Optics 3 - L.2 Telescopes

Fundamental Principles of Telescopes

• A telescope is an optical device composed of lenses designed to provide angular magnification.

• The primary objective is to increase the angle subtended by an object at the plane of the eye, which in turn creates a larger image on the retina.

• Unlike simple magnifiers used for near tasks, telescopes are concerned with the comparison of the image size with the device versus the image size without the device. The actual size of the image relative to the physical object is not the focus; rather, the factor by which the retinal image increases is the priority.

• A telescope is a relatively simple device consisting of two lenses separated by a specific distance within a tube.

• A basic functional telescope can be constructed by holding two trial lenses from a lens kit at a precise distance from one another.

Optical Components and Anatomy

• Objective Lens: This is the lens that faces the object being viewed. It is the entry point for light into the telescope. Its power is denoted in diopters as $D_{objective}$.

• Ocular Lens: This lens is positioned on the side through which the user looks (the eye side). It is the exit point for light. Its power is denoted as $D_{ocular}$.

• Tube Length ($d$): This is the physical distance between the two lenses. While usually described in centimeters ($cm$), calculations involving tube length require the value to be converted to meters ($m$).

Optical Foundations and Vergence

• The behavior of light through the system is governed by the vergence formula: $U + D = V$.

  • $U$ represents the vergence of light entering the optical system.

  • $D$ represents the power of the optical system (the lens).

  • $V$ represents the vergence of light exiting the system.

• For distance calculations in air ($n=1$), the formula applies as: $\frac{1}{u} + D = \frac{1}{v}$, where $u$ is the object distance and $v$ is the image distance.

• Real and Virtual Objects:

  • Real objects provide diverging light to the system, resulting in a negative vergence ($U$) and a negative distance ($u$).

  • Virtual objects occur when converging light enters a lens, resulting in a positive vergence ($U$) and a positive distance ($u$). Virtual objects typically sit on the right side of the lens.

• Real and Virtual Images:

  • Real images are formed by converging light leaving the lens ($V$ is positive). These rays focus at a focal point to the right of the lens.

  • Virtual images are formed by diverging light leaving the lens ($V$ is negative). The rays never meet on the right and must be traced back to the left side of the lens.

Focal Points and Lens Characteristics

• Primary Focal Point ($F_1$): Defined as the object distance conjugate to an image at infinity. For a lens in air, $F_1 = -\frac{1}{D}$.

  • For a convex (plus) lens, $F_1$ is located on the left.

  • For a concave (minus) lens, $F_1$ is located on the right.

• Secondary Focal Point ($F_2$): Defined as the image point conjugate to an object at infinity. For a lens in air, $F_2 = \frac{1}{D}$.

  • For a convex (plus) lens, $F_2$ is located on the right.

  • For a concave (minus) lens, $F_2$ is located on the left.

Keplerian and Galilean Telescopes

• Keplerian Telescope:

  • Both the objective and ocular lenses have positive (plus) power.

  • Mnemonic: The word "Keplerian" contains the letters "P" and "L," standing for "plus-plus."

  • Because it uses two plus lenses, the light converges between the lenses, resulting in an inverted image. Most Keplerian telescopes require an internal prism to re-invert the image to an upright position.

• Galilean Telescope:

  • The objective lens has a positive (plus) power, and the ocular lens has a negative (minus) power.

  • This design results in an upright image without the need for prisms.

Afocal Telescope Configuration

• A telescope is considered afocal when parallel light enters the system and parallel light exits the system.

• For a system to be afocal, the secondary focal point of the objective lens ($F_{2obj}$) must coincide exactly with the primary focal point of the ocular lens ($F_{1oc}$).

• When light from a distant object (represented as parallel rays) enters the plus-power objective lens, it converges toward $F_{2obj}$. After passing this point, the rays begin to diverge. To exit the telescope as parallel rays, this divergence must occur from the primary focal point of the ocular lens ($F_{1oc}$).

• Tube Length Calculation: The required distance between lenses for an afocal system is calculated as:     $d = \frac{1}{D_{objective}} + \frac{1}{D_{ocular}}$

  • Example (Keplerian): For a $+5D$ objective and a $+20D$ ocular:         $d = \frac{1}{+5} + \frac{1}{+20} = 0.20\,m + 0.05\,m = 0.25\,m = 25\,cm$

  • Example (Galilean): For a $+5D$ objective and a $-20D$ ocular:         $d = \frac{1}{+5} + \frac{1}{-20} = 0.20\,m - 0.05\,m = 0.15\,m = 15\,cm$

  • Galilean telescopes are generally shorter than Keplerian telescopes because the minus ocular lens must be moved closer to the objective to intercept the converging rays before they reach their focal point.

Magnification

• The magnification ($M$) of a telescope is the ratio of the powers of the two lenses:     $M = -\frac{D_{ocular}}{D_{objective}}$

• Mnemonic: To be a "top doc," the ocular power ($D_{ocular}$) must be on the top of the formula.

• Keplerian Magnification: Results in a negative value (e.g., $-4x$), indicating the image is inverted.

• Galilean Magnification: Results in a positive value (e.g., $+4x$), indicating the image is upright.

• Example: A telescope with a $+10D$ objective and a $-30D$ ocular has a magnification of:     $M = -\frac{-30}{+10} = +3x$

Impact of Uncorrected Refractive Error

• When a patient looks through a telescope without their corrective lenses, the refractive error is effectively incorporated into the power of the ocular lens.

• Hyperopes:

  • A five-diopter ($5D$) hyperope brings a $-5D$ error to the system.

  • In a Keplerian system ($+20D$ ocular), the new effective ocular power is $+15D$. This weakens the ocular, increasing its focal length.

  • Consequently, the hyperope must lengthen the tube length to keep the image clear. This results in less magnification for Keplerian and more magnification for Galilean systems.

• Myopes:

  • A five-diopter ($5D$) myope brings a $+5D$ error to the system.

  • In a Keplerian system ($+20D$ ocular), the new effective ocular power is $+25D$. This strengthens the ocular, shortening its focal length.

  • Consequently, the myope must shorten the tube length to keep the image clear. This results in more magnification for Keplerian and less magnification for Galilean systems.

• Summary of Trends:

  • Regardless of the telescope type, myopes must shorten the tube and hyperopes must lengthen it to compensate for their error.

  • If the patient is emmetropic or wearing their proper correction, the tube length and magnification remain constant for everyone using that specific telescope.

Telemicroscopes and Close-Focusing

• A telescope is naturally designed for distance (parallel light). If pointed at a near object, the entering light is diverging, which results in heavily diverging light exiting the telescope, requiring excessive accommodation from the user.

• To adapt a telescope for near tasks, a plus lens called a lens cap is added to the objective side. This creates a telemicroscope.

• The power of the lens cap is determined strictly by the desired working distance:     $D_{lens\,cap} = \frac{1}{\text{working distance in meters}}$

  • For a $25\,cm$ ($0.25\,m$) working distance, a $+4D$ lens cap is required.

• Adding a lens cap does not change the internal magnification of the telescope itself but allows the system to focus at near.

Practical and Clinical Applications

• Bio Optics: These are mini-telescopes mounted high on eyeglasses. They allow the user to adjust their head position to look through the telescope for specific tasks and look through their regular prescription for others.

• Driving: In some states, it is legal for patients with low vision to drive while using mounted telescopic devices (typically around $2.5x$ magnification).

• Field of View: The field of view in a telescope is limited and depends on factors such as lens size, lens separation, and the exit pupil of the system. This is often dictated by optical "stops" created by apertures or lens edges.