= Solution
Because the sphere lies below the optical axis, a ray passing through its upper half samples a brighter part of the <Gaussian beam> than a comparable ray through its lower half. For a particle with higher <refractive index> than its surroundings, the stronger upper ray is deflected downward overall and transfers an upward reaction to the particle. The weaker lower ray gives a smaller downward reaction. Their transverse sum is therefore upward, toward the brighter axis.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2023/paper-2-optical-forces.png]
{title=Unequal refracted rays give a transverse restoring force}
{description=Original local ray tracing near a beam waist, with chosen refractive indices. The sphere is below the optical axis. The stronger ray contributes the larger upward reaction; the diagram isolates transverse restoration rather than reconstructing the examination figure.}
{source=paper-2-optical-forces.py}
If the displacement reverses, the force reverses, so the central position is transversely stable. Summing all the rays gives an <optical gradient force> toward the intensity maximum. There is also a forward <optical scattering force>. A tightly focused beam can have an axial restoring component that balances it, generally at a small displacement downstream of the ideal focus.<Observation of a single-beam gradient force optical trap for dielectric particles by Arthur Ashkin and others>[Single-beam trapping by a sufficiently strong intensity gradient]{ref}
Three-dimensional stability requires the restoring axial component to exceed the forward force over part of the trajectory. A Gaussian transverse intensity profile alone does not prove that condition; insufficient focusing can yield transverse confinement while the particle still moves downstream.
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