Area of the Secondary Mirror Obscuration Ratio = Area of the EP Ignoring the Obscuration
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1 Compact Gregorian Telescope Design a compact 10X25 Gregorian telescope. The Gregorian telescope provides an erect image and consists of two concave mirrors followed by an eyepiece to produce an afocal system. An important design feature of a mirror-based telescope is its obscuration ratio. This ratio characterizes the fraction of the light entering the telescope that is blocked by the secondary mirror: Area of the econdary Mirror Obscuration Ratio = Area of the P Ignoring the Obscuration ince the system stop is often at the primary mirror (as it is in this design), the obscuration ratio gives the reduction in light collection efficiency of the primary mirror. Remember that since the central portion of the primary mirror is obscured, there is no penalty for placing a hole in the center of the primary mirror to get the light out the back of the telescope and into the eyepiece. An aperture is placed at the front of the telescope to define the diameter of the telescope tube. This aperture is physically located at the same plane as the secondary mirror. This aperture is not the system stop. The secondary mirror is often attached by a mechanical spider to this aperture. We will assume that this mechanical spider has no area. ection A Provide the first-order design of a Gregorian Telescope with the following specifications: Magnifying Power MP 10X ield of View OV +/- 2 deg Primary Mirror ocal Length f P 45 mm Diameter D P 25 mm top is located at the primary mirror Obscuration Ratio 50% ye Lens Diameter D 10 mm Overall ystem Length L 100 mm (from the secondary mirror or mounting aperture to the eye lens) Unvignetted Object at Infinity or vignetting calculations, the element diameters must be at the minimum size to satisfy the condition for no vignetting. In other words, the entire diameters of the primary mirror, secondary mirror, field lens and eye lens must be used. 1
2 Determine the following: Primary Mirror: Radius of Curvature R P Minimum Hole ize D H econdary Mirror Radius of Curvature R Diameter D ield Lens ocal Length f Diameter D ye Lens ocal Length f Aperture Diameter D A xit Pupil Diameter D XP ye Relief (ye Lens to XP) R All pacings or determination of the hole diameter in the primary mirror, you may assume that the primary mirror has zero thickness. A good check of the design is that the hole diameter must be smaller than the secondary mirror diameter. The field lens is located at the second intermediate image plane. Please use the variable names as defined above. The system drawing on the solution page may also help in defining the system. 2
3 ection B A Maksutov variation to the Gregorian telescope uses a thin meniscus glass shell to form the secondary mirror. This shell covers the entire entrance aperture of the telescope and eliminates the need for the mechanical spider or other mount for the secondary mirror. The central portion of the second surface of the shell is aluminized to produce the secondary mirror. In one option, both surfaces of the shell have a radius of curvature equal to that of the secondary mirror. A second option is to use a concentric shell. econdary Mirror or the purposes of this discussion, assume a 2 mm thick shell of BK7 glass (n = 1.517) with both surfaces having a radius of curvature equal to that of the secondary mirror. Discuss the effect of this glass shell on the design obtained in ection A. Assume that the mirror curvatures and mirror spacing remains fixed. What changes would be needed to the eyepiece to obtain a telescope of the desired MP. This is a discussion problem only do not redesign the telescope or the eyepiece. Note: This is a first-order design problem. All lenses can be assumed to be thin lenses in air with no aberrations and no thickness. imilarly, mirrors have radii of curvature but no sag. To aid in grading, this problem may be more completely specified than you would normally encounter. In fact, the approach specified may or may not be the best form of the solution. All of the given specifications must be met exactly. 3
4 Compact Gregorian Telescope olution ummary ection A D A = mm R P = -90 mm D H = mm f = mm D = mm D XP = 2.50 mm R = mm D = mm f = mm t 1 = mm t 2 = mm t 3 = mm R = mm L = 100 mm ection B ummary of Discussion The Maksutov shell used to form the secondary mirror introduces a small amount of power into the optical system. or the configuration given, the focal length of the shell is 4400 mm. This additional power will change MP of the telescope by changing the focal length of the telescope objective (the combination of the shell and the primary mirror). In addition, the locations of both intermediate images will shift, so that the telescope will no longer be in focus. Two changes to the eyepiece must occur: - modify the focal length of the eyepiece to get the desired MP. - shift the eyepiece to place its front focal point at the second intermediate image to present an image ay infinity to the eye. 4
