The maximum possible number of orders of diffraction is primarily determined by the properties of the diffraction grating and the wavelength of light used.
The phenomenon of diffraction occurs when light waves bend as they pass through a narrow opening or around an obstacle. For a diffraction grating, which consists of a large number of equally spaced parallel slits, the condition for constructive interference (where bright spots or maxima appear) is given by the grating equation:
$d \sin \theta = m \lambda$
Where:
- $d$ is the distance between adjacent slits (grating spacing).
- $\theta$ is the angle of the $m$-th order maximum relative to the central maximum.
- $m$ is the order of the maximum (an integer: 0, ±1, ±2, ...). $m=0$ corresponds to the central maximum.
- $\lambda$ is the wavelength of the light.
The maximum possible order ($m_{max}$) occurs when the angle $\theta$ is the largest possible, which is $90^\circ$. At this angle, $\sin \theta = \sin 90^\circ = 1$. Substituting this into the grating equation gives:
$d \cdot 1 = m_{max} \lambda$
$m_{max} = \frac{d}{\lambda}$
Since the order $m$ must be an integer, the maximum number of visible orders (on one side of the central maximum) is the largest integer less than or equal to $d/\lambda$. The total number of bright fringes observed (including the central maximum) would be $2m_{max} + 1$.
Key Factors Influencing Maximum Diffraction Orders
Based on the equation $m_{max} = d/\lambda$, the two primary factors that affect the number of maximum possible orders are:
- Grating Spacing ($d$): The distance between adjacent slits on the diffraction grating.
- Wavelength of Light ($\lambda$): The wavelength of the light incident on the grating.
Let's look at each in more detail:
1. Grating Spacing ($d$)
- How it affects $m_{max}$: According to the equation, $m_{max}$ is directly proportional to $d$.
- Practical Insight: A grating with a larger spacing (fewer lines per unit length) can produce a higher number of diffraction orders for a given wavelength. This is because the possible angles for constructive interference are larger, potentially allowing more integer values of $m$ to satisfy the $d \sin \theta = m \lambda$ condition before $\sin \theta$ exceeds 1.
2. Wavelength of Light ($\lambda$)
- How it affects $m_{max}$: According to the equation, $m_{max}$ is inversely proportional to $\lambda$.
- Practical Insight: Shorter wavelengths (like blue or violet light) can produce a higher number of diffraction orders compared to longer wavelengths (like red light) when using the same grating. This is why in a spectrum produced by a grating, blue light bends less but can potentially show more higher-order maxima than red light, although red light's maxima are spread out more.
Influence of Distance to the Screen
While the theoretical maximum number of orders ($m_{max}$) is fixed by the grating spacing and wavelength, the distance between the grating and the screen significantly influences the observed diffraction pattern.
As mentioned in the reference: "The distance between the slit and the screen significantly influences the size of the diffraction pattern. As the screen is moved further away from the slit, the diffraction pattern expands, resulting in a larger central maxima and secondary maxima."
- Effect on Observation: A larger distance to the screen spreads out the diffraction pattern. This means that the angular separation between consecutive orders results in a larger physical separation on the screen.
- Practical Implication: This expansion can make it easier to measure the positions of different orders. However, on a screen of finite size, moving the screen further away might cause higher-order maxima (which appear at larger angles) to move off the edge of the screen, making them unobservable even if they are theoretically possible. Conversely, moving the screen closer might compress the pattern, making many orders appear within the screen boundaries but potentially making them harder to distinguish if they are too close together.
In summary, the fundamental limit to the number of maximum possible orders is set by the ratio of the grating spacing to the wavelength ($d/\lambda$). The distance to the screen affects how spread out the pattern is and thus which of these possible orders are observable on a screen of a given size.
Example
Consider a diffraction grating with $d = 2 \times 10^{-6}$ m illuminated by green light with $\lambda = 500 \times 10^{-9}$ m (or $0.5 \times 10^{-6}$ m).
$m_{max} = \frac{d}{\lambda} = \frac{2 \times 10^{-6} \text{ m}}{0.5 \times 10^{-6} \text{ m}} = 4$
The maximum possible order on one side is $m=4$. This means orders $m=0, \pm 1, \pm 2, \pm 3, \pm 4$ are theoretically possible.
Now, if the grating spacing was larger, say $d = 4 \times 10^{-6}$ m:
$m_{max} = \frac{4 \times 10^{-6} \text{ m}}{0.5 \times 10^{-6} \text{ m}} = 8$
A larger grating spacing allows for more possible orders.
If the wavelength was longer, say red light $\lambda = 700 \times 10^{-9}$ m ($0.7 \times 10^{-6}$ m) with the original grating ($d = 2 \times 10^{-6}$ m):
$m_{max} = \frac{2 \times 10^{-6} \text{ m}}{0.7 \times 10^{-6} \text{ m}} \approx 2.85$
Since $m$ must be an integer, the maximum possible order is $m=2$. A longer wavelength results in fewer possible orders.
Summary Table
| Factor | Symbol | How it Affects $m_{max}$ | Explanation |
|---|---|---|---|
| Grating Spacing | $d$ | Directly Proportional | Larger spacing allows more orders before $\sin\theta=1$. |
| Wavelength | $\lambda$ | Inversely Proportional | Shorter wavelength allows more orders before $\sin\theta=1$. |
| Distance to Screen | $L$ | Affects Observability | Influences pattern size, determining which orders land on a finite screen. |
