**The Periscope Design Objective**

Perhaps the best known purpose of a periscope is to allow viewing above the surface of the ocean by an operator who is in a submerged submarine. Periscopes are also extensively used elsewhere such as in armored vehicles and tanks or, for example, outside a shielded room of a nuclear power plant.

Its physical form may be a straight tube, or some irregularly shaped housing containing multiple mirrors, prisms and lenses. These optics fold the line of sight and relay the image from an objective lens to the periscope viewer while satisfying the space constraints of the vehicle, or other installation.

A sub periscope optical design depends on the required field of view, aperture size, length and diameter. The relationships among these variables and others are discussed below.

**The Basic Periscope**

In its simplest form the periscope consists of a pair of flat mirrors on either end of a tube. This basic periscope provides a horizontal, displaced line of sight of the outside world to a viewer located at the lower end of the tube. If the size of the field of view were of no concern this would be an ideal periscope. However, the actual field of view available to the viewer would probably be too small.

With the eye fixed at the center of the lower aperture of the tube, (no lower mirror in place), the maximum field of view measured in radians would be ± (D/2) / L. The image brightness would be governed by the pupil diameter of the viewer’s eye and would be constant throughout the observed field of view.

If a camera or telescope were located at the bottom of the tube, then the field of view of the device would be ± D/L. For an objective lens aperture equal to the tube diameter, an image would be formed with maximum brightness at its center. Progressive drop off in illumination at off-axis points would occur.

The progression of light fall-off from the center of the field of view to the edge would be approximately linear falling to zero at the maximum half-field angle, D/L, as shown in the figure below.

The illumination value at off-axis image points is proportional to the area in common between two overlapping circles, one representing the full on-axis ray bundle and the other, the off-axis ray bundle. The off-axis ray bundle gets clipped progressively by the lower aperture of the tube as its angle of incidence increases.

Thus a lens, irrespective of focal length, located at the bottom of the tube would yield a field of view equal to 2 D/L with the image progressively vignetted from center to edge.

Now if an objective lens with focal length equal to the tube length were located at the* top *of the tube the FOV would be just D/L, one-half of the previous case. However, in this case there would be 100% relative image illumination at all off-axis points.The light transmission curve in the above figure would become a “top hat” extending over the range +/- D/2L.

In addition to the small field of view in this example, a focal length equal to the tube length would probably yield an unreasonably large F-number, L/D, for most applications.

Since most periscopes require larger fields of view than the one defined by the tube diameter and length additional optics are in order. An objective lens located at the top of the tube coupled to a series of relay lenses can deliver a wide field of view image with small f-number to the bottom of the tube.

**Periscope with Single Relay Lens Pair**

Periscope relay optics generally consist of a pair of lenses spaced by some distance as shown in the thin lens diagrams below. The resultant pupil patterns for three lens spacings, s = 0, f/2 and f are shown. The lens diameters and image diameters have been set equal, i.e., D = 2y.

The focal point of the lens on the left lies at the objective lens image plane or at the image of a preceding relay lens. The focal point of the lens on the right is at the periscope final image plane at the bottom of the tube or at the focal point of another relay lens .for further relaying down the tube.

.Diagrams for the on-axis and off-axis ray bundle cross-sections, at a plane located midway between the lenses, are shown above for the three different lens separations. The midway point is, in effect, the system aperture stop location.. The lens diameters on either side jointly form the physical stop. The ray bundle cross-sections or “pupils” are represented by the red and blue shadings for the off-axis and on-axis ray bundles, respectively.

**Pupils and Image Relative Illumination**

Off-axis pupil shapes and sizes are defined by the lens diameters D, the space between the lenses and the off-axis ray angle α_{s } in the space between the lenses. The ray angle α_{s} equals the objective image height y_{o} divided by the first relay lens focal length f_{1} or the final image height y divided by the second relay lens focal length f_{2} as shown in the figure which follows.

The relative illumination of the off-axis image can be calculated from the pupil vignetting factor, v, defined as the ratio of the off-axis pupil displacement to the on-axis pupil semi-diameter.at the plane located midway between the lenses. .This is shown in the diagram to the right.

The mathematical function which yields the area in common between the displaced and nominal pupils as a function of the vignetting factor, v, is not given here but it is the same function as the one for the MTF of a perfect lens of circular aperture. Its graph appears in an earlier illustration depicting ray bundle vignetting by a long thin tube containing no optics..

