Applied Physics for Engineers - Lasers, Fiber Optics, Quantum Physics, Quantum Mechanics, and Quantum Computing

LASERS AND FIBRE OPTICS

1.1 INTRODUCTION

• Objectives:
  • Explain interactions of radiation with matter.
  • Comprehend the principle of lasing systems and applications.
  • Understand the working principle of optical fibre and applications.
  • Classify different types of optical fibres.

1.1.1 Characteristics of Laser

• Laser light characteristics:
  • Monochromatic
  • Coherent
  • Directional
  • Sharply focused
• Applications: scientific research, engineering, and medicine.

1.1.2 Interaction of Radiation with Matter

• Interaction types: absorption and emission.

• Emission types (Einstein, 1917): spontaneous and stimulated.

• Absorption:

  • Photon absorption occurs when $E<em>2 – E</em>1 = hf$, where:
    • $E<em>2 – E</em>1$ is the energy difference of allowed energy states.
    • $hf$ is the photon's energy.
  • The photon disappears; the system moves to a higher energy state.

• Spontaneous Emission:

  • Photon emission occurs when a system transitions from a higher to a lower energy state without external aid.
  • Average lifetime in the excited state: ~$10^{-8}$ s.
  • The system returns to a lower energy state, emitting a photon of energy $hf = E<em>2 – E</em>1$.
  • Ordinary light source: incoherent radiation emitted in random directions.

• Stimulated Emission:

  • A stimulating photon interacts with an excited system, causing it to transition to the ground state before its lifetime.
  • Stimulated and stimulating photons have the same frequency, phase, and polarization and are emitted in the same direction (coherent).
  • Responsible for laser action.

1.1.3 Population Inversion

• From Boltzmann statistics, the ratio of populations in two energy states at equilibrium temperature T is:
  • $\frac{N<em>2}{N</em>1} = e^{-(E<em>2-E</em>1)/kT}$
    • k = Boltzmann constant
    • $N<em>1$ = density of atoms with energy $E</em>1$
    • $N<em>2$ = density of atoms with energy $E</em>2$
• Normal condition: $N<em>2 < N</em>1$. Population of atoms in upper energy state is less than that in lower energy state.
• Population inversion: $N<em>2 > N</em>1$.
  • Necessary for stimulated emission to exceed absorption.
  • Non-equilibrium condition facilitated by metastable states.
  • Metastable states have a longer average lifetime (~$10^{-3}$ s) than ordinary excited states (~$10^{-8}$ s).

1.2 EINSTEIN’S COEFFICIENTS

• Describe the probability of absorption and emission processes.
• Consider two energy states $E<em>1$ and $E</em>2$ ($E<em>2 > E</em>1$).
• $N<em>1$ and $N</em>2$ are the densities of atoms with energies $E<em>1$ and $E</em>2$ respectively.
• $I<em>f df$ is the energy incident per unit volume with frequencies in the range $f$ and $f + df$. $I</em>f$ is the energy density of frequency $f$.

Absorption

• An atom in level $E<em>1$ goes to level $E</em>2$ by absorbing a photon of frequency $f = (E<em>2 – E</em>1)/h$.
• Rate of absorption depends on:
  • Number density of lower energy state, $N_1$.
  • Energy density, $I_f$.
  • Rate of absorption = $B<em>{12} N</em>1 I_f$
    • $B_{12}$ is the Einstein coefficient of induced absorption.

Spontaneous Emission

• An atom in the higher energy level $E<em>2$ transitions to the lower energy level $E</em>1$ voluntarily, emitting a photon.
• Independent of the energy density of incident radiation.
• Rate of spontaneous emission = $A<em>{21} N</em>2$
  • $A_{21}$ is the Einstein coefficient of spontaneous emission.

Stimulated Emission

• Requires an external photon of frequency $[(E<em>2 − E</em>1)/h]$ to stimulate the atom for downward transition and emission of stimulated photons.
• Rate of stimulated emission depends on:
  • Number density of the higher energy state, $N_2$.
  • Energy density, $I_f$.
  • Rate of stimulated emission = $B<em>{21} N</em>2 I_f$
    • $B_{21}$ is the Einstein coefficient of stimulated emission.

