Figure 2.8: The focused spot of a Gaussian beam after it passes through
a lens will be located at the focal point of the lens if the input beam waist
is either very close or very far away from the lens. This is because the
input beam is approximately collimated at those points
L
|s|<<zR
wo f
|s|>>zR
wo’
wo’
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the laser’s Rayleigh range, then s << zR and (|s|-f )2<zR2. Equation 2.18
simplifies to:
2.22
This also simplifies the calculations for the output beam’s waist, and
waist location:
When s << zR, the distance from the lens to the focused spot
equals the focal length of the lens.
The other limiting situation where the lens is far outside of the Rayleigh
range and s >> zR, simplifying Equation 2.18 and the output beam waist
diameter to:
Similarly to when s << zR, the calculations for the output beam waist, and
beam waist location are also simplified:
When s >> zR, the distance from the lens to the focused spot
equals the focal length of the lens.
Both of these results intuitively make sense because the beam’s wavefront
is approximately planar both at and very far away from the beam
waist. At these locations, the beam is almost perfectly collimated (Figure
2.8). According to the standard thin lens equation, a collimated input
would have an image distance equal to the focal length of the lens.
Gaussian Focal Shift
Counterintuitively, the intensity of a focused beam on a target at a fixed
distance (L) away from the lens is not maximized when the waist is located
at the target. The intensity on the target is actually maximized
when the waist occurs at a location before the target (Figure 2.9). This
phenomenon is known as Gaussian focal shift.
The lengthy derivation is not covered in this text, but the beam radius at
the target can be described by the following expression:4
2.29
Differentiating Equation 2.29 with respect to the focal length of the
focusing lens (f ) and solving for f when d/df wL (f ) = 0 reveals the lens
focal length resulting in the minimum beam radius, and therefore highest
intensity, at the target.
2.30
2.31
As |s| approaches either zero or infinity, d/df wL (f ) = 0 when f = L. In
both of these cases, the input beam is approximately collimated, and it
thereby follows that the smallest beam radius would occur at the focal
point of the lens.
wo’ wL
wo
s’ Target
L
s
Figure 2.9: The minimum beam radius at a target occurs when the waist
of the focused beam occurs at a specific location before the target, not
when the focused waist is located at the target
2.23 2.24
2.25 2.26
2.27 2.28
References
1. Paschotta, Rüdiger. Encyclopedia of Laser Physics and Technology, RP
Photonics, October 2017, www.rp-photonics.com/encyclopedia.html.
2. Self, Sidney A. “Focusing of Spherical Gaussian Beams.” Applied Optics,
vol. 22, no. 5, Jan. 1983.
3. O'Shea, Donald C. Elements of Modern Optical Design. Wiley, 1985.
4. Katz, Joseph, and Yajun Li. “Optimum Focusing of Gaussian Laser Beams:
Beam Waist Shift in Spot Size Minimization.” Optical Engineering, vol. 33,
no. 4, Apr. 1994, pp. 1152–1155., doi:10.1117/12.158232.
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