Diffraction grating angular separation

In summary: Using this approximation,we can rewrite the equations as D*Theta1 = m*(L+dL) and D*Theta2 = m*L. Subtracting these two equations leads to D*(Theta1-Theta2) = m*dL. Now, using the small angle approximation again, we get dTheta = Theta1-Theta2 = (m/D)*dL = dL/(D/m). This is the same result as before, but it matches the desired formula of dTheta = dL / ((D/m)^2+L^2)^0.5. This is because the small angle approximation is not valid for large angles, but the desired formula takes that into account.
  • #1
charlesh
2
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Homework Statement


Light consisting of two nearly equal wavelengths L and L+dL, where dL << L, is incident on a diffraction grating. The slit separation of the grating is D. Show that the angular separation of these two wavelengths in the m'th order is dTheta = dL / ((D/m)2+L2)0.5


Homework Equations



D sin(Theta) = mL (m=0,+-1, +-2,...)

The Attempt at a Solution



D sin(Theta1)=m(L+dL)
D sin(Theta2)=m(L)
dTheta = sin(Theta1)-sin(Theta2) = (m/D)(L+dL-L) = (m/D)dL= dL/(D/m)
This answer doesn't match the formula I need to proof: dTheta = dL / ((D/m)2+L2)0.5
 
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  • #2
charlesh said:

Homework Statement


Light consisting of two nearly equal wavelengths L and L+dL, where dL << L, is incident on a diffraction grating. The slit separation of the grating is D. Show that the angular separation of these two wavelengths in the m'th order is dTheta = dL / ((D/m)2+L2)0.5


Homework Equations



D sin(Theta) = mL (m=0,+-1, +-2,...)

The Attempt at a Solution



D sin(Theta1)=m(L+dL)
D sin(Theta2)=m(L)
dTheta = sin(Theta1)-sin(Theta2) = (m/D)(L+dL-L) = (m/D)dL= dL/(D/m)
This answer doesn't match the formula I need to proof: dTheta = dL / ((D/m)2+L2)0.5

highlighted part is incorrect
 
  • #3
Thanks for the response.
For small angle, sin(Theta) is very close to Theta (in radian).
 

FAQ: Diffraction grating angular separation

What is a diffraction grating?

A diffraction grating is a device consisting of a large number of equally spaced parallel slits or grooves that are used to separate light into its component wavelengths. It works by causing light to interfere with itself, producing a pattern of bright and dark fringes known as a diffraction pattern.

How does a diffraction grating separate light?

A diffraction grating separates light by causing it to diffract, or bend, as it passes through the slits or grooves. This bending of light results in different wavelengths being separated and appearing at different angles, allowing for the identification of the components of white light.

What is angular separation in diffraction gratings?

Angular separation in diffraction gratings refers to the angle between the diffracted beams of light of different wavelengths. This angle is determined by the spacing of the slits or grooves on the grating, as well as the wavelength of the incident light.

How is the angular separation of a diffraction grating calculated?

The angular separation of a diffraction grating can be calculated using the equation θ = mλ/d, where θ is the angular separation, m is the order of diffraction, λ is the wavelength of light, and d is the spacing between the slits or grooves on the grating. This equation is known as the grating equation.

What are some applications of diffraction grating angular separation?

Diffraction grating angular separation is used in various scientific and technological applications, such as spectroscopy, laser beam analysis, and optical communication. It is also used in the production of holographic images, as well as in astronomy to study the composition of stars and galaxies.

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