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Laser Physics - CSIR NET

# Laser Physics: Comprehensive Study Notes

1. Coherence

The concept of coherence is central to understanding laser physics. A wave is defined as being perfectly coherent if it appears as a pure sine wave for an infinitely long period of time or extends across an infinitely large space. In such an ideal wave, there is a definite, predictable relationship between the phase of the wave at a given time and a certain time later, or between a given point in space and a certain distance away.

In reality, no actual light source emits a perfectly coherent wave. Real light waves are pure sine waves only for a limited period or in a limited space; thus, they are termed partially coherent waves. Coherence is broadly classified into two types based on the criteria of time and space:

  • Temporal Coherence: This refers to the correlation of phase with time at a single point in space. It exists when the phase does not vary linearly with time.
  • Spatial Coherence: This refers to the correlation of phase between different points in space at a single instant of time. It exists when the phase does not vary randomly with distance.

1.1 Temporal Coherence

Temporal coherence is intrinsically linked to the monochromaticity of the light source. It describes how long a wave train remains sinusoidal.

Key Parameters

1. Coherence Time ($\tau$):

The average time interval for which a definite phase relationship exists in the light beam.

2. Coherence Length ($L$):

The distance over which the field maintains a sinusoidal nature. It is related to the coherence time by the speed of light ($c$).

L = \tau \times c

Relation to Monochromaticity (Spectral Width)

Consider a light beam that is not perfectly monochromatic but has a wavelength spread $\Delta\lambda$. This means the beam consists of all wavelengths lying between $\lambda$ and $\lambda + \Delta\lambda$ (or $\lambda \pm \frac{1}{2}\Delta\lambda$).

In an interference experiment (like Michelson Interferometer), the interference pattern produced by wavelength $\lambda$ and wavelength $\lambda+\Delta\lambda$ would eventually fall out of step. The distance at which the interference pattern disappears is the coherence length. Mathematically, this is derived as:

L = \frac{\lambda^2}{\Delta\lambda}

Interpretation: If $\Delta\lambda = 0$ (perfectly monochromatic), then the coherence length $L$ becomes infinite, and the coherence time $\tau$ becomes infinite. This represents perfect temporal coherence.

We can also express this in terms of frequency. Since frequency $\nu = c/\lambda$, differentiating gives $|\Delta\nu| = (c/\lambda^2)\Delta\lambda$. Substituting this into the coherence length formula:

From $L = \lambda^2 / \Delta\lambda$, substitute $\Delta\lambda = (\lambda^2/c)\Delta\nu$:

L = \frac{c}{\Delta\nu}

Since $\tau = L/c$, we get:

\tau = \frac{1}{\Delta\nu}

This fundamental relation states that the longer the coherence time, the smaller the frequency spread (linewidth), and thus the more monochromatic the source.

1.2 Spatial Coherence

Spatial coherence refers to the phase relationship between radiation fields at different points in space.

Consider a light source $S$ and two points $A$ and $B$ lying on a line extending from the source. The phase relationship between $A$ and $B$ depends on the distance $AB$ relative to the coherence length $L$.

  • If $AB \ll L$: There is a definite phase relationship; high coherence.
  • If $AB \gg L$: There is no correlation; no coherence.

Now consider points $A$ and $C$ which are equidistant from source $S$.
If $S$ is a true point source, the waves reach $A$ and $C$ in exactly the same phase, implying perfect spatial coherence.
If $S$ is an extended source, waves from different parts of the source travel different path lengths to reach $A$ and $C$. The points $A$ and $C$ will no longer maintain a fixed phase relationship. Thus, extended sources have poor spatial coherence.

2. Physics of the LASER

The term LASER stands for Light Amplification by Stimulated Emission of Radiation. It is a device that emits an intense, almost perfectly monochromatic, highly directional, and coherent beam of light. Its operation is based on the quantum mechanical phenomenon of stimulated emission, first predicted by Albert Einstein in 1916.

