target is to close to create much offset and only tilts the wavefront as in the green trace (Diagram 3).
Waist Size and Position
The connection between waist size and position is more complex and is shown in figure 4.8. Finding a particular waist size at a particular position involves very careful placement of lenses and this is turn is impacted, in very practical terms, by the availability of the specific lenses needed to create a particular optical layout, and the extreme dependence of waist position on lens position. This is most easily understood by considering that small changes in lens position produce very large changes in waist position. For example, a lens will position the waist:
• at its focal length for a collimated beam which is typically of the order of a tens or hundreds of millimetres (or, if fact, for a Gaussian beam just in front of its focus, Ref [68]);
• at infinity for a beam expanding from the focus of a lens place a focal length away; • somewhere between the focal length and infinity for a beam emerging from some-
Chapter 4 Cavity Dynamics and Control
Figure 4.8 shows how waist size and position are affected by relatively small changes in the curvature of the two SLMs. The curves generates were developed using Alexei Ourjoumtsev’s modelling tool once again using the following set up:
• A beam with a waist of 0.55mm at a point 1600mm before the first SLM was es- tablished and turned into a roughly collimated beam of diameter 3.5mm using three lenses, figure 4.7, - f1479.6 at -1300mm, f50 at -640mm and f300 at -288.7mm. (The f1470.6 lens is not, of course, available commercially and is included to model char- acteristics equivalent to those of the beam created in the experiment.) ;
• Two SLMs are inserted, initially flat (i.e. finf inite) at the 0mm point and the 1200mm
points;
• The beam is reduced down using a telescope to generate a waist of 0.55mm at 1602mm, 602mm from the second SLM.
This approximates the dimensions of the experiments presented in the following chapter. The model shows how sensitive the waist position, in particular, to the the placement of the optical elements. The curves were generated by varying the focal lengths of the SLMs around the zone of interest as follows:
• SLM1 at 0mm focal lengths used -1000, -2000, -4000, -5000, -5500, -6000, -6500, -7000, -7500, -8000, -9000, -10000, -11000, -12500, -15000, -17500, -20000, -30000, -40000, 40000, 30000, 25000, 20000, 17500, 15000, 12500, 10000, 8000, 6000, 4000, 2000, 1000.
• SLM2 at 1200mm focal lengths used -8500, -12500, -17000, 17000, 125000 and 8500. giving a total of 32 data point for each of the six curves.
One of the goals of the method used in these experiments is to be able to make fine adjustments to a beam’s divergence without having to move optical components with high precision and these curves illustrate just how sensitive these adjustments are. They also show why two SLMs are required. A single actuator is insufficient to optimise both waist size and waist position.
A Gaussian Beam
The shape of the wavefront is also important. Each distortion caused by the quality of the laser or imperfections in the optical chain, potentially translates into a loss of transmitted power. Generally, traditional optical components cannot repair flaws in the beam shape. The Hartmann sensor, for example, was able to find an almost invisible flaw in a LIGO test mass Ref [32] but could do nothing to rectify the problem. This method may be able to not only identify some of these flaws but compensate for them.