**Design Trade-offs: The Periscope Equation**

The periscope physical and optical requirements are often at odds with one another. Trade-offs can be assessed using the* periscope equation *derived from the relationships between the variables shown in the illustration above.

The periscope equation is first presented for the case of a single relay lens pair, and is then followed by a treatment of multiple relay lens pairs.

**Periscope Equation: Single Relay Lens Pair **

An objective lens of focal length f_{o } forms an image of height y_{o} at half-field angle α. The objective lens image is relayed by a pair of relay lenses, focal lengths f_{1} and f_{2} to a final image plane. The final image height y is equal to the objective image height y_{o} magnified or reduced by the ratio of the focal lengths f_{2}/f_{1}. The equivalent focal length of the periscope equals the objective lens focal length multiplied by the same f_{2}/f_{1} factor.

The overall length of the periscope is the sum of the objective and relay lens focal lengths plus the space between the relay lenses. The periscope ray envelope profile is defined by the diameter of the objective lens d_{o}, the objective lens image diameter 2y_{o}, , the relay lens diameters D and the final image diameter 2y..The diameter of the collector lens equals the diameter of the objective lens image, 2y_{o}.

The periscope equation for a periscope having a single relay lens pair is:

All of the variables with dimensions in the equation are normalized to the periscope equivalent focal length f, The focal length f, therefore, does not appear explicitly in the equation.

Three design solutions to the periscope equation are illustrated below. The objective lens focal length f_{o} is set to the normalized values of 0.5, 1.0, and 2.0, while the remaining variables in the equation are held constant. The corresponding magnification of the relay lens pair in each case is 2.0, 1.0 and 0.5. The periscope equivalent focal length is maintained at unity in each case.

Solving the periscope equation, a quadratic, gives the value of the second relay lens focal length, f_{2}. From f_{2}, the relay lens focal length, f_{1, }and relay lens separation s are calculated.

The magnification m between the objective lens image and the final image, is calculated as the ratio of the second relay lens focal length f_{2} to the first relay lens focal length f_{1}.

The collector lens function is to maintain constant the location of the periscope entrance pupil in front of the objective lens. Collector lens normalized focal lengths are calculated to be -0.85, 2.16 and 1.84, for magnifications, 0,5, 1.0 and 2.0, respectively.

Relay lens diameters D are calculated from the system f-number F and the focal length of the last relay lens f_{2}, i.e., D = f_{2}/F. In this example, then, the relay lens diameters are simply one-tenth the focal length of the second relay lens, f_{2}.

Significant variation in lens diameters D can be noted in the diagrams. This is important not only with respect to the periscope physical diameter which derives from the lens diameters but also with respect to periscope aberrations and lens manufacturing costs.

The f-number of the objective lens which equals the f-number of the first relay lens is noted in the figure as well as the magnification between the objective lens image and the final periscope image. In this analysis, the collector lens location coincides with the image and therefore does not impact the objective lens and relay lens f-number.

**Multiple Relay Lens Pairs**

For a periscope length requirement where a single relay lens pair can not yield a small enough periscope diameter, additional relay lenses can be employed to reduce the diameter. The periscope equation for multiple relay lens pairs, N, becomes:

The solution of this equation for three relay lens pairs, given the same input periscope length, field of view and aperture as in the case of a single relay lens pair is illustrated below:

Three relay lens sets, not two, or any other even number has been chosen in this example since an odd number, i.e.,, 1, 3, 5 …., is required to achieve an erect image; otherwise and inverted image, i.e., an image rotated by 180 degrees from the viewed object orientation, results..

As shown in the illustration, the same variables which were held constant in the case of the single relay lens pair, are again held constant. Also, the objective lens focal lengths, f_{o} = 0.5, 1.0 and 2.0 are again chosen.

The periscope relay lens diameters now become significantly reduced: 0.129, 0.100 and 0.062 for objective lens focal lengths, f_{o} = 0.5, 1.0 and 2.0, respectively Compare to the lens diameters for the corresponding case of the single relay lens pair::2.42, 2.16, and 1.70, respectively.

For a submarine periscope application a typical length might be10 meters. Then the normalized lens diameters for the case of the single relay lens pair would be 242, 216, and 170 millimeters, much too large for a periscope with the common 190 mm overall diameter. Lenses having a diameter not exceeding 100 mm, approx., are required in order to allow space for the lens mounting means and electrical transmission lines.

