This chapter describes the very sensitive Raman heterodyne detection technique which satisfies one of the requirements discussed in the previous chapter. The experimental details are also described.
3.1
Introduction
The Raman heterodyne technique is a coherent, optical-radio-frequency (rf) double resonance method for detecting magnetic transitions. It was first reported by M lynek et al.[ 1] and since then has been successfully used for detecting nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR) in rare earth ion systems [1-14], transition metal ion systems [15,16] and colour centres [17-21]. It can be used for both cw and transient measurements. Like other conventional optically detected magnetic resonance (ODMR) techniques where the optical photon is being detected rather than the rf photon, the Raman heterodyne technique improves the sensitivity for magnetic resonance spectroscopy. But what is more significant is that there is an im portant difference between the Raman heterodyne technique and other conventional ODMR techniques which makes the Raman heterodyne technique very suitable to the present study. With the conventional ODMR techniques only the changes in the populations of the levels coupled by the optical and rf fields could be detected. As a result, the coherent information contained in the off-diagonal density matrix terms could not, in general, be observed directly [22]. The Raman heterodyne technique directly probes the coherence between
the magnetic transition levels since it relies on a coherent Raman process being stimulated simultaneously by the coherent rf and the laser fields [1,2].
3.2 Raman Heterodyne Signal
The Raman heterodyne technique is an example of a three level optical-rf double resonance process (Fig.3-1). The electric dipole transition I1><=>I3> is driven coherently by an optical (laser) field at frequency coL = CO3 j and the magnetic dipole transition 11 ><=> I2> is driven coherently by an rf field at frequency corf = co21. The resultant coherence between levels I2> and I3> produces a coherent (Stokes) Raman signal field Es at frequency cos = (03 (- co2 ] = co32 (see, Fig.3-1). In addition, due to the presence of a second subgroup of centres within the inhomogeneous line, the laser field can also be resonant with the I2><=>I3> transition, which along with the rf field, produces a coherent anti- Stokes Raman field at frequency cos = co32+ co21 = co31. The stimulated Raman signal fields beat with the laser at the rf frequency, corf, and this is detected in the beam transmitted through the crystal using a photodiode. By sweeping the rf field frequency through the I1><=>I2> transition the profiles of this transition can be obtained.
A
® L
▼
Figure 3-1. Energy-level diagram for the stimulated Raman process in a three-level system. An rf field excites 11><=>I2> magnetic transition and a laser field excites Il><s=>l3> transition.
The Raman heterodyne technique applies to any three level system where all three transitions are dipole allowed [2]. For the N-V centre the nuclear magnetic transition 11 ><=> I2> is magnetic dipole allowed and the optical transitions 11 ><=>I3> and I2><=>I3>, corresponding to the transitions between the hyperfine levels within the ground state to a common level in the excited state, are electric dipole allowed through the wave function mixing occurring in the level anti-crossing region (see, Chapter 2).
r *
For an optical thin sample, the Raman signal polarization, Ps(t), induced in the sample by the applied rf and laser fields [2,19] is given by
Ps (t) = N ( n3l< p i3(t)> + n32<p23> ) .
(3.1)
The tilde denotes the slow ly varying part and the angular brackets denote the averaging over the optical and rf transition inhomogeneous distributions. This polarization is the origin o f the Raman heterodyne signal field, Es, at frequency CÜ3 2. From Maxwell's equations
- 2 n i ks P ( t ) ,
(3.2)
p. 3 1 and p. 3 2 are the electric dipole moments o f the I1><=>I3> and I2><==>I3>
optical transitions, respectively. N is the number density o f atoms in the sample, z is distance along the sample and ks is the wave vector o f the induced
r *
Raman field Es.
The Raman signal field Es and the input laser field E0 give rise the total
From the total intensity IT = I ET I 2 , the observable heterodyne beat signal on the detector, varying at the rf frequency, is obtained [2],
(3.4)
The optical coherences p j 3 and p 23 can be calculated by solving the density matrix equations of motion describing the Raman heterodyne detection and hence the Raman signal can be calculated. The density matrix equation of motion describing the Raman heterodyne detection consists of a set of 9 x 9 linear first order differential equations [2]. However, in the experiments the optical field was kept sufficiently weak so that it could be regarded as a weak perturbation and need only be treated to first order. The three-level system is reduced to a NMR two-level system with the laser field acting as a reliable probe of the coherence within the spin levels. This is a very significant point
that was fully treated by Wong et al. [2] when the Raman heterodyne technique
was first introduced. It has been shown that, to lowest order of optical field, the optical coherences p j 3 and p 23 in Eq.(3.1) are proportional to the rf coherences p 12 and p21 respectively. Therefore the electric field component of the induced Raman beam is given by [2,19]
where k is a constant, p 21 is the off-diagonal elem ent of the density matrix which represents the coherence between nuclear hyperfine levels ll> and I2>. Substituting Eq.(3.5) into Eq.(3.4), the Raman heterodyne signal intensity becomes, E = k u Lt E R e < p ( t ) > S M 3 M 3 0 M f - i 0) t (3.5) I = k u u I E^ |2 Re < p^ ( t ) e s M 3 z3 0 k 21 - i co t rf > , (3.6) or
3 4
I = k u
u
IE I ( Re < p „ > cos co t + Im < p „ > sin co t ).
S r 1 3 r 23 0 v K 21 rf K 21 rf ’