Difference between stable and unstable resonators

Difference between Stable and Unstable Resonators:

1. The oscillating beam is converged in stable resonator while in unstable resonator is spreads out of the the resonator.


2. In stable resonator laser output is from the centre of optical axis while in unstable resonator laser output comes from the edge of the output mirror.


3. The field is confined to the axis in stable resonator while it is not so in unstable resonator.


4. Stable resonators are used for low power lasers while unstable resonators are used for high power lasers.


5. In stable resonator these remains risk of breakage of the m  irrors while it is reduced to unstable resonators.


6. The mode volume is is small in stable resonators while it is large in unstable resonators.


7. The geometrical losses are large in unstable resonator in comparison to stable resonators.


8. In unstable resonators better beam quality may be achieved in comparison to stable resonators.

Read More

Electromagnetic wave equation in non conducting media (i.e. Perfect dielectric or Lossless media)

Maxwell's Equations: Maxwell's equation of the electromagnetic wave is a collection of four equations i.e. Gauss's law of electrostatic, Gauss's law of magnetism, Faraday's law of electromotive force, and Ampere's Circuital law. Maxwell converted the integral form of these equations into the differential form of the equations. The differential form of these equations is known as Maxwell's equations for free space.

1. $\overrightarrow{\nabla}. \overrightarrow{E}= \frac{\rho}{\epsilon_{0}}$


2. $\overrightarrow{\nabla}. \overrightarrow{B}=0$


3. $\overrightarrow{\nabla} \times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}$


4. $\overrightarrow{\nabla} \times \overrightarrow{B}= \mu \overrightarrow{J}$
   
   
   Modified Form:
   
   
   $\overrightarrow{\nabla} \times \overrightarrow{B}= \mu \left(\overrightarrow{J}+ \epsilon \frac{ \partial \overrightarrow{E}}{\partial t} \right)$

For non-conducting media:

Current density $(\overrightarrow{J})=0$
Volume charge distribution $(\rho) \neq 0 $
Permittivity of non-conducting media= $\epsilon$
Permeability of non-conducting media=$\mu$

Now, Maxwell's equation for non-conducting media:

$\overrightarrow{\nabla}. \overrightarrow{E}=\frac{\rho}{\epsilon} \qquad(1)$
$\overrightarrow{\nabla}. \overrightarrow{B}=0 \qquad(2)$
$\overrightarrow{\nabla} \times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t} \qquad(3)$
$\overrightarrow{\nabla} \times \overrightarrow{B}= 0$

Modified form for non-conducting media:

$\overrightarrow{\nabla} \times \overrightarrow{B}= \mu_{\circ} \epsilon_{\circ} \frac{ \partial \overrightarrow{E}}{\partial t} \qquad(4)$

Now, On solving Maxwell's equation for a non-conducting media i.e perfect dielectric and Lossless media, we get the electromagnetic wave equation for a non-conducting media. The electromagnetic wave equation has both an electric field vector and a magnetic field vector. So Maxwell's equations for non-conducting media give two equations for electromagnetic waves i.e. one is for electric field vector($\overrightarrow{E}$) and the second is for magnetic field vector ($\overrightarrow{B}$).

Electromagnetic wave equation for non-conducting media in terms of $\overrightarrow{E}$:

Now from Maxwell's equation $(3)$

$\overrightarrow{\nabla} \times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t} $

Now take the curl on both sides of the above equation$

$\overrightarrow{\nabla} \times (\overrightarrow{\nabla} \times \overrightarrow{E})=-\overrightarrow{\nabla} \times \frac{\partial \overrightarrow{B}}{\partial t} $

$(\overrightarrow{\nabla}. \overrightarrow{E}).\overrightarrow{\nabla} - (\overrightarrow{\nabla}. \overrightarrow{\nabla}).\overrightarrow{E} \\ =-\frac{\partial}{\partial t} (\overrightarrow{\nabla} \times \overrightarrow{B}) \qquad(5)$

We know that

$\overrightarrow{\nabla}. \overrightarrow{E}=\frac{\rho}{\epsilon}$
$\overrightarrow{\nabla}.\overrightarrow{\nabla}=\nabla^{2}$
$\overrightarrow{\nabla} \times \overrightarrow{B}= \mu \epsilon \frac{\partial \overrightarrow{E}}{\partial t} $

Now substitute these values in equation $(5)$. So

$ \nabla \left(\frac{\rho}{\epsilon}\right) -\nabla^{2}.\overrightarrow{E}=-\frac{\partial}{\partial t} (\mu \epsilon \frac{\partial \overrightarrow{E}}{\partial t})$

For non conducting media, If the volume charge distribution is uniform then the gradient of volume charge density is very small (or almost zero) so neglect the term gradient of volume charge density $(\nabla \rho)$ in above eqaution.
Or
The wave propagation does not contain the charges in most of the cases. Therefore we get

$ -\nabla^{2}.\overrightarrow{E}=-\frac{\partial}{\partial t} (\mu \epsilon \frac{\partial \overrightarrow{E}}{\partial t})$

$\nabla^{2}.\overrightarrow{E}=\mu \epsilon \frac{\partial^{2} \overrightarrow{E}}{\partial t^{2}}$

The value of $\frac{1}{\sqrt{\mu \epsilon}}= v$. Where $v$ is the speed of the wave in the non-conducting media. So the above equation is often written as

$\nabla^{2}.\overrightarrow{E}=\frac{1}{v^{2}} \frac{\partial^{2} \overrightarrow{E}}{\partial t^{2}}$

This is an electromagnetic wave equation for non-conducting media in terms of electric field vector ($\overrightarrow{E}$).

