Features of using a right-angle prism to scan wide fields of view

M. V. Dorofeevaa�

NPO State Institute of Applied Optics, Kazan�Submitted October 1, 2009�Opticheski� Zhurnal 77, 30–33 �July 2010�

This paper discusses the operation of a right-angle scanning prism mounted inside a sphericalcowl. It is shown that the parameters and the relative placement of the prism and the cowl affectthe image quality of the optical system in the process of scanning the viewing zone. © 2010Optical Society of America.

In solving a number of scientific and practical problems,it becomes necessary to survey a hemispherical �or close toit� zone of space. In particular, such a survey is required invarious kinds of devices for detecting small heat-emittingobjects. A cube prism located in front of the objective androtating �or oscillating� relative to two mutually perpendicu-lar axes, one of which coincides with the optical axis of theobjective while the second is perpendicular to the principalcross section of the prism, can be used as a scanning assem-bly in such devices. With a very compact design, a cubeprism operates only in parallel pencils of rays, and when it ismounted inside a spherical cowl, a supplementary transmit-ting lens must be introduced.1 Unlike a cube prism, a right-angle prism, with certain limitations, can be used in combi-nation with a spherical cowl.

Figure 1 shows the optical layout of a detection device.It contains protective element �the cowl� 1, scanning prism 2,objective 3, and multielement photodetector 4. The center O0

of the cowl lies on the optical axis OO of the objective.When the prism rotates around the O1 axis, the principal raycoming out of the center of the photodetector and passingthrough the prism in the reverse path is displaced relative topoint O0, as a consequence of which decentering of the cowlrelative to the objective arises. Its value depends on the angleof rotation � of the prism around axis O1, the size of theprism, and the distance from point O0 to the O1 axis. Sincethe spherical cowl possesses focal power, its decentering, aswell as the altered tilt of the prism relative to the objective,affect the image quality of the optical system.

O OO0 O1

1

2 34

FIG. 1. Optical layout of the detection device.

429 J. Opt. Technol. 77 �7�, July 2010 1070-9762/2010/070

References 2 and 3 in discussing a prism that operatesunder stressed conditions and with large fields of view, rec-ommended precise formulas to be used that take into accountthe parameters of the actual pencil of rays passing through it,which in most cases has the shape of a circular cone. Theinitial data for calculating the size of a scanning right-angleprism are the viewing zone 2�0, the angular field of view 2�of the objective, the diameter d of the pencil of rays at thesite where the refracting face of the prism turned toward theobjective is located, and the refractive index n of the prismmaterial.

The range of angles of rotation of the prism around axisO1 is in general ��� ��0 /2�−�, and, when a hemisphere isbeing surveyed, ��� �45°−��, taking into account that thedeflection of a ray by the prism is a factor of 2 greater thanits angle of rotation. The path length of a ray in the prismvaries during scanning and reaches its maximum value at theminimum angle between its hypotenuse face and the opticalaxis of the system, constituting �1�� /2. The clearance ofthe prism should therefore be numerically determined whenit is in this position.

Figure 2 shows the path of the rays through the exten-sion of prism ABC in the direction from the objective. AxisO1 lies on the bisector of right angle B in order to reduce theimbalance of the prism as it moves. Besides the parametersindicated above, we designate the following: z1 is the dis-placement of an axial ray when it passes through the exten-sion of the prism; i1, i1� and i2, i2�, respectively, are the anglesof incidence and refraction of the axial and peripheral rays;O1� is the image on the extension of axis O1; and O0 and O0�are, respectively, the center of the cowl and its image on theextension. The distance between point O0� and the directionof the axial ray coming out of the extension of the prism isthe value of the decentering e1 of the cowl for a given loca-tion of the prism.

From triangles ADE and ACE, we determine segmentAE, and then the length l of the hypotenuse face of the prismand the length c of its leg.

AE = d cos �/cos�i1 + �� , �1�

l = AC = AE cos i2�/sin�45 ° − i2��

= d cos i2� cos �/�sin�45 ° − i2��cos�i1 + ��� , �2�

where i1=45°−�1, i2�=arcsin���sin�i1+���� /n�, and

�

c = AB = l/ 2. �3�

429429-03$15.00 © 2010 Optical Society of America

The path length of the ray in the prism, equal to thelength of segment GK, is determined by

lx = c/cos i1�, �4�

where i1�=arcsin��sin i1� /n�.The variation of length lx for the maximum variation of

angle i1 �from 0 to 45°� is 0.134c when n=1.5 and 0.016cwhen n=4.0.

The graphs in Fig. 3 give a clear representation of howthe ratio l /d depends on the refractive index n for variousangles �. In order to reduce the size of the prism for a coni-cal pencil of rays with a relatively large angle �, a materialwith a large n should be used.

To determine the decentering e1 of the cowl, we againturn to Fig. 2. We denote a segment as O1O0=g. We shallconsider its value to be positive when point O0 is located tothe right of point O1 and negative when it is located to theleft of O1.

