Resonant Optical Cavities

• Laser Cavity• Gain, Gain Coefficient• Optical feedback• Condition for oscillation• Losses• Standing waves• Oscillation modes and

resonant frequency• Resonance in Laser

cavity

• Resonator modes• Longitudinal modes• Laser gain bandwidth• Multi Mode and single

mode lasers• Fabry Parot Etalon• Mode locking• Resonator Types• Q Switching• Pockels Cells and Kerr

Cells

Laser Cavities / Oscillator

Gain/ Amplification

• Gain of 2 per cm= 1 photon generates 2 more photons• Gain of 0.5 /cm= 1 photon generates an average of 0.5

stimulated emission photon each cm it travels• Amplification = output / input= (1+gain) length(cm)

Essential Ingredients of a laser

light amplification + positive optical feedback

Light amplification is achieved by stimulated emission. optical materials do not amplify light. Instead, they tend to absorb or scatter the light, so that the light intensity out of the medium is less than the intensity that went in. To get amplification you have to drive the material by pumping energy into it. The amplification of the medium is defined by the gain coefficient γ, as per the following equation:

Gain Coefficient γ

( ) ( ) ( )I x dx I x I x dx

( )I x dI

Where I(x) represents the intensity at a point x within the gain medium. The differential equation can be solved as follows :

( ) (0) x

dI Idx

dII

dx

I x I e

Thus the intensity grows exponentially within the gain medium

Positive Optical Feedback

Positive optical feedback is achieved by inserting the amplifying medium inside a resonant cavity. Light in the cavity passes through the gain medium and is amplified. It then bounces off the end mirrors and passes through the gain medium again getting amplified further.

Condition for Oscillation

• The process of going back and forth continues until a stable equilibrium condition is achieved when the total round trip gain balances all the losses in the cavity.

• Condition for Oscillation is

Round –Trip gain = Round-trip loss

Losses

Useful Losses Useless losses

Output CouplingAbsorption, ScatteringImperfect Reflectivity

Condition for Oscillation -Cont

Where l is the length of the gain medium (why 2), ROC is the reflectivity of the output coupler, RHR is the reflectivity of the high reflector.

where the losses due to scattering and absorption are lumped into the distributed loss coefficient α.

The other to write this is :-

Standing Waves

• Differentiate between these words– Traveling waves– Standing Waves– Standing waves and Resonance

Standing Waves

•

Standing Waves

• Superposition of Two traveling waves moving in opposite direction

(a) Travelling toward left (b) Toward right

(c) Superposition of the two- Standing waves

Oscillation Modes & Resonant Frequencies

• Standing wave can be set up on a string of Length L by a wave such that:Nλ/2 = L, for n=1,2,3λ=2L/n

• The resonant frequencies that correspond to these wavelengthf=v/λ = nv/2L, n=1,2,3n=1: fundamental moden=2: second Harmonic

Standing Waves : Theory

• The modes of vibration associated with resonance in extended objects like strings and air columns have characteristic patterns called standing waves. These standing wave modes arise from the combination of reflection and interference such that the reflected waves interfere constructively with the incident waves. An important part of the condition for this constructive interference for stretched strings is the fact that the waves change phase upon reflection from a fixed end. Under these conditions, the medium appears to vibrate in segments or regions and the fact that these vibrations are made up of traveling waves is not apparent - hence the term "standing wave".