Thermal Equilibrium

• The total energy of the system remains unchanged.
• Rate of absorption = Rate of spontaneous emission + Rate of stimulated emission.
  • $B<em>{12} N</em>1 I<em>f = A</em>{21} N<em>2 + B</em>{21} N<em>2 I</em>f$
  • $I<em>f = \frac{A</em>{21} N<em>2}{(B</em>{12} N<em>1 − B</em>{21} N_2)}$
  • $I<em>f = \frac{A</em>{21}}{B<em>{21}} [\frac{1}{\frac{B</em>{12} N<em>1}{B</em>{21} N_2} − 1}]$
• Boltzmann's law: $\frac{N<em>2}{N</em>1} = e^{-(E<em>2-E</em>1)/kT} = e^{- hf / kT}$
• $\frac{N<em>1}{N</em>2} = e^{ hf / kT}$
• Substitute, we get
  • $I<em>f = \frac{A</em>{21}}{B<em>{21}} [\frac{1}{\frac{B</em>{12}}{B_{21}} e^{ hf / kT} − 1}]$
• Planck's law:
  • $I_f = \frac{8 \pi hf^3}{c^3} [\frac{1}{e^{ hf / kT}− 1}]$
• Comparing the equations:
  • $\frac{A<em>{21}}{B</em>{21}} = \frac{8 \pi hf^3}{c^3}$
  • $\frac{B<em>{12}}{B</em>{21}} = 1$ or $B<em>{12} = B</em>{21}$
• The probability of induced absorption equals the probability of stimulated emission.
• At thermal equilibrium:
  • $I_f = \frac{A}{B} [e^{ hf / kT}− 1]$
    • Where A and B represent $A<em>{21}$ and $B</em>{21}$ respectively.

1.2.1 Interpretation of Einstein’s Coefficients

• Dependence of emission nature on frequency:
  • $\frac{A<em>{21}}{B</em>{21}} = \frac{8 \pi hf^3}{c^3}$ implies $\frac{A<em>{21}}{B</em>{21}} ∝ f^3$
  • For higher $ΔE$ values, the probability of spontaneous emission is greater than stimulated emissions.
  • Case (i) hf >> kT:
    • Spontaneous emissions dominate ($A<em>{21} >> B</em>{21}$).
  • Case (ii) $hf ≈ kT$:
    • Stimulated emissions become significant ($A<em>{21}$ & $B</em>{21}$ are comparable).
  • Case (iii) hf << kT:
    • Stimulated emissions dominate ($B<em>{21} >> A</em>{21}$).
• Non-Equilibrium conditions leading to amplification (Laser action):
  • $\frac{Rate \space of \space emission}{Rate \space of \space absorption} = \frac{A<em>{21} N</em>2 + B<em>{21} N</em>2 I<em>f}{B</em>{12} N<em>1 I</em>f} = \frac{N<em>2}{N</em>1} [\frac{A<em>{21}}{B</em>{21} I_f} + 1]$
  • If ΔE << kT, then \frac{A{21}}{B{21} I_f} << 1
  • $\frac{Rate \space of \space emission}{Rate \space of \space absorption} = \frac{N<em>2}{N</em>1}$
  • When the population is inverted ($N<em>2 > N</em>1$),
    • Rate of emission exceeds the rate of absorption.
    • Photons emitted in a particular direction are returned into the system by reflecting them back and forth, then the rate of stimulated emission soon exceeds the absorption rate.
  • Amplification takes place at the cost of downward transitions which will soon reverse the condition, $N<em>2 > N</em>1$.
  • To sustain population inversion, energy is provided to lift atoms to a higher energy level continuously for amplification needs.

1.3 CONSTRUCTION OF LASER SYSTEM

1.3.1 Essential Components of a Laser

• Pumping system

• Lasing/active medium

• Resonant/optical cavity

• Lasing medium:

  • Atomic systems (active centers) with special energy levels.
  • Can be gas, liquid, crystal, or semiconductor.
  • Energy levels: ground state ($E<em>1$), excited state ($E</em>3$), and metastable state ($E_2$).

