# Difference between revisions of "User:Pranav Rathi/Notebook/OT/2010/08/18/CrystaLaser specifications"

Pranav Rathi (talk | contribs) (→M2) |
Pranav Rathi (talk | contribs) (→Beam propagation factor M2) |
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Here the plot beam diameter Vs Z is presented. Experimental data is presented as blue and model is red. As it can be seen that model does not fit the data. Experimental beam expands much faster than the model; this proves that the beam waist before the expansion optics must be relatively smaller. And also we are missing an important characterization parameter. The real word lasers work differently in a way that their beam do not follow the regular Gaussian formula for large propagation lengths (more than Raleigh range). So that's why we will have introduce a beam propagation factor called M<sup>2</sup>. | Here the plot beam diameter Vs Z is presented. Experimental data is presented as blue and model is red. As it can be seen that model does not fit the data. Experimental beam expands much faster than the model; this proves that the beam waist before the expansion optics must be relatively smaller. And also we are missing an important characterization parameter. The real word lasers work differently in a way that their beam do not follow the regular Gaussian formula for large propagation lengths (more than Raleigh range). So that's why we will have introduce a beam propagation factor called M<sup>2</sup>. | ||

− | [[Image:Beamwaistexp.png|700x600px| | + | [[Image:Beamwaistexp.png|700x600px|Beam waist Vs Z]] |

− | =====M<sup>2</sup>===== | + | =====Beam propagation factor M<sup>2</sup>===== |

The beam propagation factor M<sup>2</sup> was specifically introduced to enable accurate calculation of the properties of laser beams which depart from the theoretically perfect TEM<sub>00</sub> beam. This is important because it is quite literally impossible to construct a real world laser that achieves this theoretically ideal performance level. | The beam propagation factor M<sup>2</sup> was specifically introduced to enable accurate calculation of the properties of laser beams which depart from the theoretically perfect TEM<sub>00</sub> beam. This is important because it is quite literally impossible to construct a real world laser that achieves this theoretically ideal performance level. | ||

Line 92: | Line 92: | ||

::<math> | ::<math> | ||

− | \theta_{ | + | \theta_{0}=\frac{\lambda}{\pi w_0} |

</math> | </math> | ||

this is theoretical half divergence angle in radian. | this is theoretical half divergence angle in radian. | ||

::<math> | ::<math> | ||

− | + | \theta_{R}=M^2\frac{\lambda} {\pi w_0} | |

+ | </math> | ||

+ | so | ||

+ | ::<math> | ||

+ | M^2=\frac{\theta_{R}} {\theta_{0}} | ||

</math> | </math> | ||

− | |||

Where: | Where: | ||

− | λ is the laser wavelength | + | *λ is the laser wavelength |

− | + | *θ<sub>R</sub> is the far field divergence angle of the real beam. | |

− | w<sub>0</sub> is the beam waist radius and θ<sub>0</sub> is the far field divergence angle of the theoratical beam. | + | *w<sub>0</sub> is the beam waist radius and θ<sub>0</sub> is the far field divergence angle of the theoratical beam. |

− | M<sup>2</sup> is the beam propagation factor | + | *M<sup>2</sup> is the beam propagation factor |

This definition of M<sup>2</sup> allows us to make simple change to optical formulas by taking M<sup>2</sup> factor as multiplication, to account for the actual beam divergence. This is the reason why M<sub></sub>2 is also sometimes referred to as the “times diffraction limit number”. | This definition of M<sup>2</sup> allows us to make simple change to optical formulas by taking M<sup>2</sup> factor as multiplication, to account for the actual beam divergence. This is the reason why M<sub></sub>2 is also sometimes referred to as the “times diffraction limit number”. | ||

Line 109: | Line 112: | ||

My experimental beam waist is: | My experimental beam waist is: | ||

− | |||

− | |||

+ | w<sub>R</sub>=.63mm with a theoretical divergence of .69mrad. The data suggests the real far field divergence angle to be 1mrad (half angle) (w<sub>R</sub>/z at that z). This gives: | ||

