Verification of the Relationship Between Single-Slit Diffraction Stripe Width and Wavelength

In the study of wave optics, diffraction stands as one of the most compelling pieces of evidence for the wave nature of light. When a wavefront encounters an obstacle or an aperture, it does not merely cast a sharp geometric shadow; instead, it bends around the edges, spreading into the region that would otherwise be dark. This phenomenon results in a characteristic pattern of alternating light and dark fringes on a detection screen. Among the various types of diffraction, single-slit diffraction serves as the fundamental model for understanding how light interacts with small apertures.

This article explores the quantitative relationship between the width of diffraction fringes and the wavelength of the incident light. By establishing a rigorous theoretical framework and designing a controlled experimental procedure, we aim to demonstrate that the spatial distribution of light is intrinsically linked to its spectral properties.

Theoretical Foundation

Fraunhofer Diffraction and the Mathematical Model

Diffraction phenomena are generally categorized into two regimes based on the distance between the source, the aperture, and the observation screen: Fresnel diffraction (near-field) and Fraunhofer diffraction (far-field). To simplify mathematical modeling and ensure high-contrast fringe patterns, this study focuses on the Fraunhofer diffraction regime.

According to Huygens' Principle, every point on a wavefront acts as a source of secondary spherical wavelets. When monochromatic, collimated light passes through a single slit of width $a$, these secondary wavelets interfere with one another in the far field. The condition for destructive interference (the formation of dark fringes) occurs when the path difference between wavelets from the edges of the slit results in a phase difference of an odd multiple of $\pi$.

The mathematical condition for the $k$-th order dark fringe is given by:
$$ a \sin\theta = \pm k\lambda \quad (k = 1, 2, 3, \dots) $$

Where:

• $a$ is the slit width.
• $\theta$ is the angle of diffraction.
• $k$ is the order of the dark fringe.
• $\lambda$ is the wavelength of the incident light.

The Proportionality Relationship

In most laboratory setups, the distance $D$ from the slit to the screen (or the focal length $f$ of a lens) is significantly larger than the slit width $a$. Under these conditions, the diffraction angle $\theta$ is extremely small, allowing us to apply the small-angle approximation: $\sin\theta \approx \tan\theta \approx \theta$.

The distance $\Delta x$ from the center of the central maximum to the first dark fringe (or the distance between adjacent fringes) can be expressed as:
$$ \Delta x = \frac{D \lambda}{a} $$

This equation reveals the core hypothesis of our experiment: if the slit width $a$ and the distance $D$ are held constant, the width of the diffraction fringes $\Delta x$ is directly proportional to the wavelength $\lambda$.

Experimental Design and Methodology

To verify the relationship $\Delta x \propto \lambda$, we employ a comparative measurement approach. By using different monochromatic laser sources while maintaining a constant optical geometry, we can observe how the fringe spacing shifts in response to changes in wavelength.

Apparatus

1. Light Sources: High-stability semiconductor lasers of known wavelengths, specifically Red ($\lambda_1 \approx 650\text{ nm}$) and Green ($\lambda_2 \approx 532\text{ nm}$). Lasers are ideal due to their high coherence and collimation.
2. Diffraction Element: A precision adjustable single slit equipped with a micrometer head (accuracy $\pm 0.01\text{ mm}$).
3. Detection System: An optical rail, a white projection screen, and a micrometer eyepiece (or a graduated scale) for high-precision measurement of fringe positions.
4. Mounting Hardware: An optical breadboard and multi-axis adjustment stages to ensure precise alignment.

Experimental Procedure

1. Optical Alignment: Mount the laser, the slit, and the screen on the optical rail. Adjust the vertical and lateral positions to ensure all components are coaxial. The slit plane must be perfectly perpendicular to the optical axis to prevent asymmetric patterns.
2. Slit Calibration: Adjust the micrometer to set a slit width $a$ that produces clearly distinguishable fringes. If $a$ is too large, the fringes will be too narrow to resolve; if $a$ is too small, the light intensity will be insufficient for accurate measurement.
3. Measurement of Red Light ($\lambda_1$):
   • Position the screen at a distance $D$ where at least the first three orders of dark fringes ($\pm 3$) are clearly visible.
   • Use the micrometer eyepiece to record the positions of the $k$-th order dark fringes.
   • To minimize systematic error, utilize the Symmetric Measurement Method: measure the total distance $L_k$ between the $+k$ and $-k$ dark fringes. The average fringe width is then calculated as $\Delta x = L_k / (2k)$.
4. Comparative Measurement of Green Light ($\lambda_2$): Without altering the slit width $a$ or the distance $D$, replace the red laser with the green laser. Repeat the measurement process to determine $\Delta x$ for the green light.

Data Processing and Error Analysis

Sample Data and Calculation

The following table represents a typical data set obtained during the experiment, assuming a slit width $a = 0.10\text{ mm}$ and a distance $D = 800\text{ mm}$.

Theoretical Verification:
Using the formula $\Delta x_{theory} = \frac{D\lambda}{a}$:

• For Red: $\Delta x_{1\text{th}} = \frac{800 \times 650 \times 10^{-6}}{0.10} = 5.20\text{ mm}$
• For Green: $\Delta x_{2\text{th}} = \frac{800 \times 532 \times 10^{-6}}{0.10} = 4.256\text{ mm}$

The experimental results show an extremely low relative error (less than $1%$), confirming the validity of the mathematical model.

Verification of the Proportionality

To confirm the direct proportionality, we compare the ratio of the measured widths to the ratio of the wavelengths:
$$ \text{Experimental Ratio: } \frac{\Delta x_1}{\Delta x_2} = \frac{5.21}{4.26} \approx 1.223 $$
$$ \text{Theoretical Ratio: } \frac{\lambda_1}{\lambda_2} = \frac{650}{532} \approx 1.222 $$

The near-perfect agreement between these ratios provides empirical proof that $\Delta x$ scales linearly with $\lambda$.

Sources of Error

Despite the high precision, several factors may contribute to experimental uncertainty:

• Mechanical Imperfections: The physical width of the slit may deviate slightly from the micrometer reading due to mechanical backlash or edge irregularities.
• Transition Regime Effects: While we assume Fraunhofer diffraction, the finite distance $D$ may introduce minor Fresnel diffraction components, causing slight distortions in the fringe edges.
• Observational Subjectivity: Determining the exact center of a dark fringe via an eyepiece is subject to human visual error and the contrast limits of the environment.
• Thermal Drift: Prolonged laser operation can lead to temperature fluctuations, causing minute shifts in the laser's central wavelength.

Conclusion

Through a systematic application of wave optics principles, this study has successfully verified the relationship between single-slit diffraction fringe width and light wavelength. By utilizing the Fraunhofer diffraction model and a comparative experimental setup, we demonstrated that the fringe spacing is directly proportional to the wavelength. This experiment not only reinforces fundamental optical theories but also highlights the practical utility of diffraction patterns in precision tasks such as wavelength determination and the calibration of optical instrumentation.