  1.3.2 He-Ne Laser

• Glass discharge tube filled with He (80%) and Ne (20%) at low pressure.

• Helium: pumping medium.

• Neon gas: lasing medium.

• Simplified energy levels: Eo, E1, E2, and E3.

• Electrons and ions collide with He atoms, raising them to level E3 (a metastable state).

• Collisions between He and Ne atoms transfer excitation energy to Ne atoms (level E2), selectively populating E2 due to resonant energy transfer.

• Population inversion occurs between levels E2 and E1.

• Stimulated emission from level E2 to level E1 generates red laser light.

• Mirror M1: fully reflective; mirror M2: partially reflective.

• Brewster’s windows reduce reflection loss.

1.3.3 Ruby Laser

• Lasing medium is a ruby rod: $Al<em>2O</em>3$ doped with $Cr<em>2O</em>3$.
• $Cr^{3+}$ ions are active centers with energy level structure.
• Resonant cavity: parallel mirrors.
• Pumping: excites atoms in the ground state to higher energy states.
• Atoms in state E3 come down to state E1 by spontaneous emission or metastable state (E2) by collision.
• Atoms in state E2 come down to state E1 by stimulated emission.
• Reflecting ends turn a coherent beam back into the active region for regenerative process.
• Part of the light beam comes out from the partial mirror as a laser pulse
• The output is an intense beam of coherent light.

1.3.4 Semiconductor Laser

• Also called diode laser.
• Active medium: p-n junction diode.
• Principle: emission of recombination energy in the form of light in semiconducting materials.
• Stimulated emission of light is produced by heavily doping the p-n junction diode and applying a high current density.
• The diode is forward biased, with a voltage nearly equal to the band gap voltage of the material.
• Heavy doping produces population inversion in the depletion region.
• Recombination of holes and electrons takes place.
• High current density (20 kA/cm2) is applied to maintain stimulated emission.
• Output laser beam emerges from the p-n junction of the partially reflecting surface.

1.4 APPLICATIONS OF LASER

• Investigating interaction laws of atoms and molecules with high-intensity electromagnetic waves.
• Engineering: optical communication, micro welding, sealing, etc.
• Medical: bloodless surgery (retinal detachment), dental decay, tooth extraction, cosmetic surgery.

1.4.1 Bar Code Scanner

• Laser technology reads barcodes due to precision and directionality.
• Laser beam interacts with black and white stripes of barcode (e.g., Universal Product Code).
• Each digit is encoded using a pattern of seven blocks, which are either black or white, to denote numbers from 0 to 9.
• Directionality allows consistent scanning from various angles.
• The narrow focus distinguishes fine details.
• Black stripes absorb more light; white spaces reflect more light.
• Variations in reflected light intensity captured by a photodiode or CCD.
• Device converts light variations into electrical signals, which the scanner's software decodes into the barcode’s data.
• Barcode read quickly and accurately from a distance.
  Indispensable in retail, warehousing, and industrial applications.

1.4.2 Laser Printer

• Uses a laser beam to produce an image on a photosensitive drum, transferred to paper.
• Process:
  • Charging: Photosensitive drum given an overall positive charge.
  • Writing: Laser beam discharges the drum in specific areas corresponding to the image.
  • Developing: Toner applied to the drum, attracted to charged areas forming the image.
  • Transferring: Paper given a negative charge; toner transferred from drum to paper.
  • Fusing: Toner fused into place using heat.

1.4.3 Laser Cooling

• Uses dissipative light forces for reducing random motion and thus the temperature of small particles (atoms or ions).
• Achievable temperatures: milli-kelvin, micro-kelvin, or nano-kelvin regime.
• If an atom travels toward a laser beam and absorbs a photon, it slows down.
• Photon has momentum $p = E/c = h/λ$.
• Types: Doppler Cooling and Sisyphus Cooling.

1.5 OPTICAL FIBRES

• Thin, flexible strands of transparent dielectric material (glass or plastic).
• Used to guide infrared & visible light waves through curved paths.
  They are basically used to guide infrared & visible light waves through curved paths.

