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The far-field approximation neglects quadratic variation of propagation phase across an aperture of characteristic size a at observation distance D. For two narrow slits separated by s, their path difference remains $s\sin \theta$ at a finite observation angle. The additional substitution $\sin \theta \simeq Y/D$ requires a small observation angle; it does not follow from the Fraunhofer approximation itself.

In the Fraunhofer approximation, contributions from positions $\xi$ across the slit have relative phase $k\xi \sin \theta$. With uniform illumination, the field relative to its axial value isThus the single-slit diffraction intensity isOn the detector $\sin \theta =y/\sqrt{d^2+y^2}$; near the axis, $\beta \simeq \pi wy/(\lambda d)$. The central maximum takes the continuous limit $\sin \beta /\beta \to 1$, and the minima satisfy $w\sin \theta =m\lambda$ for nonzero integers $m$. The stated far-field condition justifies adding the aperture contributions with a linear phase.

Monochromatic light has one frequency, or a bandwidth sufficiently narrow for the intended measurement. A lamp can illuminate the experiment through a narrow spectral filter or monochromator, followed by a small source aperture at the focus of a collimating lens. The aperture supplies spatial coherence across the slits; the lens supplies approximately parallel incident wavefronts. Two independently illuminated incoherent slits would not provide a stationary fringe pattern.

In the Fraunhofer approximation, the optical path difference at observation angle $\theta$ is $s\sin \theta$. For equal single-slit intensities ${\mathcal{I}}_s$, adding the two wave amplitudes givesThe thin-slit idealization suppresses the individual diffraction envelope. If additionally $|Y|\ll D$, then $\sin \theta \simeq Y/D$ and $\mathcal{I}(Y)\simeq 4{\mathcal{I}}_s{\cos }^2(\pi sY/(\lambda D))$. The far-field condition does not by itself impose this small observation angle.

For order $m=10$ at $Y=0.05\mathrm{m}$,The paraxial estimate is $sY/(10D)=5.88\times 10^{-7}\mathrm{m}$, consistent to the precision of the data.

Let the upper slit contain the glass plate and take positive $Y$ towards that slit. The extra optical path length is $\Delta =(n_g-n_a)t=5.0\times 10^{-5}\mathrm{m}$. Its total upper-minus-lower path difference is $\Delta -s\sin \theta$. The order-zero fringe therefore satisfiesIt moves towards the covered slit. Using $Y_0\simeq D\Delta /s=0.425\mathrm{m}$ here is a poor small-angle approximation: $30^{\circ }$ is not small. The order-zero fringe is being tracked as the optical delay is introduced; because the pattern is periodic in phase, many other bright fringes still lie near the original centre. Neglecting unequal glass transmission changes visibility but not the zero-path-difference condition.