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412 PHYS

Lasers and their Applications

Department of Physics

Faculty of Science

Jazan University

KSA

Propagation of Gaussian Beams

& Optical Resonators

Lecure-4

Propagation of laser beams cab be described by the Helmholtz equation

one of the possible solutions to this equation is given by

2 2

0

( )( , , , ) exp { ( ) }

2 ( )

k x yx y z i p z

q z

0 Constant depends on the beam amplitude and can be determined by

boundary conditions

zkiezyxzyxU ),,(),,( 0),,()( 22 zyxUk

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( ) ln(1 / )op z i z q )(zp

( )q z complex beam parameter, given as

2

00 0( )

i wq z z iz z q z

Complex function given as

Rayleigh range at which he beam waist is given by 0z 0( ) 2w z w

0w Is the minimum beam waist

For a spherical wave

0

2 2 2 2 2

0 0 0

1 1 1

( ) ( )

zzi i

q z iz z z z z R z n w z

The real part represents the wavefront curvature, with a radius given

by 2

0( ) [1 ( / ) ]R z z z z

)(z Beam waist radius, given by 2 1/2

0 0( ) [1 ( / ) ]w z w z z

0w Is the minimum beam waist radius, given in terms of the beam

divergence as 1/2

00

log(2)

2 tan( / 2)

zw

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Laser resonators

The feedback in lasers is achieved by placing the amplifier (active medium)

between mirrors, a construction we call an optical cavity or resonator.

The resonator is the space of optical amplifier that contains the feedback elements

The resonator

When the population inversion occurs in the active medium, the spontaneous

emission produces a photon that propagates along the optical axis of the

active medium and the resonator

The photons interacts with the excited atoms and the stimulated emission will

occur and hence a wave with amplified amplitude will propagate through the

medium towards one of the mirrors

Upon reflection from the mirror, the wave will be further amplified by passing

through the medium due to the resonance with the excited atoms (because

they both have same energy )

Eventually the wave will be oscillated between mirrors and get amplified in

every pass and loss some photons in the output mirror as the output beam

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Laser resonator stability

1 2

0 1 1 1L Lr r

12 r،r Radii of mirror’s

curvatures

L Length of the

resonator

The Condition for resonator stability

• The cavity is an essential part of a laser. It provides the

positive feedback that turns an amplifier into an oscillator.

• The design of the cavity is therefore very important for

the optimal operation of the laser.

Types of resonators

Plane parallel resonator

Confocal resonator

Hemispherical resonator

Large radius resonator

Concentric resonator

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Example:

Sol.

Determine whether or not the following mirror arrangements lead to stability:

a. Two mirrors with radii of curvature of 1.8 m, separated by a distance of 2 m

b. One mirror with radius of curvature of 2m and the other with radius of 3m,

separated by a distance of 2.3m

c. One mirror with radius of curvature of 5m and the other with radius of 3m,

separated by a distance of 4m

d. Two mirrors with radius of curvature of 0.5 m, separated by a distance of 0.5m

1 20 (1 / )(1 / ) 1L r L r

a. r1=r2=1.8m, L=2m

2 2

1 2(1 / )(1 / ) (1 2 /1.8) (1 1.11) 0.121L r L r Cavity is stable

b. r1=2 m, r2=3m, L=2.3m

1 20 (1 / )(1 / ) 1L r L r

(1 2.3 / 2)(1 2.3 / 3) ( 0.15)(0.25) 0.0345 Cavity is unstable

c. r1=5 m, r2=3m, L=4m

(1 4 / 5)(1 4 / 3) (0.2)( 0.33) 0.067 Cavity is unstable

d. r1=0.5 m, r2=0.5m, L=0.5m

(1 0.5 / 0.5)(1 0.5 / 0.5) 0 Cavity is on edge of stability

---confocal cavity

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Laser modes

•The cavity determines the properties of the beam of light that is emitted

by the laser.

• This beam is characterized by its transverse and longitudinal mode

structure.

Transverse modes are created in cross section of the beam,

perpendicular to the optical axis of the laser.

Longitudinal modes only specific frequencies are possible inside

the optical cavity of a laser, according to standing wave condition.

Transverse mode structure

A transverse mode is a field configuration on the surface of one reflector that propagates

to the other reflector and back, returning in the same pattern,

apart from a complex amplitude factor (that gives the total phase shift and loss of the

round trip.

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Separating the fundamental Gaussian mode

Lasers operating with the fundamental Gaussian mode TEM00 are preferred due to

the following reasons:

1. TEM00 has a symmetrical, uniform circular configuration with the greatest

intensity at its center: this suits many applications that requires high

accuracy

2. Contains about 85% of the total output intensity

3. Can be easily separated from higher order modes by using a pin hole

aperture with a diameter that allow only photons propagated along the

optical axis to be incident on mirrors

It is preferred to get the laser operated in the fundamental mode TEM00

The intensity of the Gaussian mode is given by

2 2

0 0exp( 2 / )I r I w w

The total power 2/ 2P w I

02 /div w Divergence

0w Radius of Gaussian beam (the radius at which

the intensity reduced by 2/1 e

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Example 1:

A He-Ne Laser with a Gaussian fundamental mode TEM00 operates with a

wavelength of 632.8 nm . A lens is used to collimate the beam to pass through a

glass plate of a thickness of 12 mm and a refractive index of 1.46. suppose the

plate is place at the focal point of the lens where the beam diameter is 2m

Calculate the beam diameter at the other end of the plate?

0 0

212 0.012 , 632.8 , 2 1

2 2

d mz mm m nm d w w m

12 2

0

0

1z

w z wz

262

600 9

12 2

6 3

6

3

1.46 107.25 10

632.8 10

0.01210 1 1.66 10

7.25 10

beam dimeter 2 ( ) 3.32 10

n wz m

w z m

w z m

Sol.

