Monday, May 22, 2006

28

Telescopes

Telescopes are afocal optical systems consisting of two lenses, separated in space, in air. There are two types of telescopic systems, Keplerian and Galilean.

a. Keplerian telescopes (Figure 38) have a weak (+) objective lens and a strong (+) eyepiece lens.

The lenses are separated by the sum of their focal lengths. Keplerian (astronomical) telescopes form an inverted image so they require an erecting lens or prisms to make it a Terrestrial telescope.

Click on image to enlarge.

b. Galilean telescopes (Figure 39) has a weak (+) objective lens and a strong (-) eyepiece lens. The lenses are separated by the difference of their focal lengths. Galilean telescopes form an erect/upright image.

Click on image to enlarge.

The angular magnification of a telescope is equal to the power of the eyepiece divided by the power of the objective.

MA Telescope = (-) FE/FO

  • The eyepiece in the Galilean telescope has a negative power. Therefore, the magnification given by the equation above is positive, indicating an upright image.
  • Keplerian telescopes have both positive objective and eyepiece lenses; the magnification is negative, indicating an inverted image.
  • With any telescope, the secondary focal point of the first lens must coincide with the primary focal point of the second lens. With Galilean telescopes, the second lens is minus and so the primary focal point is virtual.
  • Galilean telescopes have several practical advantages for low vision work. The image is upright, without the need for image erecting prisms and the device is shorter. Galilean telescopes typically are 2, 3 or 4x in strength, inexpensive, light, and have a large exit pupil, which makes centering less difficult.
  • 4x telescopes and stronger are usually Keplerian in design which gives an optically superior image, but are more expensive with a smaller exit pupil requiring better centering and aiming. Keplerian binoculars, contain prisms to erect the otherwise inverted image.
  • Galilean telescopes used as surgical loupes, require an add to be combined with the objective lens. The field size is far smaller than that obtained with bifocal spectacles.
  • Telescopic loupes can produce asthenopia­­ with any type of refractive error. If binocular loupes are not aligned properly, vertical or horizontal phorias can be induced. Adopting a working distance too far inside the focal distance of the “add” can require excessive accommodation, even for a myope.
  • When viewing a near object through an afocal telescope, the telescope acts as a vergence multiplier. The approximate accommodation required is given by Aoc= M2U, where Aoc = vergence at the eyepiece = accommodation, U = object vergence at the objective = 1/u, M = the magnification of the telescope.

Question: How far apart must a +5D lens and a –10D lens be placed to form a Galilean (afocal) telescope?

Answer: With any telescope, the secondary focal point of the first lens must coincide with the primary focal point of the second lens. With Galilean telescopes, the second lens is minus and so the primary focal point is virtual. To make the secondary focal point (20cm) of the plus lens coincide with the virtual primary focal point of the minus lens (10cm), the lenses must be separated by 20 – 10 = 10cm.

Question: You are a –5.00D spectacle corrected myope stranded on a small island with your significant other. Unfortunately, your companion has broken your glasses (which had an 11mm vertex distance). The only lens available to you is a –55D Hruby lens, which your companion had.

  • How many cm from the eye should you hold the lens to fully correct your refractive error?

Answer: a) first, locate the far point of your eye. The far point of the lens is 0.211m in front of the eye (F = 1/-5 which equals 0.20m + 0.011m vertex distance = 0.211m = 211mm). The Hruby lens has a power of –55D which means, its focal point is 1/55 which equals 0.018m or 18mm away from the lens. To correct the refractive error, the focal point of the lens should coincide with the far point of the eye. Therefore, it should be 18mm away from the far point or 211 - 18 = 193mm in front of the eye.

Question: Why would you not be able to read the 20/20 line with this correction?

Answer: b) The problem is magnification. This configuration turns the combination of the eye and its corrective lens into a reverse Galilean telescope, where the eyepiece is approximately +5D (the extra power of the myopic eye) and the objective lens is –55D. The resulting magnification is (-) 5/-55, which equals 0.1x. Thus, the 20/20 line, while in focus, subtends 1/10 of the angle it would in the eye of an emmetrope. Therefore, the best distance acuity obtainable is only about 20/200, assuming an otherwise normal eye.