2.1 Interaction of Radiation with Matter

An atomic system is characterized by discrete energy states. In thermal equilibrium, most atoms occupy the lowest energy state (ground state). Atoms can transition between states by absorbing or emitting radiation. There are three fundamental processes involved:

A. Stimulated Absorption

An atom initially in a lower energy state ($E_1$) absorbs a photon of frequency $\nu$ and rises to a higher energy state ($E_2$). This can only happen if the photon energy matches the energy gap:

h\nu = E_2 - E_1

The rate of absorption depends on the spectral energy density of the radiation $u(\nu)$ and the number of atoms in the lower state $N_1$.

Rate of Stimulated Absorption: \frac{\partial N}{\partial t} = -B_{12} u(\nu) N

Where $B_{12}$ is the Einstein B coefficient for stimulated absorption.

B. Spontaneous Emission

An atom in an excited state ($E_2$) is generally unstable. It stays there for a very short duration (typically $10^{-8}$ seconds) before decaying to the lower state ($E_1$) of its own accord. In doing so, it emits a photon of frequency $\nu$. This emission is random in direction and phase.

The rate of decay depends only on the number of atoms in the excited state ($N$). It is independent of the external radiation field.

Rate of Spontaneous Emission: \frac{\partial N}{\partial t} = -A_{21} N

Where $A_{21}$ is the Einstein A coefficient for spontaneous emission.

Solving this differential equation gives the decay law:

N(t) = N_0 e^{-A_{21}t} = N_0 e^{-t/\tau_{sp}}

Here, $\tau_{sp} = 1/A_{21}$ is the average lifetime of the atom in the excited state.

C. Stimulated Emission

This is the crucial process for laser action. If an atom is already in an excited state ($E_2$), an incident photon of appropriate frequency ($\nu$) can stimulate the atom to decay to $E_1$, emitting an additional photon.

Properties of Stimulated Emission:
The emitted photon is identical to the incident photon in every way:
  • Same Frequency
  • Same Phase
  • Same Direction of Propagation
  • Same Polarization
This results in the coherent amplification of the incident light wave.

The probability of this transition depends on the energy density $u(\nu)$ and the population of the excited state $N_2$.

Rate of Stimulated Emission: \frac{\partial N}{\partial t} = -B_{21} u(\nu) N

Where $B_{21}$ is the Einstein B coefficient for stimulated emission.

2.2 Relation Between Einstein Coefficients

We can determine the relationship between $A$ and $B$ coefficients by considering an assembly of atoms in thermal equilibrium with blackbody radiation at temperature $T$.

Step 1: Equilibrium Condition

In equilibrium, the rate of upward transitions (Absorption) must equal the rate of downward transitions (Spontaneous + Stimulated Emission).

N_1 B_{12} u(\nu) = N_2 A_{21} + N_2 B_{21} u(\nu) Step 2: Solve for Energy Density

Rearranging the equation to solve for $u(\nu)$:

u(\nu) (N_1 B_{12} - N_2 B_{21}) = N_2 A_{21} u(\nu) = \frac{N_2 A_{21}}{N_1 B_{12} - N_2 B_{21}} = \frac{A_{21}}{B_{21}} \frac{1}{\left(\frac{N_1}{N_2}\frac{B_{12}}{B_{21}}\right) - 1} Step 3: Apply Boltzmann Statistics

According to Boltzmann's distribution law:

\frac{N_1}{N_2} = e^{\frac{E_2 - E_1}{k T}} = e^{\frac{h\nu}{k T}}

Substituting this back:

u(\nu) = \frac{A_{21}}{B_{21}} \frac{1}{\frac{B_{12}}{B_{21}} e^{h\nu/kT} - 1} Step 4: Comparison with Planck's Law

Planck's radiation formula is:

u(\nu) = \frac{8\pi h \nu^3}{c^3} \frac{1}{e^{h\nu/kT} - 1} Step 5: Final Relations

Comparing the coefficients, we arrive at two fundamental results:

1. \quad B_{12} = B_{21}

(Stimulated absorption and emission probabilities are identical)

2. \quad \frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3}

(The ratio of spontaneous to stimulated emission coefficients is proportional to $\nu^3$)

3. Essential Conditions for Lasing

To construct a working laser, three primary conditions must be satisfied to ensure light amplification dominates over absorption.

