2009 ao nd filters with risley prisms

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Neutral density filters with Risley prisms:

analysis and design

Virgil-Florin Duma* and Mirela Nicolov

Department of Product Design, Aurel Vlaicu University of Arad, 77 Revolutiei Avenue, 310130 Arad, Romania

*Corresponding author: [email protected]

Received 23 February 2009; revised 12 April 2009; accepted 13 April 2009;

posted 14 April 2009 (Doc. ID 107856); published 5 May 2009

We achieve the analysis and design of optical attenuators with double-prism neutral density filters. A

comparative study is performed on three possible device configurations; only two are presented in theliterature but without their design calculus. The characteristic parameters of this optical attenuator withRisley translating prisms for each of the three setups are defined and their analytical expressions arederived: adjustment scale (attenuation range) and interval, minimum transmission coefficient and sen-sitivity. The setups are compared to select the optimal device, and, from this study, the best solution fordouble-prism neutral density filters, both from a mechanical and an optical point of view, is determinedwith two identical, symmetrically movable, no mechanical contact prisms. The design calculus of thisoptimal device is developed in essential steps. The parameters of the prisms, particularly their angles,are studied to improve the design, and we demonstrate the maximum attenuation range that this type ofattenuator can provide. 2009 Optical Society of America

OCIS codes: 0120.0120, 120.4880, 220.0220, 230.0230, 230.5480, 350.2450.

1. Introduction

Double-wedge optical devices, also known as Risleyprisms, are used in a large variety of applications[1] such as optical compensators and shearing sys-tems [2], polarimeters [3], and optical attenuators[4,5], where the wedges perform translations with re-gard to each other; laser scanners with refractive ele-ments [6,7] and optical beam steering [8], where theyrotate with regard to each other; interferometers[9,10] and anamorphic optical systems for multipleholography [11]; and as beam shaping devices for la-ser diodes [12]. In several of these setups, exploring

the more general solution of double prisms instead ofdouble wedges, as we shall also do, has proved to be anatural approach that allows for improved optimiza-tion of the device, although the design calculus be-comes more complex.

We approach neutral density filters by translatingRisley prisms that are used for the attenuation oflight, while preserving the spectral distribution of

light for a certain wavelength domain, e.g., in thevisible for colorimeters. Optical attenuators are opto-mechanical devices that achieve controlled adjust-ment of the light flux, mostly without altering itsspectral distribution. An overview of the opticalattenuators can be made with regard to their func-tioning principle: (i) by modifying the diameter ofthe light fascicle, with iris diaphragms or with differ-ent shutters [13]; (ii) with rotating elements, i.e.,choppers [14]; (iii) with elements of variable trans-mittance and constant thickness: rotating semitran-sparent disks and polarizing filters; (iv) with

elements of variable thickness: sets of neutral filters,step-by-step variable neutral filters [13], and double-wedge optical filters [5]. Each device has certain ad-

vantages that make it suitable for a specific applica-tion with regard to its required characteristics: step-by-step or continuous attenuation of the flux, spec-tral range, attenuation range, and sensitivity. Theirfield of applications comprises photographic appara-tus, colorimeters, holography, and spectral andphotometric systems in various optical setups.

Attenuators with diaphragms, although light andversatile, have the disadvantage of modifying the

0003-6935/09/142678-08$15.00/0 2009 Optical Society of America

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diameter of a beam. Choppers have goodparameters, but with rotating, rapidly moving parts,therefore they are better for use in nonportable sys-tems although mechanical wear remains a problem.Sets of neutral density filters, as well as step-by-step

variable filters, are precise in flux adjustment butonly for a limited number of values. Polarizers per-form continuous attenuation but for polarized lightonly, e.g., for femtosecond laser pulses.

The scope of this paper is to achieve a rigorousanalysis and design calculus of a type of optical at-tenuator with significant advantages. Double-prismneutral density optical filters allow for good control ofthe flux, fine resolution, and are suitable for applica-tions that involve finite diameter light beams, e.g., incolorimeters [15], which is one of our fields of inter-est. A problem of this device could be its possible low-er attenuation range with regard to other solutionspreviously mentioned. The proper neutral densitymaterials and constructive parameters of the at-tenuator must, therefore, be determined to achievean optimum, as high as possible value for this essen-tial parameter.

