Context. Direct detection of exoplanets around nearby stars requires advanced adaptive optics (AO) systems. High-order systems are needed to reach a high Strehl ratio (SR) in near-infrared and optical wavelengths on future giant segmented-mirror telescopes (GSMTs). Direct detection of faint exoplanets with the European Southern Observatory (ESO) Extremely Large Telescope (ELT) will require some tens of thousands of correction modes. The resolution and sensitivity of the wavefront sensor (WFS) are key requirements for this science case. We present a new class of WFSs, the bi-orthogonal Foucault knife-edge sensors (or Bi-O edge), that is directly inspired by the Foucault knife-edge test. The idea consists of using a beam-splitter producing two foci, each of which is sensed by an edge with a direction orthogonal to the other focus. Aims. We describe two implementation concepts: The Bi-O edge sensor can be realised with a sharp edge and a tip-tilt modulation device (sharp Bi-O edge) or with a smooth gradual transmission over a grey edge (grey Bi-O edge). A comparison of the Bi-O edge concepts and the four-sided classical pyramid wavefront sensor (PWS) gives some important insights into the nature of the measurements. Methods. We analytically computed the photon noise error propagation, and we compared the results to end-to-end simulations of a closed-loop AO system. Results. Our analysis shows that the sensitivity gain of the Bi-O edge with respect to the PWS depends on the system configuration. The gain is a function of the number of control modes and the modulation angle. We found that for the sharp Bi-O edge, the gain in reduction of propagated photon noise variance approaches a theoretical factor of 2 for a large number of control modes and small modulation angle, meaning that the sharp Bi-O edge only needs half of the photons of the PWS to reach similar measurement accuracy. In contrast, the PWS is twice more sensitive than the Bi-O edge in the case of very low order correction and/or large modulation angles. Preliminary end-to-end simulations illustrate some of the results. The grey version of the Bi-O edge opens the door to advanced amplitude filtering, which replaces the need for a tip-tilt modulator while keeping the same dynamic range. We show that an additional factor of 2 in reduction of propagated photon noise variance can be obtained for high orders, such that the theoretical maximum gain of a factor of 4 in photon efficiency can be obtained. A diffractive Fourier model that accurately includes the effect of modulation and control modes shows that for the extreme AO (XAO) system configuration of the ELT, the overall gain will well exceed one magnitude in guide-star brightness when compared to the modulated PWS. Conclusions. We conclude that the Bi-O edge is an excellent candidate sensor for future very high order Adaptive Optics systems, in particular on GSMTs.