We opened this chapter by noting that there are three models of light, each useful within a certain range of circumstances. We are now at a point where we can establish an important condition that separates the wave model of light from the ray model of light.

When light passes through an opening of size $a$, the angle of the first diffraction minimum is

Equation 33.32 is for a slit, but the result is very nearly the same if $a$ is the diameter of a circular aperture. Regardless of the shape of the opening, the factor that determines how much a wave spreads out behind an opening is the ratio $\text{λ}/\mathit{a}$, the size of the wavelength compared to the size of the opening.

FIGURE 33.22 illustrates the difference between a wave whose wavelength is much smaller than the size of the opening and a second wave whose wavelength is comparable to the opening. A wave with $\text{λ}/a\approx 1$ quickly spreads to fill the region behind the opening. Light waves, because of their very short wavelength, almost always have $\text{λ}/a\ll 1$ and diffract to produce a slowly spreading “beam” of light.

FIGURE 33.22 The diffraction of a long-wavelength wave and a short-wavelength wave through the same opening.

Now we can better appreciate Newton’s dilemma. With everyday-sized openings, sound and water waves have $\text{λ}/a\approx 1$ and diffract to fill the space behind the opening. Consequently, this is what we come to expect for the behavior of waves. We see now that light really does spread out behind an opening, but the very small $\text{λ}/a$ ratio usually makes the diffraction pattern too small to see. Diffraction begins to be discernible only when the size of the opening is a fraction of a millimeter or less. If we wanted the diffracted light wave to fill the space behind the opening $({\theta }_1\approx 90\text{°})$, as a sound wave does, we would need to reduce the size of the opening to $a\approx 0.001$ mm!

FIGURE 33.23 shows light passing through a hole of diameter $D$. According to the ray model, light rays passing through the hole travel straight ahead to create a bright circular spot of diameter $D$ on a viewing screen. This is the geometric image of the slit. In reality, diffraction causes the light to spread out behind the slit, but—and this is the important point—we will not notice the spreading if it is less than the diameter $\mathit{D}$ of the geometric image. That is, we will not be aware of diffraction unless the bright spot on the screen increases in diameter.

FIGURE 33.23 Diffraction will be noticed only if the bright spot on the screen is wider than $D$.

This idea provides a reasonable criterion for when to use ray optics and when to use wave optics:

• If the spreading due to diffraction is less than the size of the opening, use the ray model and think of light as traveling in straight lines.

• If the spreading due to diffraction is greater than the size of the opening, use the wave model of light.

The crossover point between these two regimes occurs when the spreading due to diffraction is equal to the size of the opening. The central-maximum width of a circular-aperture diffraction pattern is $w=2.44\text{λ}L/D$. If we equate this diffraction width to the diameter of the aperture itself, we have

where the subscript c on $D_{\text{c}}$ indicates that this is the crossover between the ray model and the wave model. Because we’re making an estimate—the change from the ray model to the wave model is gradual, not sudden—to one significant figure, we find

This is the diameter of a circular aperture whose diffraction pattern, at distance $L$, has width $w\approx D$. We know that visible light has $\text{λ}\approx 500\text{nm,}$ and a typical distance in laboratory work is $L\approx 1\text{m}$. For these values,

Thus diffraction is significant, and you should use the wave model, when light passes through openings smaller than about 1 mm. The ray model, which we’ll study in the next chapter, is the more appropriate model when light passes through openings larger than about 1 mm. Lenses and mirrors, in particular, are almost always larger than 1 mm, so they will be analyzed with the ray model.

We can now pull all these ideas together in a more complete presentation of the wave model of light.