Diffraction grating – problems and solutions

1. A grating containing 4000 slits per centimeter is illuminated with a monochromatic light and produces the second-order bright line at a 30° angle. What is the wavelength of the light used? (1 Å = 10^{-10} m)

__Known :__

The distance between slits (d) = 1 / (4000 slits / cm) = 0.00025 cm = 2.5 x 10^{-4} cm = 2.5 x 10^{-6} meters

Order (n) = 2

Sin 30^{o} = 0.5

1 Å = 10^{-10} m

__Wanted:__ Wavelength of the light (λ)

__Solution :__

2. A monochromatic light with a wavelength of 2.5×10^{-7 }m strikes a grating containing 10,000 slits/cm. Determine the angular positions of the second-order bright line.

__Known :__

The distance between slits (d) = 1 / (10,000 slits / cm) = 0.0001 cm = 1 x 10^{-4} cm = 1 x 10^{-6} m

Order (n) = 2

Wavelength (λ) = 2.5 x 10^{-7 }m

__Wanted :__ Angle (θ)

__Solution :__

3. A monochromatic light with wavelength of 5.10^{-7} m strikes a grating. Distance between slit and the viewing screen is 2 m, distance between the third-order fringe and the central fringe is 150 cm. Determine distance between slits.

__Known :__

Wavelength (λ) = 5 x 10^{-7} m

l = 2 m

n = 3

y = 150 cm = 1.5 m

__Wanted :__ distance between slits

__Solution :__

sin θ ≈ tan θ = y / l = 1.5 / 2 = 0.75 = 75 x 10^{-2}

Distance between slits :

d sin θ = n λ

d (75.10^{-2}) = (3)(5.10^{-7})

d (75.10^{-2}) = 15.10^{-7}

d = (15.10^{-7}) / (75.10^{-2})

d = (15/75).10^{-5}

d = (1/5)(10^{-5}) m

d = (0.2)(10^{-5}) m

d = 2.10^{-6 }m

4. A monochromatic light with wavelength of 500 nm (1 nano = 10^{-9}) strikes a grating and produces the fourth-order bright line at an 30° angle. Determine the number of slits per centimeter.

__Known :__

Wavelength (λ) = 500.10^{-9} m = 5.10^{-7} m

θ = 30^{o}

n = 4

__Wanted :__ number of slits per centimeter

__Solution :__

Distance between slits :

d sin θ = n λ

d (sin 30^{o}) = (4)(5.10^{-7})

d (0.5) = 20.10^{-7}

d = (20.10^{-7}) / 0.5

d = 40.10^{-7}

d = 4.10^{-6 }m

Number of slits per centimeter :

x = 1 / 4.10^{-6 }m

x = 0.25.10^{6} / m

x = 0.25.10^{6} / 10^{2} cm

x = 0.25.10^{4} / cm

x = 25.10^{2} / cm

x = 2500 / cm

5. A monochromatic light with wavelength of 500 nm (1 nm = 10^{-9} m) strikes a grating and produces the second-order bright line at an 30° angle. Determine the number of slits per centimeter.

__Known :__

Wavelength (λ) = 500.10^{-9} m = 5.10^{-7} m

θ = 30^{o}

n = 2

__Wanted :__ number of slits per centimeter

__Solution :__

Distance between slits :

d sin θ = n λ

d (sin 30^{o}) = (2)(5.10^{-7})

d (0.5) = 10.10^{-7}

d = (10.10^{-7}) / 0.5

d = 20.10^{-7}

d = 2.10^{-6 }m

Number of slits per centimeter :

x = 1 / d

x = 1 / 2.10^{-6 }m

x = 0.5.10^{6} / 1 m

x = 0.5.10^{6} / 10^{2} cm

x = 0.5.10^{4} / cm

x = 5.10^{3} / cm

x = 5000 / cm

20 conceptual questions and answers about diffraction grating:

**1. Question:** What is a diffraction grating?

**Answer:** A diffraction grating is an optical component with a periodic structure, which splits and diffracts light into several beams traveling in different directions.

**2. Question:** How does diffraction grating differ from a double slit?

**Answer:** While both cause diffraction, a grating has many more slits (or lines) per unit length, producing more diffracted beams and clearer spectral separation.

**3. Question:** Why are the spacings between grating lines crucial?

**Answer:** The spacings determine the angles at which different wavelengths of light are diffracted, thereby affecting the resolution and clarity of the spectrum produced.

**4. Question:** How is the diffraction angle related to the wavelength of light?

**Answer:** The relationship is given by the grating equation: $mλ=dsinθ$, where $m$ is the order of diffraction, $λ$ is the wavelength, $d$ is the grating spacing, and $θ$ is the diffraction angle.

**6. Question:** Why do different colors/wavelengths diffract at different angles?

**Answer:** Because the diffraction angle ($θ$) is directly proportional to the wavelength ($λ$), longer wavelengths (like red) diffract more than shorter ones (like blue).

**7. Question:** How does increasing the number of slits (or lines) in a diffraction grating affect the pattern?

**Answer:** Increasing the number of slits sharpens the diffraction maxima, enhancing the resolution of the spectrum.

**8. Question:** What is meant by the resolving power of a grating?

**Answer:** Resolving power is the grating’s ability to distinguish between two closely spaced wavelengths. It increases with more lines and higher diffraction orders.

**9. Question:** Can diffraction gratings be used with other types of waves besides light?

**Answer:** Yes, they can be used with any wave phenomenon, including sound waves and X-rays.

**10. Question:** How is a transmission grating different from a reflection grating?

**Answer:** In a transmission grating, light passes through the grating material, whereas in a reflection grating, light is reflected off the grating surface.

**11. Question:** What are blazed gratings?

**Answer:** Blazed gratings are reflection gratings where the grooves are angled (or blazed) to direct most of the energy into a specific diffraction order, optimizing efficiency.

**12. Question:** Why is a grating’s groove density (lines per mm) significant?

**Answer:** Groove density determines the grating’s dispersion and resolution. Higher groove densities generally lead to higher resolutions.

**13. Question:** How does a grating produce a spectrum?

**Answer:** When white light interacts with a grating, each wavelength diffracts at a unique angle, spreading them out spatially to produce a spectrum.

**14. Question:** Why might higher orders of diffraction overlap with one another?

**Answer:** For different wavelengths, higher orders can diffract at angles that match lower orders of other wavelengths, causing overlapping spectra.

**15. Question:** Can a grating be used to study non-visible wavelengths?

**Answer:** Yes, gratings can be designed for different parts of the electromagnetic spectrum, including ultraviolet and infrared.

**16. Question:** What role does grating material play in diffraction patterns?

**Answer:** Material properties can affect the efficiency of different diffraction orders and determine the wavelength range for which the grating is optimized.

**17. Question:** How is a holographic diffraction grating produced?

**Answer:** Holographic gratings are created using interference patterns from laser light, rather than mechanically ruling grooves, resulting in less stray light and higher purity spectra.

**18. Question:** Why is the zero-order maximum usually much brighter?

**Answer:** The zero-order maximum corresponds to undiffracted light. Since it doesn’t undergo as much spreading as diffracted orders, its intensity is generally higher.

**19. Question:** Can gratings work with polarized light?

**Answer:** Yes, the diffraction efficiency and pattern can vary based on the polarization state of the incident light.

**20. Question:** Why do gratings have advantages over prisms for spectroscopy?

**Answer:** Gratings can offer better resolution, more linear dispersion of wavelengths, and are often more efficient at spreading out light across the spectrum compared to prisms.

Diffraction gratings are essential in optical research and applications, especially in spectroscopy, allowing for detailed study of light’s spectral components.