1.5.1 Construction of Optical Fibre

• Central cylindrical core with refractive index $n<em>1$, surrounded by cladding with refractive index $n</em>2$ ($n<em>2 < n</em>1$).
• Material continuity from core to cladding.
• Cladding enclosed in a polyurethane jacket for protection.
• Many such protected fibers are grouped to form a cable.
• Core diameter varies between 10 to 200 μm; cladding varies between 50 to 250 μm.

1.6 PRINCIPLE OF OPTICAL FIBRES

• Works on the principle of total internal reflection of light.
• When light traveling in a denser medium falls on the interface separating denser medium from relatively less dense medium, if the angle of incidence is greater than particular angle called critical angle ($\theta_C$) for the pair of media, the light undergoes total internal reflection.
• Total reflection: almost entire energy returned without energy loss.
• Optical fibers sustain light signal transmission over long distances despite infinite reflections.

1.6.1 Acceptance Angle, Acceptance Cone and Numerical Aperture

• Optical fibre with core index $n<em>1$ and cladding $n</em>2$ in a medium of index $n_0$.

• Ray AO of light enters the core at an angle $θ<em>0$. It refracts at O through an angle $θ</em>1$ and strikes the interface between the core and the cladding at the critical angle.

• Applying Snell’s law of refraction at O:

  • $n<em>0 sin θ</em>0 = n<em>1 sin θ</em>1$

• $sin θ<em>0 = \frac{n</em>1}{n<em>0} sin θ</em>1$

• Applying Snell’s law at B:

  • $n<em>1 sin θ</em>c = n_2 sin 90^\circ$
  • $sin θ<em>c = \frac{n</em>2}{n_1}$

• Since, $θ<em>c = 90 - θ</em>1$

• $sin θ<em>1 = cos θ</em>c$

  • $cos θ<em>c = \frac{n</em>2}{n_1}$

• $sin θ<em>1 = \sqrt{1- \frac{n</em>2^2}{n_1^2}}$

• Substituting the above for the sin equation at o we have

• $sin θ<em>0 = \frac{n</em>1}{n<em>0} \sqrt{1- \frac{n</em>2^2}{n_1^2}}$

• $sin θ<em>0 = \frac{\sqrt{n</em>1^2 - n<em>2^2}}{n</em>0}$

• $n<em>0 sin θ</em>0 = \sqrt{n<em>1^2 - n</em>2^2}$

• $θ_0$ = acceptance angle or half angle of the acceptance cone.

• Acceptance angle is about 5° for a single mode fibre & 10° to 15° for multimode fibres.

• Numerical Aperture (NA) = $n<em>0 sin θ</em>0$, indicates light gathering power.

• Any ray that enters the fibre at an angle less than $θ_0$, strikes the core-cladding interface at angle greater than the critical angle and undergoes total internal reflection each time it strikes the interface; and sustains the light signal transmission over a long distance.

• Fractional refractive index change ():

  • Ratio of the difference in the refractive indices ($n<em>1−n</em>2$) between the core & the cladding to the refractive index $n_1$ of the core.
  • $Δ = \frac{(n<em>1 − n</em>2)}{n_1}$
  • Since $n<em>1 > n</em>2$, $Δ$ is always positive.

• Relation between NA & :

  • Assuming $n<em>0 = 1$, $NA = \sqrt{n</em>1^2 − n_2^2}$
  • (n1 − n2) = n1  and since n1  n2, we can approximate (n1 + n2)  2 n1.
  • Therefore, $NA = n_1 \sqrt{2Δ}$

• The light accepting capacity of a fibre can be increased by making  large. But there are practical limitations to achieve this. Also a very large  may cause signal distortion.