5 olution Design a compact 10X25 Gregorian telescope. pecifications: Magnifying Power MP 10X ield of View OV +/- 2 deg Primary Mirror ocal Length f P 45 mm Diameter D P 25 mm top is located at the primary mirror Obscuration Ratio 50% ye Lens Diameter D 10 mm Overall ystem Length L 100 mm (from the secondary mirror or mounting aperture to the eye lens) Unvignetted Object at Infinity Area of the econdary Mirror Obscuration Ratio = Area of the P Ignoring the Obscuration tarting with the specifications and the layout: R 0 R 2f 90mm P P P R 0 u tan 1/2 tan The obscuration ration gives the diameter of the secondary mirror: Area Area D / 4 D Obscuration Ratio = 0.5 Area Area D / 4 D P P P P D 0.707D 0.707(25mm) 17.68mm P a 8.84mm The separation between the primary and the secondary is determined by the vignetting condition at the secondary. Trace marginal and chief rays from the primary to an arbitrary secondary location (transfer distance = -t 1 ). 5
6 At the secondary mirror: y t 0 1 y t 0 1 or no vignetting (and solving for the minimum secondary size): a y y a t t 8.84mm 1 1 t mm The next step is to obtain the proper MP for the telescope and solve for the focal lengths of the secondary mirror and the eye lens. No field lens is needed at this time. Note that a Gregorian telescope is a mirror version of a relayed Keplerian telescope, where the secondary mirror is the relay lens. The total magnifying power is f f MP m m m z /1 z P P R f f z / 1 z Remember that z is in the reflected space of the primary and has an index of -1. On the top of the next page, the system is shown with various distances. As an aside, it may be easier to design the system as the equivalent refracting relayed Keplerian telescope, and convert it back into the mirror based system. This works because a raytrace unfolds the mirror system into an air equivalent refractive system. z t f f 45mm 1 P P L z f 100mm z 23.24mm z 100mm f f z 45mm (100mm f ) P MP 10 f 16.21mm f z f 23.24mm 6
7 z 83.79mm z z f 83.79mm f f mm R 2f 36.39mm z f P t 1 = mm R = t 2 = z Ś t 3 = f L = 100 mm The spacings of all the elements are now specified, and the field lens is added at the front focal point of the eye lens. The focal length of this lens is determined by the condition of no vignetting at the eye lens (D = 10 mm). A system raytrace to the eye lens is attached as Raytrace 1. Note that the marginal ray raytrace confirms that the system MP = 10 (y = y P /10) and that D XP = 2.5 mm. At the eye lens: y 1.25mm y 16.21( ) y
8 or no vignetting at the eye lens (using the equality to utilize the entire aperture of the eye lens): a y y 5.0mm 1.25mm mm 5.0mm / mm f 26.45mm The diameter of the field lens is given by the chief ray at the field lens (the location of the second intermediate image): a y 5.665mm D 11.33mm Now that all of the optical components are specified, a final raytrace can determine the R and the sizes of the two other apertures. The entrance aperture is located t 1 in front of the primary mirror and the hole in the primary mirror is optically located t 1 behind the secondary mirror. rom the perspective of the secondary mirror, the hole looks just like an aperture that must pass all of the light. As a result, the minimum hole size in the primary mirror can be found by applying the condition for no vignetting at this location. Of course, the XP is located where the system chief ray goes to zero. Raytrace 2 is attached. R = mm Aperture: y A 12.5mm a y y 14.88mm A A A y A 2.383mm D A 29.76mm Hole: y H 1.203mm a y y 5.37mm H H H y H 4.171mm D H 10.74mm 8
9 ummary of the results: Primary Mirror: Radius of Curvature R P = -90 mm Minimum Hole ize D H = mm econdary Mirror Radius of Curvature R = mm Diameter D = mm ield Lens ocal Length f = mm Diameter D = mm ye Lens ocal Length f = mm Aperture Diameter D A = mm xit Pupil Diameter D XP = 2.50 mm ye Relief (ye Lens to XP) R = mm Primary-econdary pacing t 1 = mm econdary-ield Lens pacing t 2 = mm ield Lens-ye Lens pacing t 3 = mm ection B The Maksutov shell used to form the secondary mirror introduces a small amount of power into the optical system. or the configuration given: ( ) / mm R 36.39mm econdary Mirror t 2 t t 2.0mm n n / mm f 3760mm This additional power will change MP of the telescope by changing the focal length of the telescope objective (the combination of the shell and the primary mirror). In addition, the locations of both intermediate images will shift, so that the telescope will no longer be in focus. Two changes to the eyepiece must occur: - modify the focal length of the eyepiece to get the desired MP. - shift the eyepiece to place its front focal point at the second intermediate image to present an image ay infinity to the eye. In practice, the MP error is small enough that it might be ignored, and the focusing mechanism in the telescope can be used to adjust for the focus error. Note also that a concentric shell introduces a negative power, and that a zero power shell can be designed. 9
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