• The behavior of the waves at the points of minimum and maximum vibrations (nodes and antinodes) contributes to the constructive interference which forms the resonant standing waves. Standing waves in air columns also form nodes and antinodes, but the phase changes involved must be separately examined for the case of air columns

Web Definitions of Resonance

• the state of a system in which an abnormally large vibration is produced in response to an external stimulus, occurring when the frequency of the stimulus is the same, or nearly the same, as the natural vibration frequency of the system

• a condition in which the frequency of an external force matches the natural frequency of the object it is acting upon

• A state where the natural frequency of a body equals an applied frequency

Resonance In Laser Cavity

• Stimulated emission is coherent, so all light waves are in the same phase and that light waves add in amplitude

• In order to have constructive interference for a particular wavelength we need to have resonance condition

Nλ = 2 x Cavity Length

Cavity length should be an integral multiple of half the wavelength

Resonator Modes

• Light confined in a resonator will reflect multiple times from the mirrors, and due to the effects of interference, only certain patterns and frequencies of radiation will be sustained by the resonator, with the others being suppressed by destructive interference. In general, radiation patterns which are reproduced on every round-trip of the light through the resonator are the most stable, and these are known as the modes of the resonator.

• Resonator modes can be divided into two types: longitudinal modes, which differ in frequency from each other; and transverse modes, which may differ in both frequency and the intensity pattern of the light. The basic, or fundamental transverse mode of a resonator is a Gaussian beam

Longitudinal Modes• A longitudinal mode of a

resonant cavity is a particular standing wave pattern formed by waves confined in the cavity. The longitudinal modes correspond to the wavelengths of the wave which are reinforced by constructive interference after many reflections from the cavity's reflecting surfaces. All other wavelengths undergo destructive interference and are suppressed

• The allowed modes of the cavity are those where the mirror separation distance L is equal to an exact multiple of half the wavelength, λ:

Longitudinal modes-cont …….

L= qλ/2• Here both q and λ can vary to give rise to const L.Here q is an integer

known as mode order. In practice, the separation distance of the mirrors L is usually much greater than the wavelength of light λ, so the relevant values of q are large (around 105 to 106).

• The frequency separation between any two adjacent modes q and q+1 are given (for an empty linear resonator of length L) by ν

/ 2c L , Here c is the speed of light

Laser Gain Bandwidth• Although laser light is perhaps the purest form

of light, it is not of a single, pure frequency or wavelength. All lasers produce light over some natural bandwidth or range of frequencies. A laser's bandwidth of operation is determined primarily by the gain medium that the laser is constructed from, and the range of frequencies that a laser may operate over is known as the gain bandwidth. For example, a typical helium-neon (HeNe) gas laser has a gain bandwidth of approximately 1.5 GHz (around 0.002 nm wavelength range) (also known as linewidth =)

• These standing waves form a discrete set of frequencies, known as the longitudinal modes of the cavity. These modes are the only frequencies of light which are self-regenerating and allowed to oscillate by the resonant cavity; all other frequencies of light are suppressed by destructive interference. For a simple plane-mirror cavity, the allowed modes are those for which the separation distance of the mirrors L is an exact multiple of half the wavelength of the light λ, such that L = q λ/2, when q is an integer known as the mode order

• Using the above equation, a small laser with a mirror separation of 30 cm has a frequency separation between longitudinal modes of 0.5 GHz. Thus for the He-Ne lasers referenced above, with a 30 cm cavity the 1.5 GHz bandwidth of the HeNe laser would support up to 3 longitudinal modes

Multi Mode and Single Mode Lasers

Single Mode Laser• Shorten the cavity length : One way to achieve to single-

mode operation is to shorten the cavity so that the mode spacing exceeds the gain bandwidth. In this case only one mode will fall within the emission line of the transition and the laser will automatically go on only one mode. However, this may not be practical. For example, in the case of the HeNe laser considered above, we would need to make the cavity shorter than 10 cm. Such a laser would have very small round trip gain, and we would probably not be able to make it oscillate

• Frequency Filter : A better way to obtain single mode operation is to insert a narrow frequency filter in the cavity such as a Fabry-Perot etalon. By tuning the spacing of the etalon, the frequency of the single mode can be changed continuously, which is very useful for high resolution spectroscopy. The spectral line width of a single mode laser is a few MHz.

Fabry Perot Etalon

• Here you see the entire Etalon. The surfaces are coated with a reflective material and spaced around 100 to 200nm apart. Based on the destructive interference of light, unwanted rays are cancelled out and desired rays are passed.

