The structure of a laser diode should be such that itshould facilitate stimulated emission.The question arises, how to get the seed photons that initiate the stimulated emission process,which leads to the requirement of a feedback mechanism. The electronic equivalent of such asystem is an oscillator. The basic component of an electronic oscillator is an amplifier, say, anop-amp, and a part of the amplifier output is fed back into the input of the amplifier. In a similarmanner, the LED can be thought of as an amplifier, which when subjected to feedback, canoperate as an oscillator. With the feedback conditions maintained for sustainable oscillations, theLED can be converted into an optical oscillator, which is a laser. Thus, the difference betweenan LED and a laser diode is the presence of a feedback mechanism.In order to realize the laser, the photons generated as a consequence of spontaneous emission arefed back into the gain medium to cause stimulated emission. This optical feedback can be provided simply by using mirrors, which can trap the photons within the system, thus realizing acavity resonator. The laser is constructed by having a gain medium in an optical cavity, whichconsists of two mirrors M1 and M2 that provide the optical feedback. The gain medium is thesame as that used in an LED. It could be a ternary compound or a quaternary compound, decidedby the required wavelength of emission. The only difference is that the gain medium is nowcontained between the two-cavity mirrors. A spontaneously emitted photon now undergoesreflection from the cavity mirrors and is fed back into the gain medium, causing stimulatedemission. The cavity mirrors M1 and M2 serve two major purposes.1. They enable the feedback mechanism and facilitate stimulated emission. The reflectivities R1 and R2 decide the fraction of the light allowed to oscillate inside the cavity, and the remaining fraction to be tapped as the usable laser output. The mirror reflectivities areimportant parameters which decide the laser properties.2. They impose certain boundary conditions on the electromagnetic field in the resonator, which affects the frequency spectrum of the laser output.
The cavity resonator is constructed with two mirrors M1 and M2 with reflectivities R1 and R2respectively, and the electromagnetic field oscillates inside the cavity. This arrangement ofmirror constructs a linear cavity, also known as Fabry-Perot cavity. The reflectivity of the mirroris the ratio of reflected power to the incident power. We will now discuss the detailed operationof this cavity.Consider the amplitude of the electric field of the electromagnetic wave at the middle point ofthe cavity to be A at time t = 0, and the total distance between the mirrors forming the cavity to be d. In one round-trip over the cavity, the electric field experiences the following effects beforereaching back at the same point. Traversing a distance 2d over the entire round-trip. Undergoing reflections from the mirrors. In terms of the electric field, the reflection coefficients for the two mirrors are ඥRଵ and ඥRଶ (since power is proportional to thesquare of the electric field). Undergoing amplification in the gain medium. The gain coefficient of the gain medium,defined as per unit length quantity, is given as g. Undergoing attenuation in the rest of the cavity. The attenuation coefficient is given as α(per unit length, similar to gain coefficient).Including all the above effects, the amplitude of the electric field experiences a change by afactor of ඥRଵRଶe
(ିఈ)ଶௗ over one round-trip.
Additionally, the phase of the electric field also evolves due to propagation in the cavity. The phase of the electromagnetic field travelling as a plane wave is of the general form e(ఠ௧ିఉ௭),
where the term e
ିఉ௭ accounts for the phase accumulated by the wave after propagating througha distance z. The electromagnetic wave traverses a distance of 2d over one round-trip inside thecavity, hence the phase accumulated over the round-trip is eିఉଶ.
The condition for sustained oscillation demands that the electric field at the same location after one round-trip should be identical. This would imply,A = AඥRଵRଶe(ିఈ)ଶௗeିఉଶௗ
ඥRଵRଶe(ିఈ)ଶௗeିఉଶௗ = 1
It is essential for the electric field to satisfy the above condition at all the points in the cavity inorder to attain sustained oscillations in the resonator. The above criterion imposes the followingindependent conditions on the magnitude and the phase of the electric field. Phase condition:
eିఉଶௗ = 1
Amplitude condition:
ඥRଵRଶe(ିఈ)ଶௗ = 1
Let us consider the phase condition. The above expression implies that the accumulated round-the trip phase should be an integer multiple of 2π.
β ⋅ 2d =2πνc2d = m ⋅ 2π
NPTEL-Fiber Optic Communication Technology – Lecture 17 Page 4
v = m ⋅c2d
The above condition states that not all frequencies are allowed to be sustained in the cavity. Oncethe cavity length d is chosen and fixed, only those frequencies that are integer multiples of ଶௗ(denoted by ν) undergo constructive interference in the cavity and hence are allowed tooscillate. The frequencies that do not satisfy this condition undergo destructive interference inthe cavity.
