How can we trace one ray through a system—and use many rays to determine whether an image forms?
Model map
Represent a ray at one plane
: signed transverse height
: small signed slope, in radians
Valid when: the system is centered and rays are paraxial
Not modeled: diffraction, aberrations, and finite apertures
Height and slope make the model state visible.
Two operators
Free space moves height; a lens changes slope
For a positive lens, a ray above the axis receives a negative angular change: .
Order matters
Follow the ray. Read the product right to left.
The ray first travels from object to lens, so is on the right. It acts first.
Imaging criterion
A focused sensor plane makes .
When , final height does not depend on the ray’s initial slope. Many rays from one object point meet at one image point.
Guided calculation
Propagate one ray
Start with and . Propagate .
Reveal the calculation
The ray moves down ; its angle is unchanged.
Imaging criterion
Many rays, one image point
A focused ideal system sends every paraxial ray from one object point to the same height in the image plane.
Different slopes are allowed. Different final heights are not.
All three rays reach .
Interactive explorer
Move the sensor. Does reach zero?
Start with the Figure 1.4 values: , , , and . Predict what happens to the five sensor intercepts before moving .
Finite-focus correction
A 1 m target shifts the sensor plane.
A calibration feature is at from a surgical-imaging camera with . Its angle, , is securely paraxial. It focuses exactly at
The image height is , with . The sensor shifts beyond —an change—so is not a good approximation. These results are exact within the paraxial thin-lens model.
Independent application
Can you reproduce the Chapter 1.1 result?
Object point: . Lens: . Sensor: .
Reveal the matrix check
Every paraxial initial slope reaches , consistent with .
Takeaway
Build, order, and check the model.
Use for a paraxial ray.
Write one matrix for each propagation or optical element.
Put the first physical operation on the right.
At an image plane, all rays from one point share a final height.
Check units, paraxial validity, and the physical plausibility of the result.