It should be noted that properly corrected patients with high myopia might not be able to read 20/20 through their spectacle lenses even in the absence of other pathology. This is because the longer axial length commonly found in higher amounts of myopia, results in greater separation of the photoreceptors, which decrease the visual potential of the eye.

Question: You and a stowaway are ship wrecked on a lost island with your trial lens set, but only a few lenses survive the shipwreck. Your are left with a –20D, +4D, +5D, and a +20D. You build a viewing device to search the horizon for ships using the –20D and the +4D lens. The stowaway, Dr. Smith, uses the +20D and the +5D lens. Dr. Smith complains that his viewing device is inferior.

  • What did each of you build?
  • How did you position the lenses?
  • Why is Dr. Smith plotting to steal your telescope?

Answer: You use the –20D lens as the eyepiece and the +4D lens as the objective lens of a Galilean telescope. The secondary focal point of the plus lens should coincide with the primary focal point of the minus lens, thus the lenses are 25cm – 5cm = 20cm apart. Dr. Smith built a second telescope (astronomical) using the +20.00 diopter lens as the eyepiece and the +5D lens as the objective. The secondary focal point of the objective lens needs to coincide with the primary focal point of the eye piece lens, so he positions them 5 cm + 20 cm = 25 cm apart. Dr. Smith does not like having to stretch his arms the additional 5cm.

Question: Which telescope above will provide more magnification?

Answer: The angular magnification of a telescope is equal to the power of the eyepiece divided by the power of the objective. Magnification of the Galilean telescope is (-)-20/4 = 5x. The magnification of the astronomical telescope is (-) 20/5 = -4x. Therefore, Dr. Smith’s telescope will provide less magnification.

Question: Will the telescopes have an erect or inverted image?

Answer: The Galilean telescope will produce an upright image of the, hopefully approaching ships, while Dr. Smith’s astronomical telescope will produce an inverted image.

Question: How is the Galilean telescope modified when used as a surgical loupe?

Answer: The binocular surgical loupe is just a short Galilean telescope with an add to bring the working distance in from infinity. Powerful lenses are used so that the tube length of the telescope is kept to a minimum. A +25D object, combined with a –50D eyepiece, would provide 2x magnification. The additional add needed to focus the telescope at near is the reciprocal of the working distance in meters. Example: for a 25cm working distance, the add would be 100/25 = 4D.

Question:

  • How long is the 2x Galilean telescope described above?
  • What if it were made using a +5D objective lens and a –10D eyepiece lens?

Answer:

a) The focal length of the –50D lens is 1/50 =2cm. The +25D lens has a 100/25 = 4cm focal length. Thus, the telescope is 4 – 2 = 2cm long.

b) The +5/-10 telescope is 20 – 10 = 10cm long.

Question: You are working with a 2x afocal Galilean telescope that is fabricated with a +8D objective lens. We know that the ocular lens must be –16D and the 2 lenses are separated by 6.25cm (objective lens 1/8 = 12.5cm, ocular lens 1/16 = 6.25cm, tube length = 12.5 – 6.25 = 6.25cm).

When viewing at infinity by an uncorrected 4D hyperope, the ocular has an effective power of?

Answer: +4 is needed to correct for the hyperopic refractive error. This power must be taken from the ocular lens of the telescope and so the effective power of the ocular lens becomes -16 - 4 = - 20D. (The -20D effective ocular lens combined with the +4D correction lens gives us the -16D the ocular lens of the telescope actually has).

Question: For the telescope to remain afocal, the tube length must be?

Answer: The objective lens focal length is still 12.5cm, ocular lens is now 1/20 = 5cm. Therefore 12.5 - 5 = 7.5cm

Question: What is the telescopic power now?

Answer: MA Telescope = (-)FE/FO = (-)-20/8 = 2.5x

Question: When viewed by an uncorrected 4D myope, the ocular has an effective power of?

Answer: The uncorrected –4D of the eye must act as a correcting lens so the ocular now has an effected power of -16 + 4 = -12D. (The -12D effective ocular lens combined with the -4D correction lens gives us the -16D the ocular lens of the telescope actually has).

