In my journey to understand light better, I build a "polarized Young's interferometer". Imagine the following polarizations: horizontal, vertical, diagonal, and anti-diagonal (H, V, D, and AD). I place an H sheet polarizer over one slit, and V over the other. I use a polarized laser source. Finally, I place a D sheet polarizer "mixer" between the polarzied Young's and the observation screen. I orient the source to explore the output of the four inputs above (H, V, D, and AD). What are the results of my journey? Do I see fringes for H?, for V?, for D?, for AD? If I see fringes, are they always the same phase? (Note that D input means colinear with the mixer, and AD input means perpendicular to the mixer. Also, the lines on the polarizers refer to the transmitted electric field axis of sheet polarizers and do not denote wire polarizers. Assume I conduct this experiment at a visible laser line with slits ~1.5 mm center-to-center, and each ~0.5 mm wide. I find it safe to assume that the bare slits have a negligible effect on polarization, and that the sheet polarizers are very efficient.)
I polarize the slits (one H, the other V) of a Young’s double-slit. If the source is H or V, do I see fringes? What about if the source is D or AD
double-slit-experimentinterferometryopticspolarization
Best Answer
I am going to assume a broad audience with some who have seen complex plane-wave analysis, and many who have not. Anyone who can do the ordinary Young's double-slit experiment should have no trouble doing the polarized-mixed version. Little or no math is needed, but there is a little art in making a polarized double-slit.
(See me showing the fringes going through all four outcomes just by rotating the mixer here.)
(A note about references: As much publicly-funded information is lately behind paywalls, I also include freely-available sources – some tertiary. I do not believe the paid sources are really needed to understand this answer, but they are included for completeness and lineage.)
The experiment really can be done in one's kitchen with no math needed. I make the slits by hand with roughly 1mm width, and spaced ~1.5mm apart. I use electrical tape to mask off the boundary between two abutted sheet polarizers whose transmission axes are perpendicular, and then cut and peal away the slit windows. I may include more detail on the art of making a polarized double-slit, if people request it. Laser pointers are usually polarized, but sometimes unstable. To be sure of the source's polarization, I use a polarizer right after the laser which I call a "prefilter". I find it easier to rotate the mixer between D and AD than to rotate the laser/prefilter assembly. The mixer can even be placed right against the observation screen and rotated (or removed) at that point. It is fun to watch the fringes shift phase and disappear altogether as I spin the mixer.
From the comments, I think people are catching on that diagonal and anti-diagonal (D and AD) inputs produce fringing output and that H and V input cannot make fringes. What takes maybe all of us by surprise (certainly me) is that AD makes a dark central (zeroeth) fringe. This means that when the mixer and input polarization angles are mutually perpendicular (the meaning of AD), the fringes look like a photo-negative of the familiar ordinary Young's fringes.
I start with cartoons of the qualitative output of the following setups: single-slit$^{(7, 8)}$, double-slit$^{(9, 10)}$, polarized double-slit, and polarized double-slit with mixer (Figures 0, 1). For a reminiscent layout, see the quantum eraser of Chiao, R., P. Kwiat and Aephraim M. Steinberg$^{(11,12)}$ (see Figs. 9b, 9c of v1 of the preprint, appearing at the last pages due to a LaTeX error).
Now for an overview of the math: I frame the problem using a complex plane wave allowing for insertion or removal of the polarizers (for example, removing the mixer is equivalent to using the identity matrix in its place which does not change the wavefunction at that point). I define the problem in Figure 5. I find some sense of confidence in getting a mathematical answer aligning with my experimental journey. The following complex plane-wave analysis gives the same results for fringe existence and phase I see for experiment.
TABLE 0 (All basic experimental cases are summarized here and proved in the complex analysis.)
