Law of refraction

Geometrical optics theorem

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Law of refraction It was discovered by the Dutch mathematician Snell that Refraction of light Phenomenon, determine Refraction ray The law of direction.

(1) Refraction ray is located at Incident light And interface normal In the determined plane;

（2） Refracting line And the incoming rays are respectively on both sides of the normal;

（3） Angle of incidence I's sine and Refraction angle i The ratio of the sine of 'to Refractive index It is a constant for certain two media.

Light from light speed When a large medium enters a medium with a small light speed, the refraction angle is smaller than the incidence angle; The refraction angle is larger than the incidence angle when entering the medium with high light speed from the medium with low light speed.

Chinese name: Law of refraction
Foreign name: Snell's Law

Discipline: optics
Alias: Snell's law
Classification: Mathematical physics

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Principle concept

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Law of refraction It was discovered by the Dutch mathematician Snell that the direction of refracted light is determined in the phenomenon of light refraction Laws 。 When light from the first medium (refractive index n one ）Inject into the second medium（ Refractive index n two ）At Smooth interface Part of the light consists of the first medium Occurs when entering the second medium refraction 。

The experiment points out that:

(1) The refraction ray lies in the plane determined by the incident ray and the interface normal;

(2) The refraction line and the incoming ray are respectively on both sides of the normal;

(3) Sine and refraction angle of incidence angle i i Of sine Of ratio , is a constant for two media with a fixed refractive index.

Light enters from medium with high speed of light light speed When the medium is small, the refraction angle is smaller than the incidence angle; The refraction angle is larger than the incidence angle when entering the medium with high light speed from the medium with low light speed.

Scope of application

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This law is the basic experimental law of geometric optics. It is suitable for homogeneous isotropic media. All kinds of optical instruments used to control the optical path and image, and the structure principle of the optical path is mainly based on the law of refraction and reflection of light. This law can also be derived from the wave concept of light, so it can also be applied to the refraction of radio waves and sound waves.

The law of refraction of light is only applicable to isotropic media Static interface 。

Details

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The law of refraction is also called Snell's law（ Snell's Law ）。

The law that determines the relationship between the propagation direction of the incident light and the refraction light when the light is refracted through the interface of two media is one of the basic laws of geometric optics., The plane formed by the incident light and the normal of the interface passing through the incident point is called the incident plane, and the angle between the incident light and the refraction light and the normal is called the incidence angle and refraction angle respectively θ one and θ two express.

The law of refraction is expressed as: ① The refracted light is in the incident plane. ② The ratio of the sine of the incident angle and the refraction angle is a constant n twenty-one Means that

Where n twenty-one It is called the relative refractive index of the second medium to the first medium.

Relevant interpretation

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Explain with Fermat principle

Fermat's principle is also called "the shortest time principle"^[1] ： The path that light travels is the path that takes the least time. The more correct version of Fermat's principle should be the "stationary time principle". In some cases, the time required for the path of light propagation may not be the minimum value, but the maximum value, or even the inflection value. For example, for a plane mirror, the optical path of the reflection path of any two points is the minimum; For a semi elliptical mirror, the light reflection path of the two focal points is not unique, and the optical path is the same, which is the maximum and minimum value; For a semicircular mirror, its two endpoints Q、P The optical path of the reflection path of is the maximum value; For a mirror composed of a quarter round mirror and a flat mirror, the same two points Q、P The optical path of the reflection path of is an inflection value.

Assume that the refractive index of medium 1 and medium 2 are n one 、 n two , light from medium 1 at point O Propagation into medium 2, θ one Is the angle of incidence, θ two Is the refraction angle.

From Fermat's principle, it can be deduced that Snell's law 。 By setting the derivative of optical path with respect to time to zero, you can find the "stable path", which is the path of light propagation. The propagation speed of light in medium 1 and medium 2 is v one = c / n one ， v two = c / n two 。 Among them, c Is the speed of light in vacuum.

Since the medium will slow down the speed of light, the refractive index n one 、 n two Both are greater than 1.

