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Question #43233

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The Question

A telescope in normal adjustment consists of an objective lens of focal length $f_o$ and an eyepiece of focal length $f_e$ . It is used to observe a distant point-like star that provides a uniform photon flux $I$ (in $\text{photons} \cdot \text{m}^{-2} \cdot \text{s}^{-1}$ ) at the objective. A circular photon detector of fixed diameter $d_0$ is placed exactly at the exit pupil of the telescope. The detector integrates photons over a time interval $\tau$ . Assume the only source of noise is the inherent Poisson statistical fluctuation of the arriving photons, where the noise is defined as the standard deviation of the detected photon count. If the objective diameter $D$ can be increased arbitrarily while keeping $f_o$ and $f_e$ constant, which of the following expressions gives the maximum achievable Signal-to-Noise Ratio (SNR) for this observation?

Options

A.
$\frac{d_0 f_o}{2 f_e} \sqrt{I \pi \tau}$
B.
$\frac{\pi I \tau d_0^2 f_o^2}{4 f_e^2}$
C.
$\frac{d_0}{2} \sqrt{I \pi \tau}$
D.
$\frac{d_0 f_e}{2 f_o} \sqrt{I \pi \tau}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Telescope Normal Adjustment|Intensity and Photon Flux|Statistical Nature of Decay

Why This Question Is Tricky

Common Mistakes

This question tests the 'Variable Aperture Loading' misconception by coupling geometric optics with the quantum-statistical nature of light.

Students often assume that increasing the objective diameter $D$ indefinitely will continuously increase the light-gathering power and the Signal-to-Noise Ratio (SNR).

However, they fail to account for the geometric constraint of the exit pupil.

When the exit pupil diameter $d = D(f_e/f_o)$ exceeds the fixed detector diameter $d_0$ , the detector truncates the beam.

The photon flux density at the exit pupil depends strictly on the telescope's magnification squared $(f_o/f_e)^2$ , not on $D$ .

Furthermore, the trap exploits the misapplication of Poisson statistics; students might incorrectly assume SNR scales linearly with the photon count $N$ rather than $\sqrt{N}$ , or they might ignore the magnification factor entirely.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Telescope Normal Adjustment
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free