Gentec Beamage M2 User Manual page 21

Beamage-M² User Manual
Mathematically, it is given by the following equation:
Far from the beam waist, the beam expansion becomes linear and the theoretical divergence half-angle
��
(half of the angle shown in Figure 4-1) can be obtained by evaluating the limit of the beam radius's
��ℎ
first derivative as the position tends towards infinity:
�� = ������
��ℎ
For a laser beam that passes through a focusing lens of focal length f, the theoretical radius of the beam
��
at the focal spot of the lens can be obtained by multiplying the beam divergence half-angle with the
����ℎ
focal length f:
As mentioned, all of the equations above describe theoretical ideal Gaussian beams. However, they can
describe the propagation of real laser beams if we modify them slightly using the M
mathematically defined by the following equations:
����
������
2
�� =
With mathematics, it is easy to understand why small M
divergences and small experimental beam waist radiuses.
The experimental beam waist radius ��
experimental beam radius at the focal spot of the lens
equations:
��
�� ��ℎ
( �� )
����
��ℎ
= ������
����
��→∞ ��→∞
��
��
��ℎ
�� �� ��
0 ������ 0
������ ������
= > 1 �������������� ��
�� �� ��
��ℎ 0
��ℎ
(��), the experimental half-angle divergence ��
������
( �� ) = ��
��
������ 0
��ℎ
��
������
�� = ����
�� ������
Revision 6
2
��( )
��
0
��ℎ
=
��
�� ����
√ 1 + (
��
0
���� ��ℎ
��(��
0
����
��
= �� =
��ℎ
��
��
0
��ℎ
������
2
values correspond to low experimental
��
are therefore given by the following
�� ������
����
2 2
√ ��
+ �� (
��(��
0
��ℎ
2
�� ��
=
��
��
0
������
2
���� ��
=
������
����
0
������
2
��
) =
2
����
)
0
��ℎ
��ℎ
2
factor, which can be
��
�� > = �� ��
0 ��ℎ 0
������ �� ��ℎ
2
)
2
)
14
, and the
������