Electromagnetic wave equation for non-conducting media in terms of $\overrightarrow{B}$:

Now from Maxwell's equation $(4)$

$\overrightarrow{\nabla} \times \overrightarrow{B}= \mu_{\circ} \epsilon_{\circ} \frac{ \partial \overrightarrow{E}}{\partial t}$

Now take the curl on both sides of the above equation

$\overrightarrow{\nabla} \times (\overrightarrow{\nabla} \times \overrightarrow{B})=\overrightarrow{\nabla} \times \mu \epsilon \frac{ \partial \overrightarrow{E}}{\partial t} $

$(\overrightarrow{\nabla}. \overrightarrow{B}).\overrightarrow{\nabla} - (\overrightarrow{\nabla}. \overrightarrow{\nabla}).\overrightarrow{B} \\ =\mu \epsilon \frac{\partial}{\partial t} (\overrightarrow{\nabla} \times \overrightarrow{E}) \qquad(6)$

We know that

$\overrightarrow{\nabla}. \overrightarrow{B}=0 $
$\overrightarrow{\nabla}.\overrightarrow{\nabla}=\nabla^{2}$
$\overrightarrow{\nabla} \times \overrightarrow{E}= - \frac{\partial \overrightarrow{B}}{\partial t}$

Now substitute these values in equation $(6)$. So

$ -\nabla^{2}.\overrightarrow{B}=-\mu \epsilon \frac{\partial}{\partial t} (\frac{\partial \overrightarrow{B}}{\partial t})$

$ -\nabla^{2}.\overrightarrow{B}=-\mu \epsilon \frac{\partial^{2} \overrightarrow{B}}{\partial t^{2}} $

$\nabla^{2}.\overrightarrow{B}=\mu \epsilon \frac{\partial^{2} \overrightarrow{B}}{\partial t^{2}}$

The value of $\frac{1}{\sqrt{\mu \epsilon}}= v$. Where $v$ is the speed of the wave in the non-conducting media. So the above equation is often written as

$\nabla^{2}.\overrightarrow{B}=\frac{1}{v^{2}} \frac{\partial^{2} \overrightarrow{B}}{\partial t^{2}}$

This is an electromagnetic wave equation for non-conducting media in terms of electric field vector ($\overrightarrow{B}$).

Read More

Electromagnetic wave equation in free space

Maxwell's Equations: Maxwell's equation of the electromagnetic wave is a collection of four equations i.e. Gauss's law of electrostatic, Gauss's law of magnetism, Faraday's law of electromotive force, and Ampere's Circuital law. Maxwell converted the integral form of these equations into the differential form of the equations. The differential form of these equations is known as Maxwell's equations.

1. $\overrightarrow{\nabla}. \overrightarrow{E}= \frac{\rho}{\epsilon_{0}}$
2. $\overrightarrow{\nabla}. \overrightarrow{B}=0$
3. $\overrightarrow{\nabla} \times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}$
4. $\overrightarrow{\nabla} \times \overrightarrow{B}= \mu \overrightarrow{J}$
   
   
   Modified Form:
   
   
   $\overrightarrow{\nabla} \times \overrightarrow{B}= \mu \left(\overrightarrow{J}+ \epsilon \frac{ \partial \overrightarrow{E}}{\partial t} \right)$

For free space:

Current density ($\overrightarrow{J}$) = 0
Volume charge distribution ($\rho$) = 0
Permittivity $\epsilon = \epsilon_{0}$
Permeability $\mu = \mu_{0}$

Now, Maxwell's equation for free space:

$\overrightarrow{\nabla}. \overrightarrow{E}=0 \qquad(1)$
$\overrightarrow{\nabla}. \overrightarrow{B}=0 \qquad(2)$
$\overrightarrow{\nabla} \times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t} \qquad(3)$
$\overrightarrow{\nabla} \times \overrightarrow{B}= 0$

Modified form for free space:

$\overrightarrow{\nabla} \times \overrightarrow{B}= \mu_{\circ} \epsilon_{\circ} \frac{ \partial \overrightarrow{E}}{\partial t} \qquad(4)$

Now, On solving Maxwell's equation for free space we get the electromagnetic wave equation for free space. The electromagnetic wave equation has both an electric field vector and a magnetic field vector. So Maxwell's equations for free space give two-equation for electromagnetic wave i.e. one is for electric field vector($\overrightarrow{E}$) and the second is for magnetic field vector ($\overrightarrow{B}$).