• The type of material separating the two filters, the amount of reflectivity and most importantly, the distance between them, determine the bandpass of the filter.

Mode locking

• The opposite extreme to single-mode operation is mode-locked operation. In this situation we try to get as many longitudinal modes oscillating as possible, but with all their phases locked together. This contrasts with a multi-mode laser in which many modes are oscillating but with random phases with respect to each other.

Modelocking Theory

What will happen if there is no modelocking ?What will happen if there is no modelocking ?

• In a simple laser, each of these modes will oscillate independently, with no fixed relationship between each other, in essence like a set of independent lasers all emitting light at slightly different frequencies. The individual phase of the light waves in each mode is not fixed, and may vary randomly due to such things as thermal changes in materials of the laser. In lasers with only a few oscillating modes, interference between the modes can cause beating effects in the laser output, leading to random fluctuations in intensity;

• In lasers with many thousands of modes, these interference effects tend to average to a near-constant output intensity, and the laser operation is known as a c.w. or continuous wave

Modelocking Theory- Cont• What will be the result of modelocking ?

• If instead of oscillating independently, each mode operates with a fixed phase between it and the other modes, the laser output behaves quite differently. Instead of a random or constant output intensity, the modes of the laser will periodically all constructively interfere with one another, producing an intense burst or pulse of light. Such a laser is said to be mode-locked or phase-locked.

• These pulses occur separated in time by τ = 2L/c, which is the time taken for the light to make exactly one round trip of the laser cavity. This time corresponds to a frequency exactly equal to the mode-spacing of the laser, ν = 1/τ.

Duration of the laser pulse

• The duration of each pulse of light is determined by the number of modes which are oscillating in phase (in a real laser, it is not necessarily true that all of the laser's modes will be phase-locked). If there are N modes locked with a frequency separation Δν, the overall mode locked bandwidth is NΔν, and the wider this bandwidth, the shorter the pulse duration from the laser. In practice, the actual pulse duration is determined by the shape of each pulse, which is in turn determined by the exact amplitude and phase relationship of each longitudinal mode. For example, for a laser producing pulses with a Gaussian temporal shape, the minimum possible pulse duration Δt is given by:

Δt = 0.44/(N δν)

• The value 0.44 is known as the time-bandwidth product of the pulse, and varies depending on the pulse shape. For ultrashort pulse lasers, a hyperbolic-secant-squared (sech2) pulse shape is often assumed, giving a time-bandwidth product of 0.315.

Techniques of Mode Locking

Mode-locking is achieved by two main techniques.

• Active mode-locking : a time-dependent shutter is inserted in the cavity. The shutter is opened briefly every 2L/c seconds. Continuous operation of the laser is impossible, but the mode-locked pulses will be unaffected by the shutter.

• Passive mode-locking: a saturable absorber is inserted in the cavity. Such absorbers have strong absorption at low powers and small absorption at high powers. The peak power in the pulsed mode is much higher than in continuous operation, and thus the cavity naturally selects the pulsed mode.

Applications of Mode locked Pulses

• Mode-locked lasers are widely used in scientific research to study fast processes in physics, chemistry, and biology. For example, the typical time for a current-carrying electron in a copper wire to interact with a phonon at room temperature is about 100 fs. Similarly, the early stages of many chemical reactions or biological processes such as photosynthesis take place in less than 10-12 s.

• Mode-locked lasers are also useful to telecommunication companies, who are interested in packing as many bits of information (represented by pulses of light) as possible down their optical fibres. The shorter the pulses, the higher the data rate. There are also medical applications: it is much cleaner to use a very short low energy high peak power pulse for laser surgery, than a longer pulse with the same peak power but much higher energy.

Peak Power, Irradiance, Fluence (Laser Pulses)

• Calculate peak power, Irradiance, fluence of a laser pulse ?