Consider the example of a linear cavity of length 1 cm (d = 1 cm). Assume that it is free space inside the cavity and there is no gain or loss in the cavity. The idea of allowed (and forbidden) frequencies in the cavity are irrespective of whether or not there is a gain medium in the cavity, since it depends on the phase condition and not the amplitude condition. If there is a medium inside the cavity, the only impact it makes (in terms of the allowed frequencies) is dueto the refractive index of the medium, which alters the phase condition. Considering free space,
ν = m3 × 10଼2 × 10ିଶ = m ⋅ 15 GHz
This implies that a Fabry-Perot cavity of length 1 cm with a free space medium can support onlythose frequencies which are integral multiples of 15 GHz. In order to check whether a particular the wavelength, say 1550 nm, is supported by the cavity or not, one has to check whether the corresponding frequency (which would be in THz in this case) is an exact integer multiple of 15GHz or not.
Theoretically, the number of such allowed frequencies in the cavity is infinite. These frequencies are called the modes of the cavity, and since they are dependent on the longitudinal (in the direction of propagation of the electromagnetic wave) constraints imposed by the cavity mirrors, they are known as longitudinal modes. The frequency separation between two adjacent longitudinal modes is called the free spectral range (FSR) of the cavity. Thus, a Fabry-Perotcavity is characterized by its free spectral range νி =ଶௗ, which contains two information aboutthe cavity, the spacing between the mirrors (length of the cavity), and the speed of light in thecavity (dependent on the refractive index of the medium inside the cavity). In case of a laser,only the frequencies that are supported by the cavity, i.e., the frequencies that are integermultiples of the FSR of the cavity can be generated.
The amplitude condition for sustained oscillation states that the cavity must have a source of gain in order to compensate for the losses in the cavity. In the case of laser, the semiconductor material provides the required gain. The minimum gain required for sustained oscillations is calculated as follows.
In a single longitudinal mode laser, only one frequency is supported by the cavity, but its
spectrum is not a δ-function, it has a finite spectral width Δν. The phase condition of the Fabry-
Perot cavity yields only the allowed mode frequencies ν, so we need to derive the full-width at
half maximum (FWHM) of the modes. We now calculate the spectral width of the modes of a
Fabry-Perot cavity under stable oscillation condition. Since a fraction of the electric field in the
cavity is taken as the output, the spectral width of the cavity mode also corresponds to the
spectral width of the laser output.
Consider a Fabry-Perot cavity formed by mirrors with reflectances Rଵand Rଶ, and the initial
amplitude of the electric field at a point inside the cavity being A. The electric field undergoes
successive reflections from the cavity mirrors as it propagates back and forth inside the cavity.
Thus, at any given point, the total electric field E௧௧ is the sum total of the electric fields after
each round-trip. In steady-state, the total electric field should be equal to A. In each round-trip,
the electric field undergoes modification in the amplitude due to the mirror reflectance, and in
phase due to the propagation. Considering a cold cavity (cavity without gain, since the gain does
not play a role in deciding the modes), the total electric field after multiple reflections can be
written as follows.
For ease of understanding, we define this factor of modification as h, given as follows.
Thus, the total electric field can be written as
which is an infinite geometric progression. It is important to note here that h ≤ 1, (since
Rଵ, Rଶ ≤ 1), and hence this series is convergent, and can be expressed as follows.
If the gain element is present in the cavity, the factor h is modified as follows.
Since h is a complex number, it can be expressed as h = reିథ, where r = ඥRଵRଶe
(in absence of a gain medium), and φ is given as
where νி =
is the FSR of the cavity. The total intensity I is given as I =
ଶ
, which is
evaluated as follows.
I
Thus, the total intensity can be expressed as follows, where I = A
At frequencies ν = mνி, which correspond to the longitudinal modes of the cavity, φ = 2mπ.
Thus, at those frequencies, sin థ
Thus, the intensity inside the cavity has a variation with respect to the frequency. It has maxima
at the cavity mode frequencies and minima (I) at the frequencies between the
supported modes.
The next objective is to find the spectral width (FWHM) of the mode, which can be calculated by
finding the frequencies at which the intensity is half its maximum value (I =
intensity can be written in terms of I௫as follows.
which yields
Substituting φ = 2π
The values of ν obtained from this condition are those which correspond to the half maximum of
intensity, and the difference of those frequencies gives the FWHM. The above condition can be
solved as follows.
If the cavity modes are very narrow (which is the case with the cavities formed using mirrors of
high reflectivity), the frequencies corresponding to half maximum are very close to the mode
frequency νி. In this condition, an approximation can be made as follows.
Solving for ν, we get
The ± sign indicates that the half-maximum occurs on either side of the cavity mode, and the
difference between those two frequencies is the FWHM spectral width Δν.
The quality of the cavity is represented by the quantity finesse, which is defined as follows.
A larger value of finesse represents a cavity with sharper longitudinal modes with smallerFWHM. In this manner, finesse is indicative of the frequency selectivity of the cavity. For example, if the FWHM is large, then two adjacent modes may be indistinguishable and would result in a broad spectrum of the laser output. Thus, finesse is an important parameter for the design of a single longitudinal mode laser.
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