Engineering Physics (BT-201) Unit 4 Notes

How unit 4 is examined

This unit covers Einstein coefficients, population inversion, He-Ne, CO2 and ruby lasers, laser properties, applications, and optical fibre (acceptance angle, NA, V number, attenuation); gas lasers, Einstein A-B and fibre numericals carry the marks.

Einstein’s theory of matter radiation interaction and A and B coefficients

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Definition. Einstein showed that an atom interacts with radiation of frequency $\nu$ in three ways: absorption, spontaneous emission and stimulated emission. The rate constants $B_{12}$, $A_{21}$ and $B_{21}$ are the Einstein coefficients.

Key points.

Absorption: an atom in level 1 takes a photon $h\nu = E_2 - E_1$ and rises to level 2; rate $= B_{12} N_1 u(\nu)$.
Spontaneous emission: an excited atom drops to level 1 on its own and emits a photon in a random direction and phase; rate $= A_{21} N_2$.
Stimulated emission: an incident photon makes the excited atom emit a twin photon with the same frequency, phase, direction and polarisation; rate $= B_{21} N_2 u(\nu)$.
Here $u(\nu)$ is the energy density of radiation, and $N_1, N_2$ are the populations of the lower and upper levels.

Derivation. At thermal equilibrium upward and downward rates are equal:

$$N_1 B_{12} u(\nu) = N_2 A_{21} + N_2 B_{21} u(\nu)$$

$$u(\nu) = \frac{A_{21}}{B_{12}(N_1/N_2) - B_{21}}$$

Boltzmann gives $N_1/N_2 = e^{h\nu/kT}$, so

$$u(\nu) = \frac{A_{21}}{B_{12}e^{h\nu/kT} - B_{21}}$$

Planck's law is $u(\nu) = \dfrac{8\pi h\nu^3}{c^3}\dfrac{1}{e^{h\nu/kT}-1}$. Comparing term by term:

$$B_{12} = B_{21}, \qquad \frac{A_{21}}{B_{21}} = \frac{8\pi h\nu^3}{c^3}$$

==The probability of absorption equals that of stimulated emission ($B_{12}=B_{21}$), and $A_{21}/B_{21} = 8\pi h\nu^3/c^3$.==

Comparison.

Basis	Spontaneous emission	Stimulated emission
Trigger	Occurs on its own, no photon needed	Needs an incident photon of energy $h\nu$
Photons out	One	Two, identical
Phase	Random, incoherent	Same phase, coherent
Direction	Random	Same as incident photon
Rate	$A_{21}N_2$	$B_{21}N_2u(\nu)$
Source	Ordinary lamps	Lasers

Answer frame. Open by naming the three processes with their rates; derive in the order equilibrium equation, $u(\nu)$, Boltzmann substitution, Planck comparison; close with the two results. For the difference question, give the table and then the applications from the Applications section.

Asked: [7 marks] (Nov 2022, Jun 2025) Derive the relationship between Einstein A and B coefficients. What are the Einstein coefficients? Asked: [7 marks] (Jun 2023) Write down the difference between spontaneous and stimulated emission. Write down the applications of laser.

Amplification of light by population inversion

Definition. Population inversion is the non-equilibrium state in which the upper level holds more atoms than the lower level, $N_2 > N_1$.

Key points.

At thermal equilibrium $N_2/N_1 = e^{-\Delta E/kT}$, so the lower level is always more populated and light is absorbed more than it is emitted.
Once $N_2 > N_1$, stimulated emission beats absorption and an incident beam is amplified as it passes through.
Inversion is produced by pumping (optical, electrical discharge or collisions) and needs a metastable upper level, with a life of about $10^{-3}$ s, so that atoms stay there long enough to accumulate.
Inversion plus a resonator (two mirrors) gives laser action.

Asked: [5 marks] (Dec 2023) Explain population inversion.

Different types of lasers: gas lasers (He-Ne, CO2)

Definition. A gas laser uses an electric discharge to pump a gas mixture placed between two mirrors.