We have found, to the best of our knowledge, onlytwo solutions of the double-wedge attenuator pre-sented briefly without a design calculus in the stateof the art: #1 and #2 in our study (Section 2). We shallconsider and discuss all three possible combinationsof double prisms to determine which is best. Toachieve this, their characteristic parameters willbe defined and deduced. For a most general ap-proach, we consider double identical prisms, not onlywedges, and the useful prism angle remains a neces-sary aspect to be discussed. Based on a rigorousmathematical analysis, the design calculus of thistype of attenuator will be developed for its best de-

monstrated solution.2. Calculus of Double-Prism Attenuators

The functioning principle of the device is based onthe thickness variation of the neutral filter formed,a variation that is a consequence of the movementof two prisms. Since they are identical, the parallelincident light beam is transformed into a parallelemerging beam. Flux e that emerges from the at-tenuator by use of the law of light attenuation is

e i expd; 1where i is the incident flux, is the absorption coef-ficient, and d is the thickness of the neutral filter

formed. For solutions #1 (Figs. 1 and 2) and #3(Fig. 4), this variation of the current thickness dxis achieved by moving just one prism (2), while theother prism remains stationary. For solution #2(Fig. 3), the variation ofd is achieved by symmetricmovement of both prisms. The controlled movementis performed in all cases by use of micrometric screws(3) or, when higher precision and resolution are re-quired, by use of piezoelectric actuators. To calculateand compare the characteristic performances, thetotal transmission coefficient of the system mustbe defined:

=i; 2

where is the emergent flux, which also takes intoaccount the exit diaphragm (Figs. 3 and 4).

A. Solution #1

The two identical prisms for this device are in contactwith each other at all times (Figs. 1 and 2) because ofsprings 4 and 4 in Fig. 1(b). The mobile prism slides

along the fixed prism, with a refractive-index-match-ing liquid layer between [16]. However, in this config-uration several problems arise with regard to thislayer: (i) a possible change in transmission coefficientbecause of the lack of homogeneity of the liquid; (ii)the liquid might not always be in contact with thesame thickness on both surfaces; (iii) the tensionin the liquid because of the prism performance couldaffect the transmission coefficient. The drawbacks ofthis solution should be solved if this type of attenua-tor proved to be the best from an optical point of view;see the discussion in Section 3.

The thickness of the neutral filter formed in thiscase (Fig. 1) is

dx d0 x tan ; x l; l; 3

where d0 is the maximum thickness produced whenthe prisms are positioned on top of each other (thex 0 position), b is the width of the prisms, l isthe maximum displacement of mobile prism (2),and is the characteristic angle of the two identicalprisms. The condition for the incident beam to reachthe first facet of prism (2) completely, regardless of itsposition, is (Figs. 2(a) and 2(b))

Fig. 1. Double-prism neutral density filter for solution #1 in thecenter (x 0) position: (a) lateral view and (b) view from the top.

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a D 2l; 4

where D is the diameter of the ray bundle when itemerges from the system (in this case, equal to theincident fascicle Di). With Eqs. (1) and (2),

ln 0=d0; 5where 0 x 0. Therefore , the total transmis-sion coefficient of the system defined in Eq. (2), re-sults as

x 0 expx tan : 6

The characteristic function of the device can thus beascertained [Fig. 5(a)] as well as its sensitivity S(Table 1). From Fig. 2, considering the limiting con-dition a D 2l from Eq. (4), one has

max

l

0 exp

l tan

min l 0 expl tan

max

ffiffiffiffiffiffiffiffi0C

pmin

ffiffiffiffiffiffiffiffiffiffi30=C

q ; 7

where the following notation was introduced:

C expDi tan ; 8

in this case, as mentioned,Di D. By performing thenecessary calculus the most important parameters of

the attenuator are obtained, i.e., the adjustmentscale and attenuation range:

k max=min; 9

and the adjustment and attenuation interval:

max min:

10

Their final expressions, obtained after necessary re-placements, are provided in Table 1.