3.1 Population Inversion

Under normal thermal equilibrium, the population of the lower energy state ($N_1$) is much larger than the population of the excited state ($N_2$). Consequently, the rate of absorption ($B_{12} N_1 u(\nu)$) is far greater than the rate of stimulated emission ($B_{21} N_2 u(\nu)$). Light passing through such a medium will be attenuated.

To achieve amplification, we must create a non-equilibrium distribution where $N_2 > N_1$. This condition is called Population Inversion. It essentially means there are more atoms available to emit light than to absorb it.

3.2 Pumping

The process of supplying external energy to the medium to raise atoms from lower states to higher states and achieve population inversion is called Pumping. Common methods include:

  • Optical Pumping: Using a strong light source (like a flash lamp or arc lamp). Used in Ruby and Nd:YAG lasers.
  • Electrical Pumping: Excitation via electron impact in an electric discharge. Used in He-Ne, Argon ion, and CO2 lasers.
  • Chemical Pumping: Energy derived from exothermic chemical reactions.
  • Gas-Dynamic Pumping: Excitation achieved through the supersonic expansion of gases.

3.3 Optical Resonator

A medium with population inversion acts as an amplifier. To convert this amplifier into an oscillator (a source of light), we need positive feedback. This is achieved by placing the active medium between two mirrors to form an Optical Resonator (or Cavity).

  • One mirror is fully reflecting (Reflectivity $\approx 100\%$).
  • The other mirror is partially reflecting (Reflectivity $< 100\%$), allowing the output beam to emerge.

Light bounces back and forth between the mirrors, passing through the active medium multiple times and growing in intensity via stimulated emission.

Longitudinal Modes

For a stable standing wave pattern to form within the cavity of length $L$, the total phase change for a round trip must be an integral multiple of $2\pi$. This implies that the path length $2L$ must equal an integer number of wavelengths:

2L = m \lambda \quad (m = 1, 2, 3, \dots)

Using $\lambda = c/\nu$, the allowed frequencies of oscillation are:

\nu_m = m \frac{c}{2L}

These discrete frequencies are called longitudinal modes. The frequency spacing between adjacent modes is:

\Delta \nu = \frac{c}{2L}

4. Laser Systems

4.1 The Ruby Laser (Solid State, 3-Level)

The Ruby laser was the first successful laser, developed by T.H. Maiman in 1960.

Construction

  • Active Medium: A rod of synthetic ruby, which is Aluminium Oxide ($Al_2O_3$) doped with 0.05% Chromium Oxide ($Cr_2O_3$). The active species are the trivalent Chromium ions ($Cr^{3+}$).
  • Resonator: The ends of the rod are polished flat and parallel. One end is fully silvered, the other partially silvered.
  • Pumping Source: A helical Xenon flash lamp surrounds the ruby rod to provide intense pulses of white light.

Working Principle (Three-Level System)

  1. Pumping ($E_1 \to E_3$): The ground state ($E_1$) Cr ions absorb green/yellow light (5500 Å) from the flash lamp and are excited to a broad energy band ($E_3$).
  2. Non-Radiative Decay ($E_3 \to E_2$): Ions in the short-lived $E_3$ band ($10^{-8}$ s) rapidly lose energy to the crystal lattice and drop to the metastable state ($E_2$).
  3. Population Inversion: The metastable state $E_2$ has a very long lifetime ($\approx 3 \times 10^{-3}$ s). Ions accumulate here. Since pumping is strong, the population of $E_2$ eventually exceeds $E_1$.
  4. Lasing Transition ($E_2 \to E_1$): Stimulated emission occurs from $E_2$ back to the ground state $E_1$.

Output: Intense pulse of red light at wavelength 6943 Å.

[Image of Ruby laser energy level diagram]

4.2 The Helium-Neon Laser (Gas, 4-Level)

The He-Ne laser was the first gas laser, built by Ali Javan in 1961. It provides a continuous wave (CW) output.

