衍射光栅公式

衍射光栅是通过用多条平行线刮擦一块透明材料制成的。该材料每厘米会出现大量划痕。例如，要使用的光栅每厘米包含 6,000 条线。划痕是不透明的，但它们之间的空间允许光线通过。当光落在衍射光栅上时，它形成了具有平行狭缝的光源的多重性。

什么是衍射光栅？

A diffraction grating is a periodic optical component that separates light into many beams that go in different directions. It’s an alternative to using a prism to study spectra. When light strikes the grating, the split light will often have maxima at an angle θ.

光线将以平行束的形式落在光栅上。波前将垂直于光线并平行于光栅，因为光线和波前构成正交集合。惠更斯原理在这种情况下是相关的。

根据它，每个透明狭缝都充当一个新的源，波前上的每个点都充当一个新的源，导致圆柱形波前从每个点扩散开来。

如果一个山峰不断地落在山谷上，波浪就会抵消，那个位置就没有光。此外，如果山峰经常落在山峰上，而山谷经常落在山谷上，那么那个地方的光线会更亮。衍射是使用棱镜检测光谱的替代方法。

衍射光栅公式

考虑从与直线成 θ 角的直线发出的两条射线。如果它们的两个路径长度之差是它们波长 λ 的整数倍，则会发生相长干涉，如下所示：

nλ = d sin θ

where,

n = 1, 2, 3, …, is an integer known as order of the grating, 

λ is the wavelength, 

d is the distance between the two spectra and 

θ is the angle.

此外，光栅的两个连续狭缝（线）之间的距离称为光栅元件。光栅元件“d”计算如下：

Grating element, d = Length of grating/Number of lines

示例问题

问题1：确定宽度为2cm的衍射光栅的狭缝间距，在波长为500nm的光下产生二阶30°的偏差。

解决方案：

Given that,

The order, n = 2,

The angle of deviation, θ = 30° and

The wavelength, λ = 500 nm = 500 × 10^-9 m.

Then, by the diffraction grating formula:

nλ = d sin θ

2 × 500 × 10^-9 m = d × sin 30°

d = 2 × 10^-6 m

问题 2：求波长为 600 nm 的单色光照射光栅并在 30° 角产生四阶亮线时每厘米的狭缝数。

解决方案：

Given that,

The order, n = 4,

The angle of deviation, θ = 30° and

The wavelength, λ = 600 nm = 600 × 10^-9 m.

Then, by the diffraction grating formula:

nλ = d sin θ

4 × 600 × 10^-9 m = d × sin 30°

or

d = 4.8 × 10^-6 m

Now, the number of slits per centimeter is given as:

x = 1 / 4.8 × 10^-6 m

  = 2.08 × 10⁵/ m

  = 2.08 × 10⁵ / 10² cm

  = 2.08 × 10³ / cm

  = 2080 / cm

问题3：每厘米包含5000个狭缝的光栅用单色光照射，产生30°角的二阶亮线。确定所用光的波长？ (1 Å = 10-10 m)

解决方案：

Given that,

The order, n = 2,

The angle of deviation, θ = 30° and

Number of slits per cm, N = 5000

This implies, the distance between slits, d = 1/N = 1/5000 cm = 5 × 10^-4 cm = 5 × 10^-6 m

Then, by the diffraction grating formula:

nλ = d sin θ

2 × λ = 5 × 10^-6 m × sin 30°

λ = 1.25 × 10^-6 m 

  = 1250 Å 

问题4：宽度为1cm的衍射光栅的狭缝间距是多少，在波长为1000nm的光下产生四阶30°的偏差。

解决方案：

Given that,

The order, n = 4,

The angle of deviation, θ = 30° and

The wavelength, λ = 1000 nm = 1000 × 10^-9 m.

Then, by the diffraction grating formula:

nλ = d sin θ

4 × 1000 × 10^-9 m = d × sin 30°

d = 8 × 10^-6 m

问题5：求宽度为1cm的衍射光栅中狭缝之间的距离，用波长为300nm的光产生30°的二阶偏差。

解决方案：

Given that,

The order, n = 2,

The angle of deviation, θ = 30° and

The wavelength, λ = 300 nm = 300 × 10^-9 m.

Then, by the diffraction grating formula:

nλ = d sin θ

2 × 300 × 10^-9 m = d × sin 30°

d = 1.2 × 10^-5 m