فيزياء الليزر

# محاضرة 13 فيزياء الليزر الحصيلة مع الخسارة

## Round trip gain with losses

درسنا في محاضرات سابقة (محاضرة 10) الحصيلة الناتجة عن دورة كاملة لليزر خلال المادة وعلمنا أن انقلاب التعداد شرط اساسي لكي يستمر الحصول على ليزر ولكن الفقد الناتج عن عدة عوامل يسبب في نقصان الحصيلة.  ولكي نحصل على ليزر فإن الحصيلة لكل دورة يجب أن تكون على الاقل كبر من الخسارة في كل دورة.  في هذه المحاضرة سنركز على العلاقة بين الحصيلة والخسارة

The total losses of the laser system is due to a number of different processes these are:

اعلانات جوجل
1. Transmission at the mirrors

2. Absorption and scattering by the mirrors

3. Absorption in the laser medium

4. Diffraction losses at the mirrors

All these losses will contribute to reduce the effective gain coefficient to (go– k)

Round trip Gain (G)

Figure below show the round trip path of the radiation through the laser cavity. The path is divided to sections numbered by 1-5, while point “5” is the same point as “1”. Round trip path of the radiation through the laser cavity.

اعلانات جوجل

By definition, Round trip Gain is given by:

G = I5/ I1

G = Round trip Gain.

I1 = Intensity of radiation at the beginning of the loop.

I5 = Intensity of radiation at the end of the loop.

Gain (G) Without Losses

From lecture (10) we found that the intensity after one round trip is given by the equation

اعلانات جوجل

I5 = R1* R2*G2*I1

Gain (G) With Losses

We assume that the losses occur uniformly along the length of the cavity (L).  In analogy to the Lambert formula for losses, we define loss coefficient (a), and using it we can define absorption factor k:

اعلانات جوجل

k = exp(-2aL)

اعلانات جوجل

k = Loss factor, describe the relative part of the radiation that remain in the cavity after all the losses in a round trip loop inside the cavity.

All the losses in a round trip loop inside the cavity are 1-k(always less than 1).

a = Loss coefficient (in units of 1 over length).

2L = Path Length, which is twice the length of the cavity.

Adding the loss factor (k) to the equation of I5:

I5 = R1* R2*GA2*I1*k

From this we can calculate the round trip gain:

G = I5/I1 = R1* R2*GA2 *k

As we assumed uniform distribution of the loss coefficient (a), we now define gain coefficient (g), and assume active medium gain (GA) as distributed uniformly along the length of the cavity.

GA = exp(+gL)

Substituting the last equation in the Loop Gain:

G = R1* R2* exp(2(g-a)L)

When the loop gain (G) is greater than 1 (G > 1), the beam intensity will increase after one return pass through the laser.

When the loop gain (G) is less than 1 (G < 1), the beam intensity will decrease after one return pass through the laser. laser oscillation decay, and no beam will be emitted.

Conclusion:

There is a threshold condition for amplification, in order to create oscillation inside the laser.

This Threshold Gain is marked with index “th”.

For continuous laser , the threshold condition is:

Gth = 1 = R1 R2 GA2k = R1* R2* exp(2(g-a)L)

Example

Active medium gain in a laser is 1.05.  Reflection coefficients of the mirrors are: 0.999, and 0.95. Length of the laser is 30 cm.  Loss coefficient is: a = 1.34*10-4 cm-1Calculate:

1. The loss factor k.

2. The round trip gain G.

3. The gain coefficient (g).

Solution

1. The loss factor k:

k = exp(-2aL) = exp[-2(1.34*10-4)*30] = 0.992

2. The Loop gain G:

G = R1R2GA2k = 0.999*0.95*1.052*0.992 = 1.038

Since GL > 1, this laser operates above threshold.

3. The gain coefficient (g):

G = exp(gL)

Ln G = gL

g = Ln G/L = ln(1.05)/30 = 1.63*10-3 [cm-1]

The gain coefficient (g) is greater than the loss coefficient (a), as expected.

Example

Helium Neon laser operates in threshold condition. Reflection coefficients of the mirrors are: 0.999, and 0.97. Length of the laser is 50 cm. Active medium gain is 1.02. Calculate:

1. The loss factor k.

2. The loss coefficient a.

Solution

Since the laser operates in threshold condition, G = 1.
Using this value in the round trip gain:

G = 1 = R1R2GA2k

1. The loss factor k:

k = 1/( R1R2GA2) = 1/(0.999*0.97*1.022) = 0.9919

As expected, k < 1.
Since G > 1, this laser operates above threshold.

2. The loss coefficient (a) is calculated from the loss factor:

k = exp(-2aL)

lnk = -2aL

a = lnk/(-2L) = ln(0.9919)/(-100) = 8.13*10-5 [cm-1]

Attention:

If the loss factor was less than 0.9919, then G < 1, and the oscillation condition was not fulfilled. Example

Reflection coefficients of the mirrors are: 0.999, and 0.95. All the losses in round trip are 0.6%. Calculate the active medium gain. Solution

For finding the active medium gain G, the loss factor (k) must be found.
All the losses are 1-k.

1-k = 0.006

k = 0.994

Using this value in the threshold loop gain:

Gth = 1 = R1R2GA2 k

(GA)th = 1/sqrt( R1R2k) = 1/sqrt(0.999*0.95*0.994) = 1.03

The active medium gain must be at least 1.03 for creating continuous output from this laser.

 Summary G = round trip Gain, determines if the output power of the laser will increase, decrease, or remain constant. It include all the losses and amplifications that the beam have in a complete round trip through the laser. GL = R1R2GA2k

R1, R2 = Reflection coefficients of the laser mirrors.
GA = Active medium gain as a result stimulated emission.

GA = exp(+gL)

g = Gain coefficient.
L = Active Medium length.
k = Optical Loss Factor in a round trip path in the laser cavity.

k = exp(-2aL)

a = Loss coefficient.

• When G = 1, The laser operate in a steady state mode, meaning the output is at a constant power. This is the threshold condition for lasing, and the active medium gain is:

(GA)th = 1/sqrt( R1R2k)

The round trip Gain is:

GL = R1* R2* exp(2(g-a)L)

### الدكتور حازم فلاح سكيك

د. حازم فلاح سكيك استاذ الفيزياء المشارك في قسم الفيزياء في جامعة الازهر – غزة | مؤسس شبكة الفيزياء التعليمية | واكاديمية الفيزياء للتعليم الالكتروني | ومنتدى الفيزياء التعليمي

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