I use the Jones Calculus$^{(1, 2, 6)}$. My approach is to make the general case include the ordinary Young's for confirmation that I am solving the problem correctly. I also simplify by keeping the specific geometry implicit and looking for a multiplicative sinusoidal term which indicates fringing. The equations still look a little scary, but they would be far worse if I were to try for a profile of the intensity versus diffraction angle. Anyway, I am only studying the existence and phase of the fringes for each input of H, V, D, and AD. The Jones calculus simply treats each optical element as a linear transformation of the complex plane wave, so the J.C. is a compact way of writing out the equations that one normally encounters when writing out the polarization components and projections as vector terms. Finally, the Born rule ("squaring" = the complex result dotted with its complex conjugate) tells us the observed intensity. The results match experiment for every input polarization for the naked double-slit, the polarized double-slit, and the polarized double-slit with mixer. I model each physical setup from the general equation, reduce algebraically, and then "square" separately, after which I apply the specific input polarization vectors. (See below the unique matrix arising for each case just before applying the Born rule.)
I write an incident plane wave$^{(1,3)}$, $$\Psi_0 = \vec E_0 e^{i(\vec k \cdot \vec x-\omega t)}\tag{0}\label{eq0}$$ where, $\vec k$ is the wavevector, $ \vec E_0= \left[\begin{matrix} E_{0y} \\ E_{0z} \\ \end{matrix} \right ] $ (and $E_{0y}, E_{0z} \in \Re$), $\vec k \parallel \vec x$, and $t$ is averaged over many cycles at any given $x$, so that only relative phases are important. Therefore, I treat the exponential in $\eqref{eq0}$ as a complex constant of arbitrary phase and choose to set it to 1. I do not care about absolute intensity. I only care about the existence of output fringes, so I look for a surviving sinusoidal term, and use proportionalities liberally. The slits happen to the incident wave simultaneously, so the result is a sum. The slit illumination factors are the same while their relative complex phases differ by diffraction$^{(7, 8)}$ angle, $\theta$, slit separation$^{(9, 10)}$, $d$, and slit filter (polarizer). Removing an optical element is handled by the identity matrix$^{(1, 2, 6)}$. The final function, $\Psi_3$, is the product of the mixer$^{(1, 2, 6)}$ (if any) and the sum of the slit functions$^{(1, 2, 6)}$, $\Psi_1$ and $\Psi_2$. So, $\Psi_3 \propto M_{(identity \lor mixer)}(\Psi_1+\Psi_2)$, where $M_{(identity \lor mixer)}$ is the identity matrix (if no mixer is present) or the Jones matrix for a diagonal polarizer$^{(1, 2, 6)}$. The slit functions are $\Psi_0$ modulated by relative phase due to geometry ($\theta$ and $d$) and absolutely transformed by any polarizing filters. Expanding, $$\Psi_3 \propto M_{(identity \lor mixer)}(M_{(identity \lor H)}\Psi_1e^{i\phi_1}+M_{(identity \lor V)}\Psi_2e^{i\phi_2})\tag{1}\label{eq1}$$ where $M_{H \lor V}$ are the Jones matrices for horizontal and vertical polarizers, respectively, and $\phi_1$, $\phi_2 \equiv \phi_1(\theta, +d), \phi_2(\theta, -d)$. Any surviving sinusoidal dependence on $\phi_{1,2}$ indicates