Victor Han^{1} and Chunlei Liu^{1,2}

^{1}Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA, United States, ^{2}Helen Wills Neuroscience Institute, University of California, Berkeley, Berkeley, CA, United States

We present a fully geometric view of multiphoton excitation by taking a particular rotating frame transformation. In this rotating frame, we find that multiphoton excitations appear just like single-photon excitations again, and thus, we can readily generalize concepts already explored in standard single-photon excitation. With a homebuilt low-frequency (~ kHz) coil, we execute a standard slice-selective pulse sequence with all of its excitations replaced by their equivalent two-photon versions. With a multiphoton interpretation of oscillating gradients, we present a novel way to transform a standard slice-selective adiabatic pulse into a multiband one without modifying the RF pulse shape itself.

To analyze multiphoton excitation, consider two $$$B_1$$$ fields with frequencies $$$\omega_{xy}$$$ and $$$\omega_z$$$ along the xy-plane and z-axis respectively. The total magnetic field in the laboratory frame is $$B_z=B_0+B_{1,z}\cos{\left(\omega_zt\right)}\hspace{2em}[1]$$ $$B_x=B_{1,xy}\cos{\left(\omega_{xy}t\right)}\hspace{2em}[2]$$ $$B_y=-B_{1,xy}\sin{\left(\omega_{xy}t\right)}\hspace{2em}[3]$$ where the RF field in the xy-plane is clockwise circularly polarized; $$$B_{1,xy}$$$ and $$$B_{1,z}$$$ are amplitudes of each RF field. In a clockwise rotating frame with an angular velocity of $$$\omega_{rot}$$$, the effective $$$B_z$$$ field is $$B_{z,eff}=B_z-\frac{\omega_{rot}}{\gamma}\hspace{2em}[4]$$ To generate time-invariant $$$B_{z,eff}$$$, we choose $$\omega_{rot}=\omega_{xy}+\gamma B_{1,z}\cos{\left(\omega_zt\right)}\hspace{2em}[5]$$ We refer to this rotating frame as the phase-modulated rotating frame following

- Single-photon resonance: If $$$\omega_{xy}=\gamma B_0$$$, then $$$B_{z,eff}=0$$$ and $$$B_{x,eff}$$$ tilts magnetization from the z-axis. This is the normal on-resonance condition.
- Two-photon resonances: If $$$\omega_{xy}=\gamma B_0\pm\omega_z$$$, then $$$B_{z,eff}=\mp\frac{\omega_z}{\gamma}$$$. The linearly polarized RF field $$$B_{y,eff}$$$ oscillates with an angular frequency of $$$\omega_z$$$, matching the amplitude of $$$B_{z,eff}$$$, which induces resonances. These two new resonances correspond to state transitions where a single xy-polarized photon is absorbed and a single z-polarized photon is absorbed or emitted, depending on whether the xy-frequency is below or above the Larmor frequency. From Eq. [13], consistent with past work, the effective angular nutation frequency for two-photon excitation is $$\omega_{nut}=\frac{\gamma B_{1,xy}\gamma B_{1,z}}{2\omega_z}\hspace{2em}[14]$$ If more terms are kept in the Taylor expansion of Eqs. [10,11], we can analyze higher-order excitations with three or more photons. However, Taylor expansion requires $$$\frac{\gamma B_{1,z}}{\omega_z}\ll1$$$. Alternatively, we can exactly expand $$$B_{x,eff}$$$ and $$$B_{y,eff}$$$ using Bessel functions, which reveals that for any integer $$$n$$$, resonance occurs whenever $$\omega_{xy}=\gamma B_0+n\omega_z\hspace{2em}[15]$$ and for each integer $$$n$$$, the corresponding effective angular nutation frequency is $$\omega_{nut}=\gamma B_{1,xy}J_n\left(\frac{\gamma B_{1,z}}{\omega_z}\right)\hspace{2em}[16]$$

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