B. Solution #2

In this case [Fig. 3(a)] the prisms have symmetricmovement with regard to their center position (char-acterized by x 0) that is due to the micrometricscrews (3) of equal steps, with left-hand and right-hand threads, respectively. Still, for this solution,the air gap that is formed produces a displacementof the exit ray bundle with regard to the entrancebundle [enhanced in Fig. 3(b)]:

Fig. 2. Extreme positions of the prisms for attenuator #1:(a) x l and (b) x l.

Fig. 3. Optical attenuator with neutral density filter for solution#2: (a) lateral view and (b) spot displacement and exit diaphragm.

Fig. 4. Solution #3 of the double-prism attenuator.

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x 2x

n cos ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 n2sin2

p 1

sin2: 11

An exit diaphragm is, therefore, necessary to have anemergent, circular section ray bundle (Fig. 3), re-gardless of the position of the prisms. The diameterof the incident ray bundle, therefore, must be

Di

D

l

;

l

max:

12

In this case,

dx d0 2x tan : 13

For this solution, condition (4) becomes

a 2l D max: 14

The total transmission coefficient in this case is ob-tained from Eq. (2) by use of Eqs. (11)(13) and con-sidering condition (14) to the limit:

x r10 exp2r2x tan ; 15

where the following coefficients were used:

r1

D

D max

2

; r2

nsin2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 n2sin2

p cos

cos : 16

Using Eq. (15) with Eqs. (8) and (16) yields

max l r11r20 Cr2 ; min 0 r10: 17

Therefore, the attenuation range and intervals, de-fined by Eqs. (9) and (10), respectively, as well asthe sensitivity of the device are obtained (Table 1),and the functioning characteristic of the device ispresented in Fig. 5(b).

C. Solution #3

This is a combination (Fig. 4) of the first two deviceswith prism (1) stationary and prism (2) performing a

horizontal translation movement in this case. Thecharacteristic parameters of this optical attenuatorare deduced in a similar way by the first two solu-tions. With the same notations, one obtains the totaltransmission coefficient for this configuration:

r10 expr2x tan ; 18

from which

max l r11r22

0C

r2

2 ; min l r10: 19

Characteristic parameters S, k, and result andare presented in Table 1 in a comparative view withthose of the first two devices. A graph of the charac-teristic function is presented in Fig. 5(b).

3. Comparison of the Three Solutions

In the design of optical systems for which double-prism attenuators are a concern (e.g., colorimeters),the most important parameter is attenuation rangek max=min. However min is also important to ob-tain a satisfactory value of the exit flux. Dependingon the type of apparatus to be designed, this couldprove to be an ongoing problem. Table 1 shows that,

Fig. 5. Profile of the functioning characteristic x for three solu-tions of the double-prism attenuator: (a) #1 and (b) #2 and #3.

Table 1. Characteristic Parameters of Three Possible Configurations of Double-Prism Neutral Density Filters

No. Double-Prism Attenuator # 1 # 2 # 3

1 Transmission coefficient x 0 expx tan r10 exp2r2x tan r10 expr2x tan 2 Sensitivity Sx d=dx x tan 2r2x tan r2x tan 3 Minimum transmission coefficient min

ffiffiffiffiffiffiffiffiffiffiffi30=C

qr10 r10

4 Attenuation range k max=min C0

minC

2=3

C

0

r2

C

0

r2

5 Attenuation interval max minffiffiffiffi

0C

p C 0 r10

C

0

r2

1

r10

C

0

r2=2

1

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by analyzing the parameters of the three attenua-tors, solution #2 is better than solution #3 from allsignificant points of view, including sensitivity. Thefinal comparison should therefore be made betweensolutions #1 and #2. One can then determine if

r1 >ffiffiffiffiffiffiffiffiffiffi

0=Cp

; r2 > 1; 20

solution #2, is optimal both from an optical and a me-

chanical point of view, since all the resulting opticalparameters (min, k, and ) are better and the prismmovement is precise, with no mechanical contact,which is a plus. Based on this former, mechanicalpoint of view, solution #1 proved less reliable becauseof the drawbacks highlighted in Subsection 2.Awithregard to the liquid layer that should have beenplaced between the prisms. One can see fromEq. (22) that

C

0> 1

maxD

4

1 21

is a condition that is always fulfilled, as from Eqs. (5)and (8) one has

C=0 exp2l tan : 22It is obvious that C=0 > 1 was by all means neces-sary to obtain a good value for adjustment scale k;see Table 1. As inequalities (20) are reduced ton > 1, the final conclusion is that they are always ful-filled. Therefore, solution #2 is the best choice froman optical point of view and from a mechanical pointof view.