• Skip distance ($L_s$):

  • Distance between two successive reflections of the ray of light which propagates through the optical fibre.
    ${L<em>s = \frac{d}{ tan θ</em>1}}$
  • ${L<em>s = d \sqrt{ \frac{n</em>1^2}{n<em>0^2 sin^2 θ</em>0} - 1}}$
  • ${ θ<em>1 = 90 - θ</em>c}$

1.7 TYPES OF OPTICAL FIBRES

• Based on refractive index profile and geometry:
  • Single mode step index optical fibres
  • Multi-mode step index optical fibres
  • Multi-mode graded index optical fibres
    Multi mode graded index optical fibres
• Number of modes of transmission through an optical fibre:
  • Depending on the launch angle into the fibre, there can be hundreds of ray paths or modes by which energy can propagate down the core.
  • The ray paths corresponding to the same propagating wave front is called a mode.
  • An optical fibre permits a discrete number of modes to propagate through it.
  • Not all the rays that enter the acceptance cone sustain propagation. Only those modes that satisfy the coherent phase condition are successfully propagated.
• Number of modes supported for propagation determined by normalized frequency (V):
  • $V = \frac{n d}{λ} \sqrt{n<em>0^2 - n</em>1^2}$
    • d is the diameter of the core.
    • λ is the wavelength of the light propagated.
  • If V >> 1, then the number of propagated modes is $\frac{V^2}{2}$.

1.7.1 Single Mode Step Index Optical Fibre

• Core has a uniform refractive index $n<em>1$ that abruptly decreases to $n</em>2$ at the core-cladding interface.
• The core diameter is narrow (5-10μm).
• Supports single mode propagation because of its narrow core. Only rays nearly parallel to the fibre axis will travel through

1.7.2 Step-Index Multimode Fibre

• Core has uniform refractive index $n<em>1$ that abruptly decreases to $n</em>2$ at the core-cladding interface.
• Larger core diameter (50-200 μm).
• It also supports a large number of modes for propagation

1.7.3 Graded-Index Multimode Fibre (GRIN)

• Core refractive index decreases gradually from axis radially outward, becoming equal to the cladding index at the interface.
• The refractive index of the cladding remains uniform.
• Dimensions similar to step index multimode fibres.
• Supports a large number of modes for propagation because of its large core diameter.

1.8 ATTENUATION IN OPTICAL FIBRES

• Attenuation: loss of power of the light signal during propagation.
  • Main sources:
    • Absorption
    • Scattering
    • Other losses

1.8.1 Absorption

• Absorption of light during propagation occurs due to:
  • Impurities present in the fibre material.
    • Transition metals (iron, chromium, cobalt, copper etc.).
    • Hydroxy ions (OH-).
  • Intrinsic nature of the material itself, even if the material is free from impurities and inhomogeneities.

1.8.2 Scattering

• Glass is a heterogeneous mixture. Structural inhomogeneities act as scattering centers.
• Energy loss due to scattering resembles Rayleigh scattering proportional to $\frac{1}{λ^4}$.
• Other sources:
  • Trapped gas bubbles
  • Unreacted starting materials.

1.8.3 Other Losses

Microscopic bends(dimensional irregularities and imperfections) may not sustain total internal reflection.
Macroscopic bends (wrapping the fibre on a spool): Fibres can withstand bends of curvature up to about 10cm without significant loss. For higher curvature (smaller radius of curvature) than this, the loss increases exponentially.
Amplification needed in communication applications at regular intervals to compensate for losses. Optical repeater used to boost the signal.

1.9 DISTORTION IN OPTICAL FIBRES

• A light pulse launched into a fibre decreases in amplitude, as it travels along the fibre, due to losses in the fibre. It also spreads during its travel.
• Pulse received at the output is wider than input pulse.
• Distortion arises due to dispersion effects.
• Three mechanisms contribute to distortion (signal spreading):
  • Material dispersion
  • Waveguide dispersion
  • Intermodal dispersion

1.9.1 Material Dispersion

• Dependence of refractive indices of glass and consequently the group velocity of the optical signal.

• A LED emits a broad spectrum of light. Whenever a LED is used as a source, the broad spectrum contains a number of wavelengths.

• Different wavelengths travel through optical fiber at different speeds because the refractive indices of the glass vary with wavelengths. The short wavelength waves travel slower than long wavelength waves in a material.

• Narrow pulses of light tend to broaden.