Diagram. He-Ne: a glass tube with Brewster windows, anode and cathode, a fully reflecting mirror at one end and a partially reflecting mirror at the other, and a high-voltage supply (about 1 kV). Energy-level diagram: He 2s levels (20.61 eV) transfer energy to Ne 3s/5s (20.66 eV); Ne drops 3s to 2p.

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He-Ne laser.

Construction: a tube about 30 cm long holds He and Ne at about 10:1 (or 5:1) and low pressure, closed by a fully reflecting and a partially transmitting mirror, with Brewster windows for polarised output.
Electrons in the discharge excite He atoms to the metastable levels $2^1S$ and $2^3S$.
Excited He collides with unexcited Ne and gives up its energy by resonant transfer, since the levels match closely, which inverts the Ne population.
Ne is the lasing atom: stimulated emission between 3s and 2p gives red light at $632.8$ nm; other lines are 1.15 and 3.39 $\mu$m.
Ne then decays to the ground state by collisions with the tube wall, and output is about 1 to 10 mW.

CO2 laser.

Construction: a discharge tube holds CO2, N2 and He, with mirrors as resonator.
CO2 has three vibrational modes: symmetric stretch (100), bending (020) and asymmetric stretch (001).
Electrons excite N2 to $v=1$, which is very near the CO2 (001) level, so resonant collision transfers energy to CO2.
Inversion forms at (001); lasing goes to (100) at $10.6\ \mu$m and to (020) at $9.6\ \mu$m, in the infrared.
He empties the lower levels and cools the gas by conducting heat, so inversion is maintained. Output power is high, from watts to kilowatts, with efficiency about 10 to 20 percent.

<mark>In the He-Ne laser, helium is pumped and neon lases at 632.8 nm; in the CO2 laser, nitrogen is pumped and CO2 lases at 10.6 $\mu$m.</mark>

Answer frame. Open with "A gas laser is pumped by electric discharge"; draw the tube and then the energy-level diagram; develop construction, pumping, resonant transfer, inversion and lasing line in that order; close with wavelength and colour or infrared band.

Pitfall: In He-Ne the helium does not lase; it only passes energy to neon.

Asked: [7 marks] (Nov 2022, Dec 2023, Dec 2024) Explain the construction and working of He-Ne laser (with labelled diagram). Asked: [7 marks] (Jun 2023, Jun 2025) Explain construction and working of CO2 laser with suitable energy level diagram.

Solid-state lasers (ruby, Neodymium)

Definition. A solid-state laser uses a crystal or glass doped with ions as the active medium and is optically pumped by a flash lamp. Ruby is the first laser and a three-level system.

Diagram. A cylindrical ruby rod with one fully silvered and one partially silvered end, wound by a helical xenon flash lamp connected to a power supply, with cooling. Beside it, a three-level diagram: ground $E_1$, broad pump bands $E_3$, metastable $E_2$.

Key points.

The medium is a ruby rod, $Al_2O_3$ doped with about 0.05 percent $Cr^{3+}$ ions, which give the pink colour and do the lasing.
The rod is placed inside a helical xenon flash lamp, and its silvered ends work as the resonator.
Green and blue light from the flash lamp pumps $Cr^{3+}$ ions from the ground level $E_1$ to the broad band $E_3$.
The ions decay quickly and without radiation to the metastable level $E_2$, whose life is about 3 ms, and a population inversion builds up between $E_2$ and $E_1$.
A few ions fall spontaneously to $E_1$, and their photons stimulate the rest, so an intense coherent beam at $694.3$ nm (red) leaves through the partial mirror.
Operation is pulsed, because the lamp works in flashes and three-level lasing needs more than half the ions raised.
Nd:YAG is a four-level laser: $Nd^{3+}$ in yttrium aluminium garnet gives $1064$ nm in the infrared and works continuously at higher efficiency.

<mark>Ruby laser: optical pumping of $Cr^{3+}$, fast non-radiative decay to a metastable level, and stimulated emission at 694.3 nm.</mark>

Answer frame. Open with "Ruby laser is a three-level pulsed solid-state laser"; draw the rod-and-lamp figure and the energy diagram; develop points 1-5; close with the wavelength and that it is pulsed.

Asked: [7 marks] (Jun 2022, Jun 2025) Explain the construction and working of Ruby laser with neat diagram.

