Laser Beam Parameters
R K Rupesh - 20PHMP07
∗
MSc Physics 2020-2022
1
Aim
• To measure the spot size of a laser beam at a plane a distance z from the source.
• To Determine the divergence of a laser beam.
2
Apparatus
• Gas Laser
• Knife-Edge
• Photo Detector (LDR)
• Lens
• Bench
3
Formulae and Concepts
The irradiance distribution as a function of the distance r from the center of the beam is
given by:
2Po
2r2
I(r) =
Exp − 2
(1)
πw02
wo
where wo is the width of the beam. Thus the power P(a) transmitted past a knife-edge that
blocks all light for x ≤ a is:
√ !
Z ∞
Z ∞
Po
a 2
P (a) =
dy
dx I(x, y) = erfc
(2)
2
wo
−∞
a
where a is depth of knife-edge in the beam.
Now, with wo → 2w0 where w0 is the standard deviation, we have:
P (a)
1
a
= erfc √
(3)
P0
2
2w0
Now, since the difference between the points corresponding to relative irradiance of 25% and
75% equals to a probable error of 0.6745 · w0 , we can find the beam size as
d/2
wo = 4w0 = 4
(4)
0.6745
Where d is the distance between the 25% and 75% points.
By calculating the beam size at various distances from the laser output, we can find the
divergence angle θd as:
p
wo2 (z1) − w02 (z2)
(5)
θd =
|z1 − z2|
∗
20phmp07@uohyd.ac.in
1
Figure 1: Schematic Diagram
Observations
4.1
Spot Size
4.1.1
Z = 81 cm, Withdrawing
R e la tiv e p o w e r
7 5 %
2 5 %
1 0 0
8 0
R e la tiv e p o w e r ( % )
4
(2 .9 1 2 5 ,7 5 )
6 0
4 0
(2 .3 6 8 7 5 ,2 5 )
2 0
↔
0
0
0 .5 4 4
1
2
3
m ic r o m e te r r e a d in g ( m m )
Figure 2: Z = 81 cm, Withdrawing
2
4
x1 = x(25%) = 2.368
x2 = x(75%) = 2.913
d = |x2 − x1| = 0.544
d/2
wo = 4
= 1.61 mm
0.6745
∴ the beam width at a plane 81 cm from laser output is 1.61 mm
4.1.2
Z = 106 cm, Inserting
R e la tiv e p o w e r
7 5 %
2 5 %
1 0 0
R e la tiv e p o w e r ( % )
8 0
(1 .2 ,7 5 )
6 0
4 0
(2 .0 6 3 9 1 ,2 5 )
2 0
↔
0
0
0 .8 6 4
1
2
3
m ic r o m e te r r e a d in g ( m m )
Figure 3: Z = 106 cm, Inserting
x1 = x(25%) = 2.064
x2 = x(75%) = 1.2
d = |x2 − x1| = 0.864
d/2
wo = 4
= 2.56 mm
0.6745
∴ the beam width at a plane 106 cm from laser output is 2.56 mm
3
4
4.1.3
Z = 117 cm, Withdrawing
R e la tiv e p o w e r
7 5 %
2 5 %
1 0 0
R e la tiv e p o w e r ( % )
8 0
(3 .3 3 3 3 3 ,7 5 )
6 0
4 0
(2 .8 6 1 1 1 ,2 5 )
2 0
↔
0
0 .4 7 2
0
1
2
3
m ic r o m e te r r e a d in g ( m m )
Figure 4: Z = 117 cm, Withdrawing
x1 = x(25%) = 3.334
x2 = x(75%) = 2.861
d = |x2 − x1| = 0.472
d/2
wo = 4
= 1.40 mm
0.6745
∴ the beam width at a plane 117 cm from laser output is 1.40 mm
4.2
Divergence:
Z1 = 81 cm Z2 = 106 cm,
wo (z1 = 81cm) = 1.61 mm
wo (z1 = 106cm) = 2.56 mm
√
2.562 − 1.612
∴ θd =
= 0.0062c
1060 − 810
4
4
5
5
Error
5.1
Spot Size:
The least count of the knife-edge length scale is 0.05 mm
from Eq. (4) we have,
∆wo
∆d
=
wo
d
∆a
= 2
a
∆a
∆wo = 2
wo
a
Z = 81 cm, Withdrawing
0.05
∆wo = 2
1.61 = 0.08
2.02
Z = 106 cm, Inserting
0.05
∆wo = 2
2.56 = 0.12
2.19
Z = 117 cm, Withdrawing
0.05
∆wo = 2
1.40 = 0.06
2.18
5.2
(6b)
(6c)
Divergence:
The least count of the meter scale is 1 cm
from Eq. (5) we have,
∆θd
∆wo
∆z
=
+2
θd
wo
z
∆z
∆a
+2
= 2
a
z ∆a
∆z
∆θd =
2
+2
θd
a
z
Z1 = 81 cm Z2 = 106 cm,
0.05
1
∆θd =
2
+2
0.0062
2.11
93.5
= 0.00043
6
(6a)
Result
• Spot size at a distance z = 81cm
wo = 1.61 ± 0.08 mm
• Spot size at a distance z = 106cm
wo = 2.56 ± 0.12 mm
• Spot size at a distance z = 117cm
wo = 1.40 ± 0.06 mm
• Divergence angle of the Beam between Z1 = 81cm and Z2 = 106cm,
θd = 0.0062c ± 0.00043c
5
(7a)
(7b)
(7c)