Electrical – What are the extra lines on this farfield plot

Since you are trying to switch on battery power, you have battery power available. So use it.

Your design brief contains two completely incompatible requirements; powered from the signal, and small size. To get significant power from the signal, you need to collect significant signal; and that means a large antenna.

The classic AM band "crystal set" circuitry you are studying needs about a hundred foot wire antenna to collect enough power to run sensitive headphones. At which point you might as well run the wire to the transmitter!

At higher frequencies you can reduce the antenna size to some extent but if you need the antenna to be small you MUST expect the received power to be very low, and need amplification.

Even your transistor switch is an amplifier, so admit it; there will be amplification. Then focus on the least power you need to power that amplifier - off the battery. One way might be to use a very low power micro like the MSP430 (under 1 microamp in the right usage pattern) to turn the receiver on very briefly every few seconds, to see if a signal is being transmitted.

A standing wave occurs when two signals that have the same frequency are traveling in opposite directions through a medium simultaneously.

Most antennas, such as the simple dipole, have impedance discontinuities at their ends that cause energy to be reflected back down the conductor, creating the conditions required for standing waves.

On the other hand, traveling wave antennas such as the Beverage or rhombic are deliberately terminated at their end(s) by a resistance that matches the characteristic impedance of the conductor. This eliminates the energy reflection and the backward-traveling wave, leaving only the forward-traveling wave — and no standing wave.