The corresponding real lens diameters for the case of three relay lens pairs are 129, 100 and 62 mm, all much more feasible and desirable for manufacturing and costs.

The image and collector lens diameters are equally important. A collector lens diameter problem develops when the objective lens focal length becomes too long as illustrated in the case for f_{o} = 2.0. Here the image and collector lens diameters grow to an unacceptably large 200 mm. The problem worsens as the field of view extends beyond the very modest 0.10 radians (2.86 degrees) assumed for the example.

**Objective and Relay Lens F-number and Relay System Magnification**

The periscope equation provides clear definition of the possible trade-offs between the physical and optical system requirements. Less obvious are the trade-offs concerning aberrations, relay and objective lens design complexity, meaning the number of lens elements and glass types required.. Manufacturing costs and periscope light transmission are both impacted. Lens design experience is required to determine, for example, whether two or three lens elements are called for when the relay and objective lenses are operating at a particular f-number.

A “slower” f/10 or f/20 relay lens can be much more thoroughly corrected of aberrations than an f/5 lens. Simple doublet achromats suffice for the larger f-numbers while three or more lens elements may be required for f/5 relay and objective lenses. In any case, the correction of spherical aberration of the objective lens and the individual relay lenses must be much less than λ/4 if the total periscope optical system aberration allowance is λ/4.

The magnification of the relay optics is very important in the trade-off process. An aberration generated by the objective lens appears at the final image plane either magnified or reduced in magnitude by the magnification of the relay optics. Therefore relay optics which reduce rather than magnify aberrations created “upstream” are most desirable.

**Aberration Predictions From Thin-lens Periscope Design **

The thin-lens designs above do not allow complete and exact determination of the aberrations that a detailed real lens periscope design would provide. However, accurate predictions of two of the most dominant aberrations, field curvature and secondary spectrum, remarkably, can be gleaned from just the thin lens design.

Even when all the periscope lenses are color aberration corrected ordinary achromats there remains a residual color aberration, secondary spectrum. This aberration limits resolution, particularly for broadband image sensors, such as electronic imaging arrays or panchromatic photographic film. Secondary spectrum is particularly troublesome in periscope design because its contribution from the objective lens and the numerous relay lenses is additive. The relative amount of secondary spectrum for a given design configuration can be expressed as follows:

The secondary spectrum aberration, Δα, is expressed here as angular blur in object space, measured in radians. The constant k, 1/2400, and system f-number F apply to all candidate configurations. The aberration will be the least when the number of relay lens pairs is least, objective focal length f_{o} is the longest and last relay lens focal length f_{2} is the shortest.

Field curvature aberration which degrades off-axis image quality is particularly bothersome in periscopes also because it is proportional to the total dioptric power of the optics forming the image. There is much dioptric power in a periscope due to the necessary addition of relay optics.

The dioptric power of the periscope increases directly with the number and individual dioptric power of the objective, collector and relay lenses. Thus, a design configuration employing many short focal length relay lenses is more problematic than one with fewer longer focal length lenses.

The Petzval Sum, ΣP, a measure of the periscope field curvature aberration which results from the total dioptric power of the periscope is given by:

The quantity n is the presumed refractive index of all the lenses in the periscope optical train. The factor in brackets following the 1/f_{o} quantity is a measure of how much the Petzval curvature of the entire periscope exceeds the Petzval curvature of the objective lens alone. Periscope Petzval curvature is strongly influenced by the number of relay lens pairs N, exceeding that of the objective lens alone by 10x for the case of three relay lens pairs and unity focal length objective and relay lenses. The increase in Petzval curvature translates to a corresponding increase in off-axis image defocus and astigmatism.

**References**

The above matter is based on a presentation made at the 18th Annual Technical Meeting of the Society of Photo-Optical Instrumentation Engineers, August 19-23, 1974, San Diego, CA, “Periscope Design Trade-off Analysis”, William H. Taylor, Kollmorgen Corporation

** Acknowledgements**

The work carried out in the above referenced paper was an unsolicited individual effort related to but completely independent of an IR&D periscope development project by the Electro-Optical Division of Kollmorgen Corporation circa 1970. .

The author wishes to acknowledge the efforts of Harry S. Friedman, Kollmorgen Chief Optical Engineer at the time, whose first order optical calculations were very supportive and in concurrence with the author’s work. His work corroborated the findings of Dr. Robert E. Hopkins of the University of Rochester, Institute of Optics, who earlier performed, independently, a similar analysis.