fringing, so $\theta$ and $d$ remain implicit from here on out. For slits only $\eqref{eq1}$ reduces to, $$\Psi_3 \propto I(I\vec E_0e^{i\phi_1}+I\vec E_0e^{i\phi_2})$$, where $I$ is the identity matrix. Therefore, $$\Psi_3 \propto \vec E_0(e^{i\phi_1}+ e^{i\phi_2})$$ and taking the complex vector dot product$^{(4,5)}$ (aka “squaring”) gives, $$\lvert\Psi_3\rvert^2 \propto \lvert E_0\rvert^2(e^{i\phi_1}+ e^{i\phi_2})(e^{-i\phi_1}+ e^{-i\phi_2})$$ which reduces to, $$\lvert\Psi_3\rvert^2 \propto \lvert E_0\rvert^2(1+\cos(\phi_2 - \phi_1))\tag{2}\label{eq2}$$ The sinusoidal term indicates fringing. Note that the basis components of $\vec E_0$ are unimportant, so any input polarization gives fringes and the whole “ordinary double-slit” first column of Table 0 is proven. Now, for polarized slits,
$$ \Psi_3 \propto I \bigl ( \left(\begin{matrix} 1 & 0 \\ 0 & 0 \\ \end{matrix} \right ) \vec E_0 e^{i \phi_1} + \left(\begin{matrix} 0 & 0 \\ 0 & 1 \\ \end{matrix} \right ) \vec E_0 e^{i \phi_2} \bigr) $$
$$ \Psi_3 \propto \left( \begin{matrix} e^{i \phi_1} & 0 \\ 0 & e^{i \phi_2} \\ \end{matrix} \right ) \left[\begin{matrix} E_{0y} \\ E_{0z} \\ \end{matrix} \right ] $$ $$ \Psi_3 \propto \left [ \begin{matrix} E_{0y}e^{i\phi_1} \\ E_{0z}e^{i\phi_2} \\ \end{matrix} \right ] $$ $$ \lvert\Psi_3\rvert^2 \propto \left [ \begin{matrix} E_{0y}e^{-i\phi_1} & E_{0z}e^{-i\phi_2} \\ \end{matrix} \right ] \left [ \begin{matrix} E_{0y}e^{i\phi_1} \\ E_{0z}e^{i\phi_2} \\ \end{matrix} \right ] $$ $$ \lvert\Psi_3\rvert^2 \propto \lvert\vec E_{0y}\rvert^2 + \lvert\vec E_{0z}\rvert^2 = \lvert\vec E_{0}\rvert^2 \tag{3}\label{eq3} $$ So that any input does not give fringing, and the second “polarized Young’s” column of Table 0 is proven. Including the mixer, $\eqref{eq1}$ becomes $$ \Psi_3 \propto \left(\begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix} \right ) \bigl ( \left(\begin{matrix} 1 & 0 \\ 0 & 0 \\ \end{matrix} \right ) \vec E_0 e^{i \phi_1} + \left(\begin{matrix} 0 & 0 \\ 0 & 1 \\ \end{matrix} \right ) \vec E_0 e^{i \phi_2} \bigr) $$ $$ \Psi_3 \propto \left(\begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix} \right ) \left ( \begin{matrix} e^{i \phi_1} & 0 \\ 0 & e^{i \phi_2} \\ \end{matrix} \right ) \vec E_0 $$ $$ \Psi_3 \propto \left (\begin{matrix} e^{i \phi_1} & e^{i \phi_2} \\ e^{i \phi_1} & e^{i \phi_2} \\ \end{matrix} \right ) \vec E_0 $$ $$ \Psi_3 \propto \left [ \begin{matrix} E_{0y}e^{i\phi_1}+E_{0z}e^{i\phi_2} \\ E_{0y}e^{i\phi_1}+E_{0z}e^{i\phi_2} \\ \end{matrix} \right ] $$ $$ \lvert\Psi_3\rvert^2 \propto \left [ \begin{matrix} E_{0y}e^{-i\phi_1}+E_{0z}e^{-i\phi_2} & E_{0y}e^{-i\phi_1}+E_{0z}e^{-i\phi_2} \end{matrix} \\ \right ] \left [ \begin{matrix} E_{0y}e^{i\phi_1}+E_{0z}e^{i\phi_2} \\ E_{0y}e^{i\phi_1}+E_{0z}e^{i\phi_2} \\ \end{matrix} \right ] $$ $$ \lvert\Psi_3\rvert^2 \propto \left [ \begin{matrix} E_{0y}e^{-i\phi_1}+E_{0z}e^{-i\phi_2} & E_{0y}e^{-i\phi_1}+E_{0z}e^{-i\phi_2} \end{matrix} \\ \right ] \left [ \begin{matrix} E_{0y}e^{i\phi_1}+E_{0z}e^{i\phi_2} \\ E_{0y}e^{i\phi_1}+E_{0z}e^{i\phi_2} \\ \end{matrix} \right ] $$ $$ \lvert\Psi_3\rvert^2 \propto \lvert\vec E_0\rvert^2+2 E_{0y} E_{0z}\cos(\phi_2-\phi_1) \tag{4}\label{eq4} $$