• Material dispersion is present in both single mode and multimode fibres

• $Dm = \frac{λ (Δλ) c}{L} \frac{d^2 n}{dλ^2}$

  • λ = peak wavelength
  • Δλ = spectral width
  • L = length of the core
  • n = refractive index of the core.

1.9.2 Waveguide Dispersion

• Arises from the guiding properties of the fibre. Pulses at different wavelengths, through propagating in the same mode, travel at slightly different angles.
• Present in both single mode and multimode fibres

1.9.3 Intermodal Dispersion

• A ray of light launched into a fibre follows different zigzag paths. When numerous modes are propagating in a fibre, they travel with different net velocities with respect to fibre axis.
• Parts of the wave arrive at the output before other parts leading to a spread of the input pulse.
• Does not depend on the spectral width of the source. Light pulse from an ideal monochromatic source can still exhibit spreading.
• In single- mode fibres, since there in only one mode, there is no intermodal dispersion.

1.10 APPLICATIONS OF OPTICAL FIBRES

1.10.1 Fibre Optic Networking

• Telecommunication network with optical fibre.
• Optical fibre has become the preferred medium for voice, video and data transmission.
• An optical node is a multifunctional element which basically acts as a transceiver unit capable of receiving, transmitting and processing the optical signal.
• A signal carried on a dedicated wavelength from source to a destination node is known as a light path.
• The data can be sent over these light paths once the connections are set up.

1.10.2 Fibre Optic Communication

• Communication at a distance using light to carry information.
• Transmitter encodes a message into an optical signal, a channel (optical fibre) carriers the signal to its destination, and receiver reproduces the message from the received optical signal.
• Optical communication:
• Can carry large data in digital form.
• Interference and noise free.
• Used in sensors, flexible fibrescope (endoscope) and other industrial applications.

2 QUANTUM PHYSICS

2.1 BLACKBODY RADIATION AND PLANCK’S HYPOTHESIS

• A black body is an object that absorbs all incident radiation.
• A small hole cut into a cavity is the most popular and realistic example. None of the incident radiation escapes.
• The electromagnetic radiation emitted by the black body is called black-body radiation.
• A black-body reaches thermal equilibrium with the surroundings when the incident radiation power is balanced by the power re-radiated.
• The emitted "thermal" radiation from a black body characterizes the equilibrium temperature of the black-body.
• The nature of radiation from a blackbody does not depend on the material of which the walls are made.

2.1.1 Basic Laws of Radiation

• All objects emit radiant energy.
• Hotter objects emit more energy (per unit area) than colder objects. The total power of the emitted radiation is proportional to the fourth power of temperature. This is called Stefan’s Law and is given by,
  • P =  A e T^4
    • P is power radiated from the surface of the object (W).
    • T is equilibrium surface temperature (K).
    • σ is Stefan-Boltzmann constant (= 5.670 x 10−8 W/m2K4 ).
    • A is surface area of the object (m2).
    • e is emissivity of the surface (e =1 for a perfect blackbody).
• The peak of the wavelength distribution shifts to shorter wavelengths as the black body temperature increases. This is Wien’s Displacement Law and is given by,
  • $λ<em>m T = constant = 2.898 × 10^{−3} m.K$, or λm  T^{−1}
    • $λ_m$ is the wavelength corresponding to peak intensity.
    • T is equilibrium temperature of the blackbody.
• Rayleigh-Jeans Law:
• ${I (λ,T) dλ = \frac{2 \pi c k_B T}{λ^4}}$ * kB is Boltzmann’s constant. * c is speed of light in vacuum. * T is equilibrium blackbody temperature.
  • It agrees with experimental measurements only for long wavelengths. It predicts an energy output that diverges towards infinity as wavelengths become smaller and is known as the ultraviolet catastrophe.
• Planck‘s Law:
  • $I (λ,T) dλ = \frac{2 \pi hc^2}{λ^5 (e^{hc / λk_B T} -1)}$
    • ${I (λ,T) dλ}$ is the intensity or power per unit area emitted in the wavelength interval $dλ$
    • $h$ is Planck’s constant
      *$k_B$ is Boltzmann's constant