Source http://www.maritime.org/fleetsub/pscope/chap1.htm
1A8. Limits of periscope design. It is seen from the preceding section that there are definite limits in periscope design. The vital factors, as in a telescope, are: 1) length of tube, 2) diameter, 3) illumination, 4)magnification, and 5) size of field. If a periscope favoring any one of these factors is to be produced, such favoring can be only at the expense of the other factors; hence, the final design generally is a compromise. 1A9. Examples of periscope design. The following requirements are for periscopes which have been used in submarines: field, at least 40 degrees to 45 degrees;magnification, between 1.2x and 1.5x; exit pupil, at least 5 millimeters in diameter;length, not specified; external diameter, 5 inches; thickness of walls, about 1/4 inch. Let us find possible periscope lengths under these conditions for the two magnifications given, 1.2x and 1.5x. The inside diameter of the tube is 5 inches minus 1/2 inch, or 4 1/2 inches. The lens, lensholding ring, supporting tube, and so forth take up another 1/2 inch of diameter, leaving about 4 inches free for the objective. 4 inches = 101.6 mm, which is close to 100 mm In order to obtain an exit pupil of 5 millimeters, the magnification of the telescope must be: Diameter of objective / Diameter of exit pupil = 

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Figure 11. Section through submarine with periscope elevated.  
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If the magnification of the final periscope is to be 1.2x, the reduction of the upper telescope must be:
20 / 1.2 = 16.67, or 16.67x Since the field must be 40 degrees / 16.67, or 2.4 degrees = 2 degrees 24′, this limits the length between the objectives of the two telescopes, since the entire beam of light must fall on the lower objective. From Figure 13, it can be seen that the permissible length equals 2 / tan θ, where 2 is half the 
diameter of the lower objective lens in inches and θ is half the angle of beam. θ equals 2 degrees 24′ / 2, or 1 degrees 12′.
log 2 = 10.30103 – 10 log tan 1 degree 12′ = antilog 1.97991 = 95.58 inches = The upper and lower telescope systems enter into the total length, and if it were possible to increase the focal length of their objective lenses 
Figure 12. Detail of encircled section in Figure 11.  
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indefinitely, the periscope could be lengthened. Increasing this is limited, however, by the same considerations of diameter and cannot exceed the same length; that is, about 7 feet 11 1/2 inches for each telescope system. Hence, the total possible length is roughly 3 times 7 feet 11 1/2 inches, or about 23 feet 10 1/2 inches. Since this length is greater than is required, the diameter of the periscope may be reduced, the magnification increased, or the size of the exit pupil increased without sacrifice.If the magnification is to be 1.5x, the reduction of the upper telescope must be:
20 / 1.5 = 13 1/3x For a field of 40 degrees, the angle of beam is: 40 / 13 1/3 = 30 degrees The interobjective distance is: log 2 = 10.30103 – 10 log tan 1 degrees 30′ = antilog 1.88296 = 76.37 inches = 6 feet 4.4 inches 
The total length possible is 3 times 6 feet 4.4 inches, or 19 feet 1.2 inches.To increase the length of tube beyond these limits, more telescopes may be placed in the tube. If astronomical telescopes are used, two more must be employed to keep the image erect, making a total of four telescope systems. One Galilean telescope could be used. The objection to adding more telescopes lies in the fact that each lens through which the beam must pass absorbs light, and if more are added, the illumination is seriously reduced.
Figure 14 shows a periscope designed as a straight instrument, and Figure 15 shows it with prisms introduced. The prisms may be placed at any point where the angle of the rays does not exceed the critical angle which results in total reflection. In this particular case, the prisms are placed at the focal planes. Both periscopes produce an erect image, since the two astronomical telescopes and the two prisms counteract each other in inverting the object. Prisms should not be placed exactly in a focal plane. Doing so is faulty design, since any minute imperfections 
Figure 13. Example of periscope design. 
Figure 14. Example of periscope design. 
Figure 15. Example of periscope design.  
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that may be present in or on the reflecting surface are reproduced as part of the final image, whereas a lens or glass plate which is not in a focal plane, or near one, may be dirty without affecting the resulting image. Periscope specifications often state that no lens or glass plate should be in or near a focal plane except the crosswire reticle, which must of necessity be placed in a focal plane.Since the backs of the prisms, which are the reflecting surfaces, are silvered, the critical angle for reflection is raised to more than 20 degrees; thus the two eyepieces may be placed between the prisms and the objectives. Both forms of construction are used in various periscopes. However, the best position for a prism is at a point at which the rays are approximately parallel; in erecting telescopes, this point lies between the two erecting lenses.
The chief function of a telescope system in a periscope is to take an object appearing from the point of vision under narrow angular view, and produce it to the eye at a wide angle. The ratio of these two angles is the magnification of the telescope. 1A10. Altiscopes. The only difference between a periscope and an altiscope is that in an altiscope the upper prism is omitted and the view is directly upward toward the zenith. The field of an altiscope is 100 degrees. To obtain this field, some sacrifice must be made in other characteristics. The magnification is necessarily less than unity. The only type of periscope used in the Navy today which permits observation of the zenith 
is the Type II design (Design Designations 89KA40T/1.414HA, 91KA40T/1.414HA, and 92KA40T/1.4HA built by the Kollmorgen Optical Corp., Brooklyn, N.Y., which is of the highangle type. The prism has a maximum elevation of the line of sight above horizontal of 74.5 degrees. The entire sky is observed with the line of sight set respectively at 14 degrees, 44 degrees, and 74.5 degrees or full elevation, giving complete zenith at the edge of the field in low power. The periscope is rotated 360 degrees in each zone with a minimum of overlap between the zones.1A11. Types of periscopes. Periscopes under Bureau of Ships Specifications R20 P5 of 15 June 1940, are of the following types:
1. Type I. Outer diameter of taper section, 1.414 inches. The line of sight can be moved through all angles between 10 degrees depression and 45 degrees elevation. 2. Type II. Outer diameter of taper section, 1.414 inches. The line of sight can be moved through all angles between 10 degrees depression and 74 degrees elevation. 3. Type III. Outer diameter of taper section, 1.99 inches. The line of sight can be moved through all angles between 10 degrees depression and 45 degrees elevation. 4. Type IV. Outer diameter of taper section, 3.750 inches. The line of sight can be moved through all angles between 10 degrees depression and 45 degrees elevation. The periscope is designed for night use with an installed antenna array and waveguide for the attachment of an electronic range device. 