Unveiling non-monochromatic modes and nonlinearity in Piet Hein quantum semiconductor waveguides

We investigate the propagation characteristics of transverse electric (TE) and transverse magnetic (TM) modes in a semiconductor quantum plasma-filled coaxial waveguide with a Piet Hein cross-section. The unique geometry of the Piet Hein cross-section offers intriguing possibilities for tailoring modal properties and exploiting novel nonlinear phenomena. Using analytical and numerical methods, we unveil the non-monochromatic behaviour of TE and TM modes, characterized by distinct peaks and troughs in their field components. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>E</mml:mi> <mml:mi>z</mml:mi></mml:msub> </mml:math> component of the TM mode exhibits a suppressed field within the central core and higher concentration in the cladding region, potentially minimizing energy loss within the waveguide. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>E</mml:mi> <mml:mi>ρ</mml:mi></mml:msub> </mml:math> component exhibits pronounced oscillations near the waveguide boundaries, suggesting constructive and destructive interference patterns. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>E</mml:mi> <mml:mi>ξ</mml:mi></mml:msub> </mml:math> component displays a non-monotonic behaviour with multiple pits and bumps, highlighting the interplay between the TE mode, the Piet Hein geometry, and the semiconductor quantum plasma properties. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>B</mml:mi> <mml:mi>ρ</mml:mi></mml:msub> </mml:math> component of the TM mode exhibits a non-monochromatic behaviour with multiple maxima and minima, attributed to the complex interactions between propagating waves with different phase shifts. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>B</mml:mi> <mml:mi>ξ</mml:mi></mml:msub> </mml:math> component exhibits a non-uniform distribution with distinct pits and bumps, with a resurgence towards the waveguide edges potentially due to higher-order mode contributions and local field enhancement effects. Our findings pave the way for the development of novel photonic devices with enhanced functionalities based on Piet Hein semiconductor quantum plasma waveguides.