Properties of laser beams: mono-chromaticity, coherence, directionality and brightness

Definition. LASER stands for Light Amplification by Stimulated Emission of Radiation; its light differs from ordinary light in four ways.

Key points.

Monochromaticity: the light has a very narrow line width $\Delta\nu$, essentially a single colour, whereas a lamp emits a wide band of wavelengths.
Coherence: all waves are in phase, with temporal coherence (fixed phase along the beam over the coherence length) and spatial coherence (fixed phase across the wavefront).
Directionality: the beam is almost parallel, with divergence of milliradians or less, so it travels far without spreading; an ordinary source spreads in all directions.
Brightness: the power is packed into a tiny area and solid angle, so intensity is enormously higher than the brightest lamp of the same power.
These properties follow from stimulated emission, where every new photon copies the phase, direction and frequency of the one that caused it.

<mark>Laser light is highly monochromatic, coherent, directional and bright.</mark>

Answer frame. Open with the expansion of LASER; give the four properties in the order above with one contrast to ordinary light each; close with the stimulated-emission reason.

Asked: [7 marks] (Jun 2022, Dec 2023, Dec 2024) Discuss the properties of laser light. What is LASER?

Laser speckles

Definition. A speckle pattern is a grainy pattern of bright and dark spots seen when laser light is scattered from a rough surface.

Key points.

The rough surface scatters coherent waves with random phases, and these interfere at the eye or screen.
Bright spots are constructive interference and dark spots destructive interference.
Speckle is useful in speckle interferometry for measuring strain and vibration, but it is noise in imaging and holography.

Applications of lasers in science, engineering and medicine

Definition. Lasers are used wherever an intense, narrow, coherent beam is needed.

Key points.

Engineering: cutting, drilling and welding of metals (CO2 laser), precision measurement of distance (LIDAR), barcode scanners and holography.
Communication: information travels through optical fibres carried by laser light.
Medicine: eye surgery (LASIK, retinal detachment), bloodless surgery, endoscopy, dermatology and cancer treatment by photodynamic therapy.
Science: spectroscopy, laser cooling of atoms, and measuring the Earth-Moon distance.

Asked: [5 marks] (Dec 2023) Write down the applications of LASER in engineering and medicine.

Introduction to Optical fiber

Definition. An optical fibre is a thin flexible glass or plastic strand that guides light by total internal reflection (TIR); it has a core of index $n_1$ surrounded by a cladding of lower index $n_2$.

Diagram. A fibre cross-section: core ($n_1$), cladding ($n_2$), outside medium ($n_0$), light entering at angle $\theta_0$ within the acceptance cone and zig-zagging by TIR.

Key points.

Principle: when light goes from a denser to a rarer medium at an angle of incidence above the critical angle $\phi_c = \sin^{-1}(n_2/n_1)$, it is totally reflected.
Condition: $n_1 > n_2$, and the ray must meet the core-cladding boundary at more than $\phi_c$.
Propagation: a ray entering within the acceptance cone is reflected again and again at the boundary and reaches the far end with almost no loss.
TIR is important because it traps the light in the core over long distances, with low attenuation and high bandwidth.
Fibres are immune to electromagnetic interference, light, thin and flexible, and can be routed freely.
Related terms: population inversion is $N_2 > N_1$ (see above), NA and V number are defined below.

<mark>An optical fibre works on total internal reflection at the core-cladding boundary, which needs $n_1 > n_2$.</mark>

Answer frame. Open with the principle; draw the fibre with core, cladding and acceptance cone; explain propagation, then TIR importance; for the definitions question give NA, population inversion and V number in a line each with formula.

Asked: [5 marks] (Dec 2023) Write down the importance of total internal reflection in optical fiber. Asked: [7 marks] (Jun 2023) What is the working principle of an optical fibre? How light propagates through it? Define: numerical aperture, population inversion, V-number.

Acceptance angle and cone

Definition. The acceptance angle $\theta_0$ is the maximum angle at the fibre end, to the axis, for which light is still guided by TIR; rotating it about the axis gives the acceptance cone.