Question: To make the telescope afocal, the tube length must be?

Answer: The objective lens focal length is still 12.5cm, ocular lens is now 1/12 = 8.33cm. Therefore 12.5 – 8.33 = 4.17cm.

Question: What is the telescopic power now?

Answer: MA Telescope = (-)FE/FO = (-)-12/8 = 1.5x.

Question: An afocal Keplerian telescope has an objective lens that is +7D and an eyepiece lens that is +17.50D. What is the separation between the lenses?

Answer: The focal length of the objective lens is 1/7 = 14.3cm. The focal length of the eyepiece lens is 1/17.5 = 5.7cm. Therefore, the lens separation is 14.3 + 5.7 = 20cm

Question: What is the power of the lenses?

Answer: MA Telescope = (-)FE/FO = (-)17.5/7 = -2.5x

Question: A patient uses a focusable 2x Keplerian telescope that has a +8D objective lens. What is the power and tube length of the afocal telescope when used by an emmetropic patient and focused for distance viewing?

Answer: The power is 2x because it is being used by an emmetrope.

The eyepiece lens power would be +16D (MA Telescope = (-)FE/FO = (-)X/8 = -2x)

To find the tube length, the focal length of would be 1/8 = 12.5mm for the objective lens and 1/16 = 6.25mm for the eyepiece lens. Therefore, the tube length would be 12.5 + 6.25 = 18.75mm.

Question: When used by a 4D hyperope in a similar fashion?

Answer: For the uncorrected 4D hyperope, the eyepiece lens now has an effective power of 16 – 4 = 12D. (The +12D effective ocular lens combined with the +4D correction lens gives us the +16D the ocular lens of the telescope actually has). The power of the telescope would become (-) 12/8 = -1.5x. The tube length would be 20.83mm. (1/12 = 8.33mm + 12.5 = 20.83mm)

Question: When used by a 4D myope in a similar fashion?

Answer: For the uncorrected 4D myope, the eyepiece lens now has an effective power of 16 + 4 = 20D. (The +20D effective ocular lens combined with the -4D correction lens gives us the +16D the ocular lens of the telescope actually has). The power of the telescope would be (-)20/8 = -2.5x. The tube length would be 17.5mm (1/20 = 5mm + 12.5mm = 17.5)

Question: A focusable Galilean telescope with a +20D objective lens with a –40D ocular lens is dispensed to a patient for a variety of tasks.

a) What is the magnification of the telescope at distance?

Answer: M = (-) -40/20 = 2x

b) What tube length is required for viewing distance objects?

Answer: 1/20 = 5cm, 1/40 = 2.5cm, 5 - 2.5 = 2.5cm

c) What is the tube length required for viewing numbers that are 50cm away in an elevator?

Answer: The objective power would now be +20 + (-2) = +18D. 100/+18 = +5.55, +5.55 – 2.5 = 3.05cm.

Important to remember – 20 inches = 50 cm. To find the vergences when working in inches, use the formula V = 40/distance (inches) = 100/distance (cm)

Question: A 3x afocal Galilean telescope has a separation between the objective and ocular lens of 2cm. When viewing an object 25cm in front of the objective lens, what power reading cap would eliminate the need to accommodate for this target distance?

Answer: 100/25 = +4D

Question: A low vision patient needs a 10D add to read the text on a computer monitor but the 10cm working distance is too close. He wants to work at a 25cm distance. What theoretical telescope and reading cap combination would be needed?

Answer: 25/10 = 2.5x, 100/25 = +4D, therefore you would need a 2.5x telescope with a +4D reading cap.

Question: What is the equivalent lens that should be prescribed to replace a 4x telescope with a +2.50D reading cap (FRC) so the patient has the same resolution ability through the lens that he has through the telemicroscopic system?

Answer: Fe = (FRC) (power of telescope) = 2.5D (4) = 10D

Question: If a patient is able to read enlarged sheet music with a 3x telescope and a cap focus for 16 inches, what telescope and cap are needed to read the same sheet music set at 32 inches?

Answer: 32/16 = 2x additional magnification, 40/32 = 1.25D, Therefore you would need a 6x telescope with 1.25D cap



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