+ | |||

+ | '''M<sup>2</sup>≈1.4''' | ||

+ | |||

+ | Now using beam propagation formula with M<sup>2</sup> correction: | ||

+ | |||

+ | :<math>w_R(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z M^2}{z_\mathrm{R}} \right)}^2 } \ . </math> | ||

+ | instead of: | ||

+ | :<math>w_R(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2 } \ . </math> | ||

+ | The result is obvious. Plot shows the real experimental data with theoretical data fit with and without M^2 correction. | ||

+ | |||

+ | [[Image:Beam waist experiment with M2.jpg|700x600px|Beam waist Vs Z]] | ||

+ | M<sup>2</sup> is an important parameter and it is good to know it to complete the characterization of a laser. | ||

[[Category:1064crystalaser]] | [[Category:1064crystalaser]] |

## Latest revision as of 12:39, 8 February 2013

## Contents

## Specifications

We are expecting our laser any time. To know the laser more we are looking forward to investigate number of things. These specifications are already given by the maker, but we will verify them.

### Polarization

Laser is TM (transverse magnetic) or P or Horizontal linearly polarized (in the specimen plane laser is still TM polarized; when looking into the sample plane from the front of the microscope). We investigated these two ways: 1) by putting a glass interface at Brewster’s angle and measured the reflected and transmitted power. At this angle all the light is transmitted because the laser is P-polarized, 2) by putting a polarizing beam splitter which uses birefringence to separate the two polarizations; P is reflected and S is transmitted, by measuring and comparing the powers, the desired polarizability is determined. We performed the experiment at 1.8 W where P is 1.77 W and S is less than .03 W*

### Beam waist at the output window

We used knife edge method (this method is used to determine the beam waist (not the beam diameter) directly); measure the input power of 1.86W at 86.5 and 13.5 % at the laser head (15mm). It gave us the beam waist (Wo) of .82mm (beam diameter =1.64mm).

### Possible power fluctuations if any

The power supply temperature is really critical. Laser starts at roughly 1.8 W but if the temperature of the power supply is controlled very well it reaches to 2 W in few minutes and stay there. It’s really stupid of manufacturer that they do not have any fans inside so we put two chopper fans on the top of it to cool it and keep it cool. If no fans are used then within an hour the power supply reaches above 50 degrees of Celsius and then, not only the laser output falls but also the power supply turns itself off after every few minutes.

### Mode Profile

Higher order modes had been a serious problem in our old laser, which compelled us to buy this one. The success of our experiments depends on the requirement of TEM_{00} profile, efficiency of trap and stiffness is a function of profile.So mode profiling is critical; we want our laser to be in TEM00. I am not going to discuss the technique of mode profiling; it can be learned from this link: [1]
[2].

As a result it’s confirmed that this laser is TEM00 mode. Check out the pics:

A LabView program is written to show a 3D Gaussian profile, it also contains a MatLab code[3].

## Specs by the Manufacturer

All the laser specs and the manual are in the document: [Specs[4]]

## Beam Profile

The original beam waist of the laser is .2mm, but since we requested the 4x beam expansion option, the resultant beam waist is .84 at the output aperture of the laser. As the nature of Gaussian beam it still converges in the far field. We do not know where? So there is a beam waist somewhere in the far field. There are two ways to solve the problem; by using Gaussian formal but, for that we need the beam parameters before expansion optics and information about the expansion optics, which we do not have. So the only way we have, is experimentally measure the beam waist along the z-axis at many points and verify its location for the minimum. Once this is found we put the AOM there. So the experimental data gives us the beam waist and its distance from the laser in the z-direction. We use scanning knife edge method to measure the beam waist.

### Method

- In this method we used a knife blade on a translation stage with 10 micron accuracy. The blade is moved transverse to the beam and the power of the uneclipsed portion is recorded with a power meter. The cross section of a Gaussian beam is given by:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle I(r)=I_0 exp(\frac {-2r^2}{w_L^2}) }**

Where **I(r)** is the Intensity as function of radius (distance in transverse direction), **I _{0}** is the input intensity at r = 0, and w

_{L}is the beam radius. Here the beam radius is defined as the radius where the intensity is reduced to 1/e

^{2}of the value at r = 0. This can be seen by letting r = w

_{L}.