Now, $\lvert\vec E_0\rvert^2 = \lvert\vec E_{0y}\rvert^2 + \lvert\vec E_{0z}\rvert^2 = E_{0y}^2 + E_{0z}^2 $ , and for D, AD, H, and V input, $\vec E_0= \left[\begin{matrix} \pm1 \\ \pm1 \\ \end{matrix} \right ] $, $ \left[\begin{matrix} \pm1 \\ \mp1 \\ \end{matrix} \right ] $, $ \left[\begin{matrix} 0 \\ \pm1 \\ \end{matrix} \right ] $, $ \left[\begin{matrix} \pm1 \\ 0 \\ \end{matrix} \right ] $, respectively. Plugging in to $\eqref{eq4}$ shows that the sinusoidal fringing disappears for H and V. For D and AD, the sinusoid remains, but is dark for AD wherever D is bright due to the sign inversion against the cosine term. So, the last column of Table 0 is proven.
So, I guess my theory skills are "correct" because I get the "right answer" for all cases, but that might be a long way from answering why this effect happens. In English, I would say that the complex phase of D is 180 degrees away from that for AD, so all else being equal (along the central fringe), the input amplitude has zero transmission through the mixer. That statement seems to make sense to me, if I imagine that the light vector components really are rotating in and out of imaginary space. For anyone not liking that image, I am not sure what I can do for you.
Acknowledgements
My gratitude to those who have supported and inspired my enthusiasm for science; there is not space here to mention them all. Zeroth, I thank Being One. I acknowledge myself for resolving to initiate this line of research at my personal expense which is not an immediate outgrowth of previous projects I have been blessed to work on. Of those not known to me in the flesh, Socrates who reminds me that I am allowed to start from zero, Huygens who proves more than he guesses, Tesla who trusts in the ether, B. Franklin who shows the way, J. C. Bose who sees life, Heaviside with his dogged quest for understanding, application, and proper credit, Newton and Kepler for sweeping away epicycles, Euler showing unity of cycle, Gauss empowering in surfaces and distributions,… Of those known personally: the late Robert Lin, Davin Larson, and Janet Luhmann who watched over my undergraduate scholarship at Space Sciences Lab, and my then other co-authors S. W. Kahler and N. U. Crooker; Jerry Edelstein, leader and mentor on SPEAR (Spectroscopy of Plasma Evolution from Astrophysical Radiation) and T-EDI (TripleSpec Exoplanet Discovery Instrument) projects, David Erskine, inventor and pioneer of EDI and X-EDI (Ordinary and Crossfading Externally Dispersed Interferometry); Julia Kregenow, Dae-Hee Lee, Mario Marckwordt, Matthew Ward Muterspaugh, Kaori Nishikida, Kwangsun Ryu, Michael Scholl, Kwang-Il Seon, Martin Sirk, the late C. H. Townes, Barry Welsh, and Ed Wishnow for many meaningful interactions. My late parents Esther and Bill Feuerstein, as well as being my teachers, always supported me in my sense of purpose.
W. Michael Feuerstein is an independent researcher investigating orthogonal projection interferometry and interferometric basis discrimination. He has professional research experience in space physics (particularly the relationship between the interplanetary magnetic field and electron pitch-angle distributions), space-born extreme and far ultraviolet astronomy, and externally-dispersed interferometry for Doppler planet-finding.