Galilean vs Keplerian Telescopes: Optical Geometries, Exit Pupils & Low Vision Fields

A low vision telescope consists of an Objective Lens ($F_o$) and an Eyepiece Ocular ($F_e$) separated by a physical tube length ($d$):

1. Galilean Telescope: Combines a positive objective lens ($F_o > 0$) with a high-minus negative eyepiece ($F_e < 0$):

$$M = -\frac{F_e}{F_o} \quad | \quad \text{Tube Length: } d = f_o + f_e = f_o - |f_e|$$

• Produces a naturally erect, non-inverted image without requiring internal erecting prisms.
• Ultra-compact physical tube length and light weight ($15\text{--}30\ \text{grams}$); ideal for spectacle mounting.

2. Keplerian (Terrestrial) Telescope: Combines a positive objective ($F_o > 0$) with a positive eyepiece ($F_e > 0$):

$$M = -\frac{F_e}{F_o} \quad | \quad \text{Tube Length: } d = f_o + f_e$$

• Produces an inverted, reversed image requiring an internal Porro prism or Schmidt-Pechan roof prism assembly to erect the image.
• Longer, heavier physical body ($60\text{--}120\ \text{grams}$), but delivers an exceptionally wide, sharp panoramic field of view.

The Exit Pupil ($EP$) is the optical image of the objective lens aperture formed by the eyepiece:

$$EP_{\text{diameter}} = \frac{\text{Objective Lens Diameter } (A_o)}{\text{Magnification } (M)}$$

• Galilean Exit Pupil: Resides inside the telescope tube between the lenses. Because the eye's entrance pupil cannot physically enter the tube, light rays are clipped at the edges, restricting field of view to a narrow tunnel ($8^\circ\text{ to } 12^\circ$).
• Keplerian Exit Pupil: Projects outside the eyepiece at a comfortable Eye Relief distance ($10\text{ to } 15\ \text{mm}$). The patient's pupil aligns perfectly with the exit pupil, doubling or tripling the usable field of view ($15^\circ\text{ to } 22^\circ$) with edge-to-edge optical sharpness.

Dispensing telescopes requires balancing magnification against field of view and movement jitter:

$$\textbf{Optical Law: } \text{Field of View (FOV)} \propto \frac{1}{M} \quad | \quad \text{Hand Tremor Jitter} \propto M^2$$

• Galilean ($1.7\times\text{ to } 2.5\times$): Best for bioptic driving spectacles and television viewing where lightweight comfort and modest magnification suffice.
• Keplerian ($3.0\times\text{ to } 6.0\times$): Mandatory for reading distant street names, sporting events, or high-magnification spotting tasks where wide fields of view are essential.