New Physics of Dipole and Quadrupole Brewster Angles in Thin Films

Abstract

The Brewster angle is a well-known phenomenon that describes the angle dependent reflection minimum resulting from the difference in the refractive index between two media for p-polarized light. From multipole decomposition, we show this well-known Brewster angle for the electric dipole term, but also new Brew ster angles for the magnetic dipole, electric quadrupole, and magnetic quadrupole terms for p-and s-polarizations. Our new equations show that the complex dipole and quadrupole moments must become real valued for the Brewster angle to occur. After applying our new theory to the reflection of a thin-dielectric SiN film, we identify a previously undiscovered interference effect required to observe the Brewster angle in thin films, the destructive interference between magnetic dipole and electric quadrupole terms. The electric dipole Brewster angle occurs for all wavelengths studied but those of the magnetic dipole, electric quadrupole and magnetic quadrupole occur over a very narrow wavelength range.

Figures (5)

• Figure 1: Illustration of geometry for incident (s- and p- polarized), reflected and transmitted electric fields from a thin film considering each multipole on its own. Subplot (a) illustrates the traditional Brewster angle, the electric dipole moment, $\vec{p}$, aligns with the reflected ray,$\vec{n}$, and the reflected ray is perfectly s-polarized. However, the Brewster angle can be observed for all multipoles as we show below, as any multipole moment may align with the reflected ray. Subplots (b) illustrate this for the magnetic dipole, where the reflected ray is perfectly p-polarized. Subplots (c)-(d) illustrate this for the electric/ magnetic quadrupoles, where the reflected ray may be perfectly s- or p-polarized depending on the direction of the incident ray.Brewster angles are denoted with various subscripts e.g. MD and complex multipoles are also indicated
• Figure 2: Demonstration of the Brewster angle condition, $\theta_{ED,real}-\theta_{I}+i\theta_{ED,imag}=0^{\circ}+0i$ being satisfied for p-polarized reflected rays for the (a) electric dipole, (b) electric quadrupole, (c) magnetic quadrupole and s-polarized reflected rays, (d) magnetic dipole, (e) electric quadrupole, (f) magnetic quadrupole. The Brewster angle condition is satisfied when the real part of the angle of the multipole aligns with the angle of incidence, as marked by the intersection point with the red line, as well as the imaginary part of the angle of the multipole being zero, marked by the dark blue colour. The colour represents the imaginary angle. The imaginary angle can be related to the magnitude of the imaginary part of $\vec{p}$ through the relation $\theta_{ED,i}=\mathop{\mathrm{arcsinh}}\nolimits({\frac{|imag(\vec{p})|}{|(\vec{p})|}})$. Both conditions must be satisfied for a Brewster angle to occur in the multipole model.
• Figure 3: Comparison of our multipole model with the S-parameter simulation and experiment via a study of (a) free standing SiN thin film of thickness 455 nm at a wavelength of 1500 nm. (b) Schematic of experiment to measure reflection. (c)-(d) Comparison between measured (circles) and modelled reflection with S-parameter (blue line) and multipole decomposition (dashed cyan line) for both s-polarization (c) and p-polarization (d). (e)-(f) Amplitude of individual multipole reflection coefficients, ED (black line), MD (dashed blue line), EQ (orange line), MQ (green line) in the multipole model for both s-polarization (e) and p-polarization (f). (g)-(h) Phase of individual multipole reflection coefficients in the multipole model, phases are divided by $\pi$.(i)-(j) Angle of multipoles as previously shown in Figure 2 for both s-polarization (i) and p-polarization (j), Vertical black, green and orange lines mark the ED, MQ and EQ Brewster angles.
• Figure 4: Broader comparison between simulations and experiment from wavelengths 750-2200 nm, (a) and (f) measured reflection, (b) and (g) multipole reflection, (c) and (h) S-parameter reflection for both s- and p-polarizations respectively,magenta lines represent the first 2 Fabry Perot modes plotted by equation S.74. (d)-(j) Individual reflection coefficients for electric dipole (d), magnetic dipole, (e) and (j) electric quadrupoles for p- and s-polarization over a limited range of reflection from 0 to 0.1 to identify the Brewster angles. (k)-(n) Investigation of SiN thin film at 850 nm, where the MD and EQ BAs are present similar to Figure \ref{['fig2']}. (k) Comparison between measured and modelled reflection. (l) Magnitude of individual reflection coefficients. (m) Phase of individual reflection coefficients. (n) Angle of MD,EQ and MQ multipoles.
• Figure 5: Magnitude of reflection coefficients for individual multipoles for p- and s-polarization. Colour map is restricted to reflection in the range 0-0.1 to highlight reflection minima corresponding to the multipole.(a) p-polarized ED ,(b) s-polarized ED. (b) p-polarized MD and (c) s-polarized MD. (e) p-polarized EQ and (f) s-polarized EQ .(g) p-polarized MQ and (h) s-polarized MQ.