From point Q To point P The propagation time of is

。

According to Fermat's principle, The path of light propagation is the path with extreme time required , take the propagation time T To variable x And set it to zero. Available after sorting

d T /dx=sin θ one / v one -sin θ two / v two =0。

Substitute the relation between propagation speed and refractive index to get the refractive law:

n one sin θ one = n two sin θ two 。

Explanation by the particle nature of light

If the system remains unchanged after a translation of the whole system, the system is said to have translational symmetry. From the translational symmetry, Snell's law can be deduced. This is based on The transverse uniform interface cannot change the transverse momentum The truth. Because of the wave vector

Therefore, k one sin θ one = k two sin θ two 。（1）

According to the definition formula of refractive index: n = c / v = ck / ω ，

Among them, ω Is the angular frequency of the light wave.

Bring it into equation (1) to get the refraction law: n one sin θ one = n two sin θ two 。

From micro to atomic size, although no interface is completely uniform, if the propagation area can be estimated to be uniform if it is fine to the wavelength size of light wave, the translational symmetry is still a good approximation.

Explanation by Maxwell's electromagnetic field theory

The three basic laws of geometric optics are:

The first law: the wave vector of incident wave, reflected wave and refracted wave is contained in the "incident plane" together with the normal of the interface.
The second law: the reflection angle is equal to the incidence angle. This law is called "reflection law".
The third law: This law is called "Snell's law", also called "refraction law".

Since light waves are electromagnetic radiation in a specific frequency band light Maxwell equations and accompanying boundary conditions must be satisfied. One of the boundary conditions is, In the vicinity of the boundary, the components of the electric field parallel to the boundary must be continuous 。 Assume the boundary is xOy Plane, then at the boundary, there is

E ∥，i ( x ， y ，0)+ E ∥，r ( x ， y ，0)= E ∥，t ( x ， y ，0)。

Among them, E ∥，i 、 E ∥，r 、 E ∥，t They are the components of the electric field of the incident wave, reflected wave and refracted wave (transmission wave) parallel to the boundary.

Assume that the incident wave has a frequency of ω In order to meet the boundary conditions at any time, the frequency of reflected and refracted waves must be ω 。 set up E ∥，i 、 E ∥，r 、 E ∥，t In the form of

E ∥，i = E ∥，i0 exp(i k i · r - ωt )、

E ∥，r = E ∥，r0 exp(i k r · r - ωt )、

E ∥，t = E ∥，t0 exp(i k t · r - ωt )。

Among them, k i 、 k r 、 k t Are the wave vectors of incident wave, reflected wave and refracted wave, E ∥，i0 、 E ∥，r0 、 E ∥，t0 The amplitude of incident wave, reflected wave and refracted wave (possibly complex value).

To be anywhere on the boundary（ x , y , 0) Meet the boundary conditions, the phase change must be the same, and must be set

k i x x + k i y y = k r x x + k r y y = k t x x + k t y y 。

Therefore, k i x = k r x = k t x ， k i y = k r y = k t y 。

Without losing generality, assumption k i y = k r y = k t y = zero , it can be immediately inferred that the first law is true, and the wave vectors of the incident wave, reflected wave and refracted wave are contained in the incident plane together with the normal of the interface.

Slave wave vector x -The equality of components can be obtained k i sin θ i = k r sin θ r 。

In the same medium, k i = k r 。 So, the second law holds, the angle of incidence θ i Equal to reflection angle θ r 。

The definition formula of applied refractive index: n = c / v = ck / ω ，

It can be inferred that the third law holds: n i sin θ i = n t sin θ t 。

Among them, n t 、 θ t They are the refractive index and refractive angle of the refractive medium.

From the phase relationship among incident wave, reflected wave and refracted wave, three basic laws of geometric optics can be deduced.

theoretical development

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The first person who quantitatively studied the refraction phenomenon was C. Ptolemy, a Greek in the 2nd century AD, who measured the corresponding relationship between the incidence angle and the refraction angle when light refracted from air to water. Although the experimental results were not accurate, he was the first person who quantitatively studied the refraction law through experiments. In 1621, the Dutch mathematician W. Snell accurately determined the rule that the ratio of the cosecant of the incident angle and the refraction angle is a constant through experiments, namely

csc θ i /csc θ t =Constant

Therefore, the law of refraction is also called Snell's law. In 1637, the French R. Descartes first published the law of sine ratio in modern form in his book "Refractive Optics". Like the reflection law of light, the refraction law initially determined by experiments can be proved according to Fermat's principle, Huygens's principle or the electromagnetic theory of light.

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