Fresnel Biprism

Obtaining light interference using Fresnel biprism.

Fresnel biprism is an optical device for obtaining a pair of coherent light beams, proposed by Auguste Fresnel . Biprism is two identical triangular rectangular prisms, with a very small refractive angle, folded by their bases. In practice, biprism is usually made from a plate of glass.

Using biprism, interference of light beams can be observed ^[1] ^[2] .

The use of Fresnel biprisms to obtain coherent light beams is one of the variants of the wavefront division method . In accordance with the number of interfering light beams, the interference obtained using the Fresnel biprism is referred to as two-beam interference ^[1] .

Allows you to measure the wavelength of light radiation by the known refractive index of glass and by the known angle of the prism.

Content

Principle of Operation

To obtain interference, the light source S is arranged symmetrically with respect to the prisms constituting the biprism. The angles of incidence of the rays on the surface of the prism are small, so all the rays are deflected by it at the same angle $\delta$ ${\ displaystyle \ delta}$ equal to $(n-1)\alpha$ ${\ displaystyle (n-1) \ alpha}$ where $n$ ${\ displaystyle n}$ - the refractive index of the material from which the prism is made, and $\alpha$ ${\ displaystyle \ alpha}$ - the refracting angle of the prism.

As a result of such refraction, two coherent light beams are formed, the vertices of which S ₁ and S ₂ can be considered as points of location of imaginary images of the source S. On the screen, coherent rays from sources S ₁ and S ₂ overlap and form an interference pattern, which is a set of alternating light and dark bands ^[3] .

Usually, a narrow slit is used as the light source, parallel to the edge of the biprism and illuminated by a bright source with monochromatic light . In this case, the interference pattern on the screen is a system of alternating light and dark stripes parallel to the gap. In practical demonstration experiments, a high degree of monochromatic radiation is not required, and to obtain an interference pattern, it is enough to cover the white light source with a filter made, for example, of colored glass. If white light is used without a filter, then the interference pattern will consist of multi-colored stripes, and there will be no completely dark stripes, since the places of minimum illumination for light with one wavelength will coincide with the places of maximum illumination for light with a different wavelength ^[3] . As the slit width increases, the illumination of the screen increases, but at the same time, the contrast of the interference pattern decreases, up to its complete disappearance.

In experiments with Fresnel biprism, interference fringes are observed in the region of overlapping beams on the screen at any distance from the screen to the biprism. These bands are said to be not localized ^[1] .

Theory

Distance $d$ ${\ displaystyle d}$ between imaginary sources is determined by the angle of rotation $\delta$ ${\ displaystyle \ delta}$ and distance $a$ ${\ displaystyle a}$ between light source S and prism; at small $\delta$ ${\ displaystyle \ delta}$ for the distance is fulfilled ^[4] :

d=2a(n-1)\alpha .

{\ displaystyle d = 2a (n-1) \ alpha.}

From the general theory of two-beam interference, it is known that the maximums of illumination on the screen are formed at distances $x_{m}$ ${\ displaystyle x_ {m}}$ from the center of the screen satisfying the condition ^[1]

x_{m}=m{\frac {(a+b)}{d}}\lambda ,

{\ displaystyle x_ {m} = m {\ frac {(a + b)} {d}} \ lambda,}

Where

b

{\ displaystyle b}

- the distance between the prism and the screen,

\lambda

{\ displaystyle \ lambda}

Is the wavelength of light, and

m

{\ displaystyle m}

- an integer taking the values 0, ± 1, ± 2, ...

It follows that in the case of a biprism, for the positions of the maxima

x_{m}=m{\frac {(a+b)}{2a(n-1)\alpha }}\lambda .

{\ displaystyle x_ {m} = m {\ frac {(a + b)} {2a (n-1) \ alpha}} \ lambda.}

Accordingly, for distances $\Delta x_{m}$ ${\ displaystyle \ Delta x_ {m}}$ between the maxima the relation is true ^[4]

\Delta x_{m}={\frac {(a+b)}{2a(n-1)\alpha }}\lambda .