Electromagnetic wave equation for free space in term of $\overrightarrow{E}$:

Now from Maxwell's equation $(3)$ in free space

$\overrightarrow{\nabla} \times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t} $

Now take the curl on both sides of the above equation

$\overrightarrow{\nabla} \times (\overrightarrow{\nabla} \times \overrightarrow{E})=-\overrightarrow{\nabla} \times \frac{\partial \overrightarrow{B}}{\partial t} $

$(\overrightarrow{\nabla}. \overrightarrow{E}).\overrightarrow{\nabla} - (\overrightarrow{\nabla}. \overrightarrow{\nabla}).\overrightarrow{E} \\ =-\frac{\partial}{\partial t} (\overrightarrow{\nabla} \times \overrightarrow{B}) \qquad(5)$

We know that

$\overrightarrow{\nabla}. \overrightarrow{E}=0 \qquad$
$\overrightarrow{\nabla}.\overrightarrow{\nabla}=\nabla^{2}$
$\overrightarrow{\nabla} \times \overrightarrow{B}= \mu_{\circ} \epsilon_{\circ} \frac{ \partial \overrightarrow{E}}{\partial t}$

Now substitute these values in equation $(5)$. So

$ -\nabla^{2}.\overrightarrow{E}=-\frac{\partial}{\partial t} ( \mu_{\circ} \epsilon_{\circ} \frac{ \partial \overrightarrow{E}}{\partial t} )$

$ -\nabla^{2}.\overrightarrow{E}=-\mu_{0} \epsilon_{\circ} \frac{\partial^{2} \overrightarrow{E}}{\partial t^{2}} $

$\nabla^{2}.\overrightarrow{E}=\mu_{0} \epsilon_{0} \frac{\partial^{2} \overrightarrow{E}}{\partial t^{2}}$

The value of $\frac{1}{\sqrt{\mu_{0} \epsilon_{0}}}= c$. Where $c$ is the speed of the wave in free space. So the above equation is often written as

$\nabla^{2}.\overrightarrow{E}=\frac{1}{c^{2}} \frac{\partial^{2} \overrightarrow{E}}{\partial t^{2}}$

This is an electromagnetic wave equation for free space in terms of electric field vector ($\overrightarrow{E}$).

Electromagnetic wave equation for free space in term of $\overrightarrow{B}$:

Now from Maxwell's equation $(4)$

$\overrightarrow{\nabla} \times \overrightarrow{B}= \mu_{\circ} \epsilon_{\circ} \frac{ \partial \overrightarrow{E}}{\partial t}$

Now take the curl on both sides of the above equation

$\overrightarrow{\nabla} \times (\overrightarrow{\nabla} \times \overrightarrow{B})=\overrightarrow{\nabla} \times \mu_{\circ} \epsilon_{\circ} \frac{ \partial \overrightarrow{E}}{\partial t} $

$(\overrightarrow{\nabla}. \overrightarrow{B}).\overrightarrow{\nabla} - (\overrightarrow{\nabla}. \overrightarrow{\nabla}).\overrightarrow{B} \\ =\mu_{\circ} \epsilon_{\circ} \frac{\partial}{\partial t} (\overrightarrow{\nabla} \times \overrightarrow{E}) \qquad(6)$

But for free space:

$\overrightarrow{\nabla}. \overrightarrow{B}=0 \qquad $
$\overrightarrow{\nabla}.\overrightarrow{\nabla}=\nabla^{2}$
$\overrightarrow{\nabla} \times \overrightarrow{E} = - \frac{\partial \overrightarrow{B}}{\partial t} \qquad$

Now substitute these values in equation $(6)$. So

$ -\nabla^{2}\overrightarrow{B}=-\mu_{\circ}\frac{\partial}{\partial t} (\epsilon_{0} \frac{\partial \overrightarrow{B}}{\partial t})$

$ -\nabla^{2}\overrightarrow{B}=- \mu_{\circ} \epsilon_{0} \frac{\partial^{2} \overrightarrow{B}}{\partial t^{2}} $

$\nabla^{2}\overrightarrow{B}=\mu_{0} \epsilon_{0} \frac{\partial^{2} \overrightarrow{B}}{\partial t^{2}}$

The value of $\frac{1}{\sqrt{\mu_{0} \epsilon_{0}}}= c$. Where $c$ is the speed of the wave in free space. So the above equation is often written as

$\nabla^{2}\overrightarrow{B}=\frac{1}{c^{2}} \frac{\partial^{2} \overrightarrow{B}}{\partial t^{2}}$

This is the electromagnetic wave equation for free space in terms of electric field vector ($\overrightarrow{B}$).

Read More

Waves: MCQ

Waves: MCQ

You'll have 60 seconds to answer each question.

Time's Up

score:

Test Result

Total Questions:

Attempt:

Correct:

Wrong:

Percentage:

Read More

Alternating Current: MCQ

Alternating Current: MCQ

You'll have 60 seconds to answer each question.

Time's Up

score:

Test Result

Total Questions:

Attempt:

Correct:

Wrong:

Percentage:

Read More

Quantum Mechanics: MCQ-1

Quantum Mechanics: MCQ-1

You'll have 60 seconds to answer each question.

Time's Up

score:

Test Result

Total Questions:

Attempt:

Correct:

Wrong:

Percentage:

Read More