Probing microwave fields and enabling in-situ experiments in a transmission electron microscope

A technique is presented whereby the performance of a microwave device is evaluated by mapping local field distributions using Lorentz transmission electron microscopy (L-TEM). We demonstrate the method by measuring the polarisation state of the electromagnetic fields produced by a microstrip waveguide as a function of its gigahertz operating frequency. The forward and backward propagating electromagnetic fields produced by the waveguide, in a specimen-free experiment, exert Lorentz forces on the propagating electron beam. Importantly, in addition to the mapping of dynamic fields, this novel method allows detection of effects of microwave fields on specimens, such as observing ferromagnetic materials at resonance.


II: Theoretical considerations
At the core of the electron beam interaction with EM fields lies the Aharonov-Bohm effect. A beam of coherent electrons propagating along the optical axis z of an electron microscope can be interpreted as a monochromatic plane wave Ψ = Ae ik z . z with complex amplitude A = A 0 (r, z)e iφ (r,z) . The effects of propagating the electron beam in a region with non-zero electric and magnetic potentials can be expressed as a phase gradient ∇ φ (r, z), which is given by [1][2][3][4][5] : where the constant C U is given by The value of C U is 7.28 × 10 6 rad/Vm at 200 keV electron beam 1,6 . The term λ corresponds to the electron wavelength. The terms U and U 0 correspond to the kinetic energy and rest mass energy of the electrons, respectively. The term t corresponds to the distance over which the EM field is non-zero and n z the unit vector parallel to the propagation direction of the electrons. The first term on the right accounts for the gradient of the electric potential V (r) ( E = −∇ V ) and the second term accounts for the induction field components perpendicular to propagation direction.
The intensity distribution of an image obtained with SAES corresponds to a magnified view of the pattern formed in the back focal plane of the image forming lens 3 . In this plane, the electron wave is given by the Fourier transform F [A 0 e i∇φ . z ] of the wave transmitted through the region with non-zero EM fields. Thus the intensity pattern imaged in SAES corresponds to the magnitude of |F A 0 e i∇φ . z | 2 . Figure 1 illustrates the formation of a diffraction pattern resulting from an 'ideal specimen with two magnetic domains, whose magnetic moment is in opposite direction, and separated by a domain wall (neglected). The SAES pattern was obtained following the implications of Eq. 1.

Figure 2.
Illustration of the effect of two magnetic domains on the electron beam. Two magnetic domains opposed to each other (left) induce a phase variation on the transmitted electron beam. When the image is acquired in the back focal plane of the image forming lens, the pattern observed resembles the image shown on the right, which is known as an SAES pattern. From a classical perspective, the electron beam experiences a Lorentz deflection angle β proportional to the layer thickness and the magnetic induction, as expressed in Eq. 2. The separation between the two spots on the image corresponds to 2β L, with L being the camera length. The low intensity signal at the centre of the SAES pattern corresponds to the undeflected beam which crossed the domain wall where the net induction field is assumed negligible.
In cases where the magnetic induction can be considered uniform across the film thickness and the gradient of the electric potential may be neglected, the diffraction pattern can be described as two intensity spots deflected by an angle β . The deflection angle, β , is directly proportional to the thickness, t, of the region with non-zero magnetic induction, B 0 , as expressed in Eq. 2.
When both the gradient of the electric potential and the magnetic induction field are considered, complex SAES patterns emerge, which is the case of the present study. When calculating the simulated SAES patterns one required the approach given by Eq. 1, as explained in the following section.

2/4 III: Simulated SAES patterns
Having discussed an example of the static field distribution, it is now relevant to discuss the nature of dynamic EM fields in the microwave frequency range. EM fields in the microwave frequency range are time and space varying waves whose electric (and magnetic) field component can be expressed as 7 : where E 0 is the wave amplitude, ω is the angular frequency and k represents the propagation vector. For simplicity, assume that E and H are transverse wave modes so the field components are alongx andŷ. If only the time varying components of Eq. 3 are considered (fixed z= 0), the electric field can be written as Note that at a fixed position, take z = 0, the electric field direction rotates with ωt following its orthogonal field components in what is defined as a polarisation state. The relative amplitude of the E x 0 and E y 0 as well as the phase, given by will define the polarisation of the electric field. EM waves can be described as linearly polarized if either E x 0 or E y 0 components are zero or φ = π/4 radians. The waves will have circular polarisation if E x 0 = E y 0 and φ = π/2 radians. Elliptical polarisation will be obtained for E x 0 = E y 0 . In a microstrip with non-linear shape, such as loops or bends, the polarisation state will not be a linear polarisation since propagating waves cannot be approximated to transverse EM modes and their propagation direction cannot be trivially determined since the propagating modes in the loop shaped region of the microstrip are hybrid transverse electric (TE) and transverse magnetic TM) modes. As a consequence, when probing the EM field distribution in the SAES experiments one should expect a superposition of polarisation states. Figures 3(a) and 3(b) show examples of the calculated spatial distribution of the in-plane electric and magnetic field components as well as the corresponding field magnitude at 1 GHz and 8.7 GHz, respectively, in the region imaged by the electron beam. Having obtained the distribution of the electric and magnetic field from the microwave simulations we calculated the corresponding SAES patterns by applying Eq. 1 and integrating the phase along the propagation direction of the electron beam. The final SAES patterns were obtained by integration over a 2π rotation of the phase term corresponding to ωt. Figure 4 shows the evolution of the SAES pattern as the phase varies from 0 to 2π in phase steps of 0.3 radians at 18 GHz. The calculated SAES pattern shown in Fig. 4(u) corresponds to the time-integrated intensity profile, resembling the SAES profiles obtained in the experiments.

IV: Future perspectives
The simple extension of this technique to scanning-TEM has the potential to improve the spatial resolution needed to quantify properties of condensed matter which are hardly accessible. For example, one can (1) characterise the ground state of the specimen while driving its local or non-local precessional modes; (2) measure deformations in the SAES patterns while the specimen in at resonance and correlate these with the lineshape of the resonance response and (3) study microwave field switching on nano-structured magnetic materials. From a different view point, the control over the polarisation state would enable the study of materials whose slow and fast response is very sensitive to the driving microwave field polarisation 8 .