{\ displaystyle \ Delta x_ {m} = {\ frac {(a + b)} {2a (n-1) \ alpha}} \ lambda.}

The brightness of the screen at a point with a coordinate $x$ ${\ displaystyle x}$ depends on phase difference $\Delta \phi (x)$ ${\ displaystyle \ Delta \ phi (x)}$ beams interfering at this point:

E(x)=2E_{0}(1+\cos \Delta \phi (x))=4E_{0}\cos ^{2}{\frac {\Delta \phi (x)}{2}},

{\ displaystyle E (x) = 2E_ {0} (1+ \ cos \ Delta \ phi (x)) = 4E_ {0} \ cos ^ {2} {\ frac {\ Delta \ phi (x)} {2 }},}

Where

E_{0}

{\ displaystyle E_ {0}}

- illumination created by one of the interfering beams, and the phase difference has the form

\Delta \phi (x)={\frac {4\pi }{\lambda }}{\frac {a(n-1)\alpha }{(a+b)}}x.

{\ displaystyle \ Delta \ phi (x) = {\ frac {4 \ pi} {\ lambda}} {\ frac {a (n-1) \ alpha} {(a + b)}} x.}

Thus, the brightness of the screen changes from the minimum value $E_{min}=0$ ${\ displaystyle E_ {min} = 0}$ to the maximum $E_{max}=4E_{0}.$ ${\ displaystyle E_ {max} = 4E_ {0}.}$

The area covered by the light beams on the screen in the coordinate direction $x$ ${\ displaystyle x}$ approximately equal in length $2b(n-1)\alpha$ ${\ displaystyle 2b (n-1) \ alpha}$ . Hence, using the above expression for the distance between the maximums of illumination $\Delta x_{m}$ ${\ displaystyle \ Delta x_ {m}}$ , we find that the number of interference bands observed in experiments with Fresnel biprism is equal to:

N={\frac {4ab}{a+b}}{\frac {(n-1)^{2}\alpha ^{2}}{\lambda }}.

{\ displaystyle N = {\ frac {4ab} {a + b}} {\ frac {(n-1) ^ {2} \ alpha ^ {2}} {\ lambda}}.}

Notes

↑ ¹ ² ³ ⁴ Born M. , Wolf E. Fundamentals of Optics. Ed. 2nd. - M .: "Science" , 1973. - S. 245-248. - 720 s.
↑ Fresnel Biprism // Brockhaus and Efron Encyclopedic Dictionary : in 86 volumes (82 volumes and 4 additional). - SPb. , 1890-1907.
↑ ¹ ² Landsberg G.S. Optics. - M .: Fizmatlit , 2003 .-- 848 p. - ISBN 5-9221-0314-8 .
↑ ¹ ² Sivukhin D.V. General course of physics. - 3rd ed., Stereo. - M .: Fizmatlit , MIPT , 2005. - T. IV. Optics. - S. 212-213. - 792 p. - ISBN 5-9221-0228-1 .

Literature

Sivukhin D.V. General course of physics. - 3rd edition, stereotyped. - M .: Fizmatlit , MIPT , 2002. - T. IV. Optics. - 792 p. - ISBN 5-9221-0228-1 .

[Борн-1] ¹ ² ³ ⁴ Born M. , Wolf E. Fundamentals of Optics. Ed. 2nd. - M .: "Science" , 1973. - S. 245-248. - 720 s.

[2] Fresnel Biprism // Brockhaus and Efron Encyclopedic Dictionary : in 86 volumes (82 volumes and 4 additional). - SPb. , 1890-1907.

[Ландсберг-3] ¹ ² Landsberg G.S. Optics. - M .: Fizmatlit , 2003 .-- 848 p. - ISBN 5-9221-0314-8 .

[Сивухин-4] ¹ ² Sivukhin D.V. General course of physics. - 3rd ed., Stereo. - M .: Fizmatlit , MIPT , 2005. - T. IV. Optics. - S. 212-213. - 792 p. - ISBN 5-9221-0228-1 .