Diagram. Draw the core with the ray entering from medium $n_0$ at $\theta_0$, refracting at angle $r$, and meeting the core-cladding boundary at angle $\phi_c$ to the normal, so that $r = 90^\circ - \phi_c$.

Derivation.

Snell's law at the entry face: $n_0\sin\theta_0 = n_1\sin r$.
For the limiting ray, TIR at the boundary needs $\sin\phi_c = n_2/n_1$.
Since $r = 90^\circ - \phi_c$, $\sin r = \cos\phi_c = \sqrt{1 - n_2^2/n_1^2}$.
So $n_0\sin\theta_0 = n_1\sqrt{1 - n_2^2/n_1^2} = \sqrt{n_1^2 - n_2^2}$.

$$\theta_0 = \sin^{-1}\!\left(\frac{\sqrt{n_1^2 - n_2^2}}{n_0}\right), \qquad n_0 = 1 \text{ (air)}$$

Any ray at an angle less than $\theta_0$ is guided; any larger angle refracts into the cladding and is lost.

Example. Given $n_1 = 1.55$, $n_2 = 1.50$, $n_0 = 1$.

Step	Working
$\sin\theta_0$	$\sqrt{1.55^2 - 1.50^2} = \sqrt{2.4025 - 2.25} = \sqrt{0.1525} = 0.3905$
$\theta_0$	$\sin^{-1}(0.3905)$

Answer: $\theta_0 \approx 22.98^\circ$. For $n_1 = 1.75$, $n_2 = 1.70$: $\sqrt{3.0625 - 2.89} = \sqrt{0.1725} = 0.4153$, so $\theta_0 \approx 24.54^\circ$.

Answer frame. Open with the definition of $\theta_0$; draw the ray diagram; write the four derivation steps; substitute the numbers; close with the boxed angle.

Asked: [7 marks] (Jun 2022, Dec 2024) Deduce the expression for acceptance angle of an optical fiber. Find the acceptance angle of a fiber with core and cladding indices 1.55 and 1.50 (Dec 2024: 1.75 and 1.70).

Numerical aperture

Definition. Numerical aperture (NA) is the light-gathering ability of a fibre, equal to the sine of the acceptance angle (in air): $\text{NA} = n_0\sin\theta_0$.

Formula.

$$\text{NA} = \sqrt{n_1^2 - n_2^2}, \qquad \theta_0 = \sin^{-1}(\text{NA})$$

Key points.

A larger NA means a larger acceptance cone, so the fibre collects more light from the source.
NA depends only on the core and cladding indices, not on the fibre size.
Typical values are 0.1 to 0.5.
The relative index difference is $\Delta = (n_1 - n_2)/n_1$, and $\text{NA} \approx n_1\sqrt{2\Delta}$.

Example 1. $n_1 = 1.55$, $n_2 = 1.50$: $\text{NA} = \sqrt{2.4025 - 2.25} = \sqrt{0.1525}$, NA $\approx 0.3905$.

Example 2. $n_1 = 1.45$, $n_2 = 1.41$: $\text{NA} = \sqrt{2.1025 - 1.9881} = \sqrt{0.1144} = 0.338$, then $\theta_0 = \sin^{-1}(0.338)$, NA $\approx 0.338$, $\theta_0 \approx 19.77^\circ$.

Answer frame. Open with the definition and $\text{NA} = \sin\theta_0$; draw the ray diagram with $n_0, n_1, n_2$; give the formulas, then the numerical; close with the value and unit-less NA.

Asked: [7 marks] (Nov 2022) Explain the numerical aperture of an optical fiber. Calculate NA for core and cladding indices 1.55 and 1.50. Asked: [7 marks] (Jun 2025) Explain the Numerical aperture and acceptance angle of Optical Fiber. Asked: [4 marks] (Dec 2023) Calculate the numerical aperture and acceptance angle for a fibre with core and cladding indices 1.45 and 1.41.

V number

Definition. The V number (normalised frequency) decides how many modes a fibre carries:

$$V = \frac{2\pi a}{\lambda}\,\text{NA}$$

Key points.

