3.2.1 Fundamental operation
We will first consider four-wave mixing via a reflection grating. Figure 3.1 shows the formation and reading of the gain grating. Here the forward and backward pump beams (FP and BP respectively) write the grating which is then read by the signal beam (sig) producing a phase conjugate of the backward pump beam (PC).
FP
BP
sig
PC
Figure 3.1: Writing a reflection grating with the forward pump (FP) and backward pump (BP) with diffraction of a signal beam (sig) generating a phase conjugate (PC) of the backward pump.
The reflection grating can be written by passing a beam through the gain medium twice such that it intersects with itself (as shown in figure 3.2). At the point of intersection the beams interfere forming a gain grating. This grating is encoded with information about any phase distortions that the beam sees from passing through the crystal.
Seed beam
Distorted seed
beam
Doubly distorted
seed beam
Figure 3.2: Writing a gain grating between a seed beam and a distorted version of the seed beam.
Fluorescence from the amplifier will then scatter off the grating. When the fluorescence scatters it Bragg matches with the grating, counter and co-propagating with the writing beams as shown in figure 3.3.
Figure 3.3: Fluorescence scattering off the grating.
The scatter that propagates along the path of each beam will have the same wavefront as that beam. The scattered light that counter-propagates along the path of the beams will have a wavefront which is the reverse of that beam, in other words its phase conjugate.
Scattered fluorescence will propagate around the loop in the opposite di- rection to the initial writing beam (following the path of the backward pump). When it reaches the amplifier again a portion will scatter off the grating back along the path of the backward pump again (figure 3.4).
If more of the beam is scattered than is incident on the grating (this is possible due to the gain seen in the medium) then the power in the loop will build until threshold is reached (the gain is equal to the losses) and it reaches a stable oscillation. The wavefront of this beam is defined by the spatial information encoded in the gain grating and is, as previously stated, the phase conjugate of the backward pump.
The portion of the oscillating beam that is not scattered by the grating will leave the crystal re-tracing the path of the seed beam as its amplified phase conjugate (Figure 3.5).
Figure 3.4: Build up of fluorescence into a steady state oscillation.
Figure 3.5: The phase conjugate oscillator emitting a phase conjugate of the signal beam.
3.2.2 Longitudinal modal structure
In the previous chapter the formation of gain gratings in a laser medium was described. Figure 2.25 showed that a gain grating is formed with a
π phase shift relative to the interference pattern. This phase shift can be used to describe the longitudinal modal structure of the oscillation [1].
3.2.2.1 Writing the grating
F
1F
3Path length = L
-F
3F
1F
interfereFigure 3.6: The phase diagram of the interference pattern formed whilst writing a reflection gain grating
The phase of a gain grating can be calculated from the phase of its writing beams. Figure 3.6 shows the writing of a gain grating using a self inter- secting geometry. Here the seed beam has an instantaneous phase of Φ1
and a wave vectork1. When this is passed around the loop and back into
the amplifier it has a phaseΦ3given by
Φ3 = Φ1+k1L+ Φother (3.1)
where Lis the length of the loop and Φother accounts for any other phase effects present in the loop.
The interference pattern at the intersection of the two beams will have a phase ofΦinterf erewhich can be calculated from
Φinterf ere = Φ1−Φ3 (3.2)
= Φ1−(Φ1+k1L+ Φother) (3.3)
Φinterf ere = −(k1L+ Φother) (3.4)
The phase of the gain grating can now be calculated by adding aπ phase shift to the phase of the interference pattern
Φgrating = Φinterf ere+π (3.5)
Φgrating = π−(k1L+ Φother) (3.6)
3.2.2.2 Reading the gain grating
The gain grating is then read by scattered ASE which has built up to form a stable resonant loop. A stable beam is produced passing around the loop anticlockwise and reflecting off the gain grating as shown in figure 3.7.
F
2F
4Path length = L
F
gratingF
2F
4Figure 3.7: The phase diagram for reading a gain grating
In the reading process beam 2 (with phase Φ2 at the grating) enters the
amplifier with wave vector k2. It diffracts off the gain grating to produce
beam 4 which has a phaseΦ4.
Φ4 = Φ2+ Φgrating (3.7)
Φ4 = Φ2+π−(k1L+ Φother) (3.8)
Beam 4 then passes around the loop re-entering the amplifier as beam 2∗ (With phaseΦ∗
2).
Φ∗
2 = Φ4+k2L+ Φother (3.9)
Φ∗2 = Φ2+ (k2−k1)L+π (3.10)
If the grating writing beam has the same wavelength as the reading beams (k1 =k2) thenΦ∗2 = Φ2+π. This will cause destructive interference to occur
between the two beams leading to an unstable laser mode.
In order to maintain a stable oscillation constructive interference is needed. This can be achieved by settingΦ∗
2 = Φ2+ 2nπ. Hence
(k2−k1)L= (2n−1)π (3.11)
This allows the phase conjugator to oscillate over a range of frequencies given by ∆ν = ¡ n+1 2 ¢ c L (3.12)
For a loop length of 1 metre a frequency shift of∆ν = 0.15GHz from the writing beam is needed, which is well within the 254 GHz gain bandwidth of Nd:YVO4 [2].
Lasing will then occur on the frequencies which see the highest gain. Emis- sion on two longitudinal modes either side of the writing beam’s frequency has been observed in pulsed experiments in Nd:YAG [3]. Emission over multiple wavelengths is possible if each mode can see enough gain to res- onate.