We find from triangles AFG and BGO1 that

BO1 = �c sin�45 ° + �1� − d/2�/cos �1. �5�

Moreover, it follows from a consideration of the figurethat

O O0O1

e 1

z 1

i1

i2

1

OН

A

d

D

F G

E

C

B

B

O1

O0

i1

i2

FIG. 2. Extension of the prism.

2

4

6

8

10

2 3 4n

l/d

1

2

3

4

FIG. 3. Ratio l /d vs refractive index n for various angles �. 1—0°, 2—5°,3—10°, 4—15°.

430 J. Opt. Technol. 77 �7�, July 2010

O1O1� = 2�BH − BO1� = 2�l/2 − BO1� , �6�

O0O0� = O1O1� − 2g sin �1, �7�

e1 = z1 − O0O0� cos �1. �8�

It is well known that the transverse displacement of a raypassing through a tilted plane-parallel plate �in this casethrough the extension of the prism, equal to the length c ofits leg�, is determined from3

z1 = c sin i1�1 − cos i1/�n2 − sin2 i1� . �9�

In general, when the prism rotates by an arbitrary angle�, Eqs. �7�–�9� will have the form

O0O0� = O1O1� − 2g sin � , �10�

e = z − O0O0� cos � , �11�

z = c sin i�1 − cos i/�n2 − sin2 i� , �12�

where i=45°−�.Let us consider how the value of e varies as a function of

angle �, using the specific example of the following initialdata: 2�0=180°, 2�=26°, d=12 mm, and n=3.424. We find�1�6.5°. Assuming �1=5° �taking into account the overlapof the angular fields during scanning�, we get ��=35° andc=26 mm. Figure 4 shows graphs of the dependence of e on� for various values of g. As can be seen, the decenteringwhen the center of the cowl is displaced toward the positiveside increases somewhat toward the edge of the viewingzone �with �=40°�, appreciably decreases in the center ofthe zone �at �=5°�, and can equal zero for some value of g.The height of the cowl needed for the transmission of pencilsof rays toward the edge of the viewing zone without vignett-ing is increased in this case, and it consequently becomesharder to fabricate. It is expedient in a number of cases toselect a value of g for which decentering has about the sameeffect at the center and at the edge of the viewing zone.

4

0

�4

�810 20 30

e,mm

, deg

1

2

3

4

5

6

FIG. 4. Decentering e of cowl vs angle � for various displacements g.1—−1.5, 2—0, 3—1.5, 4—3.0, 5—4.5, 6—6.0 mm.

430M. V. Dorofeeva

The decentering of the cowl and the tilt of the prismcause additional aberrations to appear at the point of theimage on the optical axis of the objective—namely, comaand astigmatism. As an illustration, we consider an opticalsystem with cowls made from various materials having radiiof curvature of the surfaces of 40 and 35 mm, a prism whoseparameters and angles of rotation are given above, and anobjective in the form of an aberration-free paraxial lens withfocal length 20 mm and relative aperture 1:2. We assume thevalue g=1.2 mm, for which the absolute value of the decen-tering is identical for the extreme angles of rotation of theprism and equals 4.1 mm. Table I shows the results of cal-culations of the coma �yk�, the astigmatism �xs�−xm� �, and theenergy concentration F concentrated in a circle equal to theAiry circle from an infinitely remote point source for theextreme positions of the prism. The calculations were carriedout by means of the ZEMAX software package. As can beseen, the coma depends on the focal power of the cowl and isabout identical at the center and at the edges of the viewingzone. As the tilt angle of the prism to the optical axis of theobjective increases and the modulus of the focal length �f�� of

TABLE I. Results of calculating the coma, astigmatithe prism for cowls that differ in focal power.

Material of cowl f0�, mm ��=4.5 �m�

KO1 ceramic −1090.5

Synthetic sapphire −709.7

Germanium −372.7

0

431 J. Opt. Technol. 77 �7�, July 2010

the cowl decreases, the astigmatism increases, while the en-ergy concentration F decreases.

The variation of the path length of the ray in the prismduring scanning in the example under consideration is0.018c=0.46 mm, and this produces virtually no defocusingof the image on the photodetector, since the cowl is a weaknegative lens.

The final choice of parameters of the spherical cowl andits position relative to the scanning prism can be made on thebasis of an estimate of the image quality with an actual ob-jective in accordance with the requirements imposed on theoptical system as a whole.

a�Email: postmaster@gipo.kazan.ru

1V. N. Churilovski�, The Theory of Optical Devices �Mashinostroenie, Len-ingrad, 1966�.

2M. M. Rusinov, Size Calculations of Optical Systems �Gosgeoltekhizdat,Moscow, 1963�.

3Yu. G. Kozhevnikov, Optical Prisms �Mashinostroenie, Moscow, 1975�.

nd energy concentration for the extreme positions of

eg �yk�, �m �xs�−xm� �, �m F, %

0 −5.1 −10.7 82.65.5 −11.9 82.3

0 −6.7 −13.3 81.86.8 −16.6 80.8

0 −8.4 −11.6 80.98.9 −27.6 75.4

sm, a

�, d

454545

431M. V. Dorofeeva