Here $a$ is the core radius and $\lambda$ the wavelength of light.
A step-index fibre is single-mode when $V < 2.405$; for larger $V$ it is multimode.
The number of modes is about $V^2/2$.
A small core radius or a long wavelength gives a small $V$.

Attenuation

Definition. Attenuation is the loss of optical power as light travels along a fibre, measured in decibels per kilometre:

$$\alpha = \frac{10}{L}\log_{10}\frac{P_{in}}{P_{out}}\ \text{dB/km}$$

Key points.

Absorption by impurities (such as OH ions) and by the glass itself turns light into heat.
Scattering, mainly Rayleigh scattering from small density variations, varies as $1/\lambda^4$.
Bending losses arise from macro-bends and micro-bends that let light escape into the cladding.
Silica fibre is best at 1.55 $\mu$m, with loss about 0.2 dB/km.

Last-minute revision

LASER = Light Amplification by Stimulated Emission of Radiation.
Rates: absorption $B_{12}N_1u$, spontaneous $A_{21}N_2$, stimulated $B_{21}N_2u$.
$B_{12} = B_{21}$ and $A_{21}/B_{21} = 8\pi h\nu^3/c^3$.
Population inversion means $N_2 > N_1$ and needs pumping and a metastable level.
He-Ne: 632.8 nm red, He:Ne about 10:1, He pumped, Ne lases.
CO2: 10.6 and 9.6 $\mu$m; N2 pumps, He depopulates the lower levels.
Ruby: $Al_2O_3$ with $Cr^{3+}$, 694.3 nm, three-level, pulsed; Nd:YAG 1064 nm, four-level.
Laser properties: monochromatic, coherent, directional, bright.
$\text{NA} = \sqrt{n_1^2 - n_2^2} = \sin\theta_0$ (air).
1.55/1.50 gives NA 0.3905, $\theta_0 = 22.98^\circ$; 1.75/1.70 gives 0.4153, $24.54^\circ$; 1.45/1.41 gives 0.338, $19.77^\circ$.
$V = 2\pi a\,\text{NA}/\lambda$; single-mode if $V < 2.405$.

Memory hooks

ABB: A for spontaneous, B and B for absorption and stimulated, and the two B's are equal.
He-Ne: "Helium helps, Neon does the lasing", red at 632.8.
CO2: N2 pumps, He cools, CO2 lases in the infrared at 10.6.
Ruby is red, pulsed and three-level: 694.3 nm.
Sine of the acceptance angle is the NA: "NA = sin of the angle".

Coverage checklist

Einstein’s theory of matter radiation interaction and A and B coefficients: A-B derivation (Nov 2022, Jun 2025); spontaneous vs stimulated (Jun 2023).
amplification of light by population inversion: population inversion (Dec 2023).
different types of lasers: gas lasers (He-Ne, CO2): He-Ne (Nov 2022, Dec 2023, Dec 2024); CO2 (Jun 2023, Jun 2025).
solid-state lasers(ruby, Neodymium): ruby laser (Jun 2022, Jun 2025).
Properties of laser beams: mono-chromaticity, coherence, directionality and brightness: properties of laser light (Jun 2022, Dec 2023, Dec 2024).
laser speckles: covered, not asked.
applications of lasers in science, engineering and medicine: engineering and medicine (Dec 2023).
Introduction to Optical fiber: TIR importance (Dec 2023); working principle and terms (Jun 2023).
acceptance angle and cone: derivation and numerical (Jun 2022, Dec 2024).
Numerical aperture: NA explain and numerical (Nov 2022, Dec 2023, Jun 2025).
V number: covered, not asked.
attenuation: covered, not asked.

Account

Engineering Physics (BT-201) - Unit 4 Short Notes

How unit 4 is examined

Einstein’s theory of matter radiation interaction and A and B coefficients

Amplification of light by population inversion

Different types of lasers: gas lasers (He-Ne, CO2)

Solid-state lasers (ruby, Neodymium)

Properties of laser beams: mono-chromaticity, coherence, directionality and brightness

Laser speckles

Applications of lasers in science, engineering and medicine

Introduction to Optical fiber

Acceptance angle and cone

Numerical aperture

V number

Attenuation

Last-minute revision

Memory hooks

Coverage checklist