Construction

  • Medium: A mixture of Helium (He) and Neon (Ne) gases in a ratio of approximately 10:1 inside a narrow discharge tube at low pressure (~1 mm Hg).
  • Pumping: Electrical discharge using a high-frequency source.

Working Principle (Four-Level System)

  1. Excitation of He: Electrons from the discharge collide with Helium atoms, exciting them to metastable states at 20.61 eV.
  2. Resonant Energy Transfer: Excited He atoms collide with ground-state Neon atoms. Neon has excited states at 20.66 eV, very close to Helium's 20.61 eV. Energy is transferred to Neon resonantly (kinetic energy supplies the 0.05 eV difference).
  3. Population Inversion in Ne: Neon atoms collect in the 20.66 eV state (Upper Laser Level). The lower laser level at 18.70 eV is initially empty.
  4. Lasing ($20.66 \text{ eV} \to 18.70 \text{ eV}$): Stimulated emission occurs between these levels.
  5. Decay: The Neon atom then decays spontaneously to a lower metastable state and finally to the ground state via wall collisions.

Output: Continuous beam of coherent red light at 6328 Å.

4.3 The Nd:YAG Laser (Solid State, 4-Level)

Active Medium: Yttrium Aluminium Garnet ($Y_3Al_5O_{12}$) crystal where some Yttrium ions are replaced by Neodymium ($Nd^{3+}$) ions.

Mechanism: It operates as a four-level system. Electrons are optically pumped from the ground state to pump bands using a flash lamp. They decay non-radiatively to the upper laser level ($^4F_{3/2}$). The laser transition occurs to a lower level ($^4I_{11/2}$) which is above the ground state.

Because the lower laser level is not the ground state, it is effectively empty at thermal equilibrium, making population inversion much easier to sustain than in the 3-level Ruby laser.

Output: Infrared radiation at 1.06 µm. Power is typically 20-30 Watts.

4.4 The Carbon Dioxide ($CO_2$) Laser (Molecular Gas)

This is a high-efficiency molecular gas laser. The energy levels correspond to vibrational and rotational states of the molecule.

Mixture: $CO_2 : N_2 : He$ in a ratio of 1:4:5.

Working: Nitrogen molecules are excited by electric discharge to a metastable vibrational level. They transfer this energy to \(CO_2\) molecules via collision, exciting them to the (001) vibrational level. Lasing transitions occur to lower vibrational levels like (100) and (020).

Output: High power Infrared radiation at 10.06 µm and 9.6 µm.

4.5 Semiconductor (Diode) Laser

Uses a p-n junction of a direct band-gap semiconductor like Gallium Arsenide (GaAs). When forward biased, electrons and holes recombine at the junction, emitting photons.

Wavelength: The wavelength is determined by the band gap energy \(E_g\):

\[ \lambda = \frac{hc}{E_g} = \frac{1242}{E_g \text{ (in eV)}} \text{ nm}\]

Tunability: By doping the semiconductor (e.g., with Aluminium or Phosphorous), the band gap \(E_g0)\ can be altered, thereby tuning the laser wavelength.

4.6 Other Lasers

  • He-Cd Laser: Uses Cadmium vapor as the active medium.
  • Dye Laser: Uses organic dyes dissolved in liquid solvents (water, methanol). These are tunable lasers because the dye emits over a broad range of wavelengths.

5. Applications of Lasers

The unique properties of laser light make it indispensable in modern technology.

  • Scientific Research: Spectroscopy, interferometry, and fundamental physics experiments.
  • Isotope Separation: Precise wavelengths can selectively ionize specific isotopes for separation.
  • Nuclear Fusion: High-power lasers are used in Inertial Confinement Fusion (ICF) to compress fuel pellets.
  • Medical Applications:
    • Eye surgery (LASIK, retinal welding).
    • Treatment of cancer and tumours.
    • Breaking kidney stones (Lithotripsy).
  • Communication: Fiber optic data transmission (using semiconductor lasers).
  • Industry: Cutting, welding, and drilling of materials.