We have

A fundamental Gaussian beam from a Ti:Sapphire laser with a wavelength of 759nm

and a power of 1mW incident on a target far from the minimum waist point by

100m. If the radius of the minimum waist is 2mm, find the beam waist at the

target?

Calculate the radius of wavefront curvature and peak intensity of the beam

Sol.

Waist radius at the target

1/22

2 3 2

00 0 9

0

(2 10 )( ) 1 , ( ) 3.14 16.55

759 10

wzw z w z Rayleigh range

z

1/22

3 100( ) 2 10 1 12

16.55w z mm

Example 2:

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Usually the spot size is represented by the beam waist and the area of the laser

spot (if it is circular) is give by 2 3 2( ) 3.14 12 10 0.0314A w z m

The radius of wavefront curvature is

2

2

0

16.55( ) [1 ( / ) ] 100 1 102.74

100R z z z z m

peak intensity 3

3 2

2 3 2

1 104.4 10 /

( ) / 2 3.14 (12 10 ) / 2p

P PI W m

A w z

Note: is the effective area 2{ ( ) / 2}A w z

The longitudinal modes determine the emission spectrum of the laser.

The light bouncing repeatedly off the end mirrors sets up standing waves inside the

cavity.

LONGITUDINAL MODES

More general

where n is the average refractive index of the cavity.

The last Equation implies that only certain frequencies which satisfy will oscillate

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Plane-mirror resonators

• This type of cavity consists of two flat, parallel mirrors separated by a distance L . It is also called

a Fabry-Perot resonator (F-P).

• In this type of cavity, the beam fills the space between the mirrors nearly uniformly, unlike in the

spherical mirror cavities where the beam is focused somewhere inside (or outside of) the cavity.

The finesse coefficient of the Fabry-Perot cavity Is given as

1 2

1/4 1/4

1/2 1/2

1 21c

R Rf

R R

If the two mirrors have similar reflectivities R1=R2 then

1c

Rf

R

The finesse coefficient has great importance for determining the fluorescence

line shape

/cf Frequency spacing between modes

Laser fluorescence line width

2 (2 )kL m

2 42 (2 )

n L LkL m

C

Note, that the condition of standing wave is

2

mL

2m

cm

n L

Frequency spacing between modes

or Free Spectral Range 1

2m m

c

n L

Phase displacement for one round trip oscillation

Frequency of the mode of order m

Derivation of the number of Longitudinal modes

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2

22

cor

n Ln L

The number of operating

longitudinal modes is given by

( / 2 )m

c n L

The quality factor of Fabry-Perot

resonator is

0Q

Calculate the width of the frequency mode for a Fabry-Perot resonator

consists of two similar plane mirrors separated by a distance of 1 cm

Assume the reflectivities 70% 99.9% 95%

Calculate the finesse factor for the wavelength of 800 nm from a GaAs laser

And find also the value of the resonator quality factor for each case?

8103 10

1.5 102 2 1 0.01

cHz

n L

814

9

3 103.75 10

800 10

cHz

1 0.9999R For 10

4 5

4

0.9999 1.5 103.141 10 4.78 10

1 0.9999 3.14 10c

c

f Hzf

Example 3:

Sol.

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Resonator quality 14

80

5

3.75 107.85 10

3.78 10Q

2 0.95R For 10

80.95 1.5 1061.2 2.45 10

1 0.95 61.2c

c

f Hzf

1460

8

3.75 101.53 10

2.45 10Q

For 3 0.75R

109

145

9

0.75 1.5 1010.87 1.38 10

1 0.75 10.87

3.75 102.72 10

1.38 10

c

c

f Hzf

Q

Example 4:

For a He-Ne laser with a wavelength of 632.8 nm, if the length of the

resonator is 30 cm, find:

a. Frequency difference between longitudinal modes (mode spacing

b. Number of modes

c. Frequency of the laser light

Sol. 83 10

0.52 2 0.3

cGHz

L

a. Mode spacing

b. Number of modes 6

6

2 2 2 0.30.948 10

0.6328 10m

L Lm

m

c. Frequency of the laser

6 9 140.948 10 0.5 10 4.74 10m Hz

or 8

14

6

3 104.74 10

0.6328 10

cHz

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Note: 1) and 2) allow only low powers to be obtained (of no practical use)

1. Reducing laser cavity length to make the mode

spacing large and hence allow only one mode to

operate 2

c

n L

2. Reduce pumping power and hence allow the

amplification only for the central mode

Laser operation in single longitudinal mode

There are many methods that can be used to force the laser oscillation in

single longitudinal mode

4. Using Prism to select

one longitudinal mode

5. Using grating to select

one longitudinal mode

3. Using Etalon

Generate additional losses for the extra modes by

placing frequency selective optical elements in the

laser resonator

The lasing mode gets some of the gain of the killed

modes higher power/mode

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Example 5: An ion Argon laser with a wavelength of 514.5 nm with a spectral

bandwidth of 2GHz and a cavity length of 50 cm. find the number of possible

longitudinal modes?

Sol. 9

8 2

2 106mod

/ 2 3 10 / (2 50 10 )m es

c L

Refractive index of

argon gas =1

Example 6: If a spectral filter with a bandwidth of 0.1 nm is used to obtain a single

longitudinal mode from a He-Ne laser, what should be the length of the laser cavity?

89 10

2 9 2

3 10(0.1 10 ) 7.5 10

(632.8 10 )

cHz

Sol. The fluorescence frequency bandwidth is given by

For a single mode oscillation,

we should have

8

10

3 100.002

2 2 7.5 10

cL L m

L