The experiment data is obtained by gradually moving the blade across from point A to B, and recording the power. Without going into the math the intensity at the points can be obtained. For starting point A

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathbf{I_A(r=0)}=I_0 exp(-2)=I_0*.865 }**

For stopping point B

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathbf{I_B}=I_0 *(1-.865) }**

By measuring this distance the beam waist can be measured and beam diameter is just twice of it:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathbf{\omega_0}=r_{.135}-r_{.865} }**

this is the method we used below.

- Beam waist can also be measured the same way in terms of the power. The power transmitted by a partially occluding knife edge:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathbf {p(r)}=\frac{P_0}{\omega_0} \sqrt{\frac{2}{\pi}} \int\limits_r^\infty exp(-\frac{2r^2}{\omega^2}) dr }**

After integrating for transmitted power:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathbf {p(r)}=\frac{P_0}{2}{erfc}(2^{1/2}\frac{r}{\omega_0})}**

Now the power of 10% and 90% is measured at two points and the value of the points substituted here:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathbf{\omega_0}=.783(r_{.1} - r_{.9})}**

The difference between the methods is; the first method measures the value little higher than the second method (power), but the difference is still under 13%. So either method is GOOD but the second is more accurate. Here is a link of a LabView code to calculate the beam waist with knife edge method[5].

#### Data

We measured the beam waist at every 12.5, 15 and 25mm, over a range of 2000mm from the output aperture of the laser head. The measurement is minimum at 612.5 mm from the laser, thus the beam waist is at 612.5±12.5mm from the laser. And it is to be 1.26±.1 mm.

{{#widget:Google Spreadsheet |key=0ApjWjFYiQdkfdEhhMkIxR2tVSlpVVzh1TEx1OFhHcUE |width=1100 |height=400 }}

#### Analysis

Here the plot beam diameter Vs Z is presented. Experimental data is presented as blue and model is red. As it can be seen that model does not fit the data. Experimental beam expands much faster than the model; this proves that the beam waist before the expansion optics must be relatively smaller. And also we are missing an important characterization parameter. The real word lasers work differently in a way that their beam do not follow the regular Gaussian formula for large propagation lengths (more than Raleigh range). So that's why we will have introduce a beam propagation factor called M^{2}.

##### Beam propagation factor M^{2}

The beam propagation factor M^{2} was specifically introduced to enable accurate calculation of the properties of laser beams which depart from the theoretically perfect TEM_{00} beam. This is important because it is quite literally impossible to construct a real world laser that achieves this theoretically ideal performance level.

M^{2} is defined as the ratio of a beam’s actual divergence to the divergence of an ideal, diffraction limited, Gaussian, TEM_{00} beam having the same waist size and location. Specifically, beam divergence for an ideal, diffraction limited beam is given by:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \theta_{0}=\frac{\lambda}{\pi w_0} }**

this is theoretical half divergence angle in radian.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \theta_{R}=M^2\frac{\lambda} {\pi w_0} }**

so

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle M^2=\frac{\theta_{R}} {\theta_{0}} }**

Where:

- λ is the laser wavelength
- θ
_{R}is the far field divergence angle of the real beam. - w
_{0}is the beam waist radius and θ_{0}is the far field divergence angle of the theoratical beam. - M
^{2}is the beam propagation factor

This definition of M^{2} allows us to make simple change to optical formulas by taking M^{2} factor as multiplication, to account for the actual beam divergence. This is the reason why M_{}2 is also sometimes referred to as the “times diffraction limit number”.
The more information about M^{2} is available in these links:[6][7]

My experimental beam waist is:

w_{R}=.63mm with a theoretical divergence of .69mrad. The data suggests the real far field divergence angle to be 1mrad (half angle) (w_{R}/z at that z). This gives:

**M ^{2}≈1.4**

Now using beam propagation formula with M^{2} correction:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle w_R(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z M^2}{z_\mathrm{R}} \right)}^2 } \ . }**

instead of:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle w_R(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2 } \ . }**

The result is obvious. Plot shows the real experimental data with theoretical data fit with and without M^2 correction.

M^{2} is an important parameter and it is good to know it to complete the characterization of a laser.