Laser optics is a crucial field in modern photonics, with applications spanning industries such as telecommunications, medicine, and manufacturing. To answer this question in depth, we need to consider several optical principles, including Gaussian beam propagation, lens focusing, and wavefront behavior.

Understanding Laser Beam Propagation

A laser beam is a coherent, collimated light source that propagates in free space. The fundamental mode of laser beams is often modeled as a Gaussian beam, which means its intensity distribution follows a bell curve.

When this beam encounters optical elements like lenses, mirrors, or diffraction gratings, its characteristics change. The way it transforms depends on factors such as:

1. Focal Length of the Lenses
2. Lens Positioning Relative to the Beam Waist
3. Wavefront Curvature of the Incident Beam
4. Aberrations Introduced by the Optical System

Each of these factors contributes to how the beam profile alters as it passes through the lenses.

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Effect of Two Lenses with Different Focal Lengths

When a laser beam enters a system with two lenses of different focal lengths, a few key optical transformations occur:

1. First Lens – Initial Beam Focusing or Divergence

When the beam enters the first lens, several possibilities arise:

• If the beam is collimated and the first lens is converging, the beam will focus to a spot at a distance determined by the lens’s focal length.
• If the beam is already converging or diverging before reaching the lens, the focal spot shifts according to lens-maker’s equations.
• A diverging lens will cause the beam to spread out further, altering its divergence angle before it encounters the second lens.

Mathematically, the transformation of the beam can be described using the thin lens equation:

1f=1do+1difrac{1}{f} = frac{1}{d_o} + frac{1}{d_i}f1​=do​1​+di​1​

where:

• fff = focal length of the lens
• dod_odo​ = object distance (distance from laser source or previous optical element)
• did_idi​ = image distance (where the beam focuses)

Depending on these distances, the beam could be collimated, focused, or diverging before it enters the second lens.

2. Second Lens – Further Modification of Beam Profile

Now that the beam has been altered by the first lens, it enters the second lens with a modified divergence. The second lens's effect depends on its relative focal length compared to the first lens.

• If the second lens has a shorter focal length, it will focus the beam more tightly, reducing the beam waist.
• If the second lens has a longer focal length, it will result in a wider beam waist, effectively reducing the beam's divergence.
• If the lenses are arranged as a beam expander (concave-convex pair), the beam will emerge with an altered divergence but a maintained wavefront curvature.

This phenomenon follows the Fourier optics principles, where spatial transformations occur due to changes in lens curvature and distance.

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Mathematical Modeling of the Beam Profile Change

To fully understand how the beam profile changes, we must consider the Gaussian beam transformation equations.

1. Beam Waist Transformation
   The beam waist after passing through a lens is given by:

   w′=λfπw0w' = frac{lambda f}{pi w_0}w′=πw0​λf​

   where:

   • w′w'w′ = new beam waist
   • λlambdaλ = laser wavelength
   • fff = focal length of the lens
   • w0w_0w0​ = initial beam waist

   This equation demonstrates that a shorter focal length results in a smaller beam waist, while a longer focal length produces a wider beam waist.

2. Ray Transfer Matrix Analysis
   The transformation of a beam through an optical system can be described using ABC-D matrices:

   [ABCD]begin{bmatrix} A & B \ C & D end{bmatrix}[AC​BD​]

   where each matrix represents a different optical component, such as a free-space propagation matrix or a lens matrix.

3. Wavefront Curvature Adjustment
   When a laser beam moves through multiple lenses, its wavefront curvature is modified. This curvature affects how it converges or diverges in space. The curvature is defined as:

   R′=R1−dRR' = frac{R}{1 - frac{d}{R}}R′=1−Rd​R​

   where:

   • RRR = initial radius of curvature
   • ddd = distance from the lens

   The combination of these transformations determines how the final laser beam appears after the second lens.

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Experimental Validation & Practical Implications

To verify this effect, researchers use beam profilers to measure the intensity distribution before and after the lenses. Some common experimental setups include:

• Focal Spot Measurement: Placing a sensor at different distances to capture beam waist changes.
• Interferometry: Observing wavefront distortions caused by lens-induced aberrations.
• Fourier Transform Lens Systems: Used in optical data processing to alter beam properties.

Understanding these transformations is critical in designing optical fiber coupling systems, laser machining tools, and high-precision optical instruments.

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Conclusion

In summary, when a laser beam passes through two lenses with different focal lengths, its profile changes based on:

• First Lens Effect: Alters the initial divergence/convergence.
• Second Lens Effect: Further refines beam width, divergence, and wavefront curvature.
• Mathematical Transformations: Described using Gaussian beam optics and ABCD matrices.
• Practical Applications: Used in laser processing, microscopy, and optical communication.

By carefully selecting focal lengths and lens positions, engineers and physicists can manipulate laser beams for precision applications without degrading beam quality.