4. Design Calculus of the Optimal DeviceIn Fig 5 we present the functioning characteristicsx of the three types of attenuator. Their obviousnonlinearity must be evaluated by a scale factor that,taking into account Eq. (9), is

Smax

Smin max

min k; 23

As attenuation range k must be increased as much aspossible from the original design (this is one of itsmain features), the consequence of significant nonli-nearity of the characteristic of the device is inevita-

ble. Here we address only solution #2 of the Risleyprism attenuator, since we have demonstrated thatit has superior performance compared with the othertwo possible devices. This also eliminates the me-chanical difficulties that solution #1 would haveraised if it were better from an optical point of view.

The design theme usually imposes (i) exit diameterD of the fascicle; (ii) the minimum necessary trans-mission coefficient min to have a sufficient value ofthe exit flux; (iii) the attenuation interval or itsequivalent, attenuation range k, as the two arelinked to each other by Eqs. (9) and (10), by equation

mink 1; 24

and eventually (iv) the minimum allowed sensitivitySmin. As can be seen from the characteristic para-meters in Table 1, the best way to perform the designcalculus would be to choose certain parameters, suchas , the material, and also refractive index n, and tocalculate the others, the most important of whichwould be prism angle . However, is difficult to ob-

tain from the characteristic equations of the device(Table 1).A rather complicated design method is reported in

Ref. [5]. The simplest way to perform the design cal-culus is to obtain, from Eq. (17) and attenuationrange k in Table 1, with Eqs. (5) and (8),

min r10 r1 expd0;k C=0r2 exp2r2l tan : 25

From Eq. (14) and from Figs. 2 and 3, one has

a 2l D max d0= tan : 26

These are the three design equations for the device.From Eqs. (25) and (26) one can, therefore, obtain theabsorption coefficient

l 1D 2r2l tan ln

1ffiffiffiffiffiffiffiffiffimin

p 1 2r2 1l=D:

27

The graph of Eq. (27) is presented in Fig. 6(a), for thefollowing parameters: min 0:1, D 10mm, andn 1:517; for three values of the prism angle theparameters are

10, 20, and 30. For the same

parameters, the graph of the kl xmax functionwas achieved in Fig. 6(b). Thus, from Fig. 6(b), forthe necessary value of adjustment scale k, with a me-chanically optimum displacement l, prism angle re-sults by the proper curve choice ofkl xmax. Then,from Fig. 6(a), with these values of l and , the opti-mum coefficient is obtained. The minimum trans-mission coefficient min must then be verified to allowfor the minimum necessary illumination in the opti-cal path. In Fig. 7(a) such a multiple example is con-sidered for three minl functions drawn for threedifferent pairs of values for and . Ifmin is not sa-tisfactory, this entire calculus must be considered as

an iteration.Several conclusions can be drawn from the discus-sion: the adjusment scale and attenuation range kcan be obtained between 2 and 2.5 for 10 (forthe specific parameters considered in Fig. 6) andfor proper values of l (820 mm). For smaller prismangles k does not increase significantly; for higher, k begins to decrease, especially for largedisplacements l (e.g., k becomes 2.4 for 10 andl 20mm). These values of k are rather small incomparison with other devices, e.g., diaphragms,choppers, or polarizers. This is a disadvantage, but

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it is the best range to be obtained with this type ofattenuator. For > 48 (an approximate value, alsoconsidered for the parameters in Fig. 6), the devicecan no longer be used: k decreases below 1.5, whereasit could be achieved for a maximum displacement ofapproximately 3mm. Interesting, higher values ofkare obtained for smaller prism angles . Since issmaller, max in Eq. (11) decreases; therefore, r1 inEq. (16) increases. Thus, for the same min and d0, ab-

sorption coefficient in Eq. (27) is also higher forsmall [Fig. 6(a)], i.e., for double wedges.

An advantage of this type of optical attenuator isits good sensitivity: Sx 2r2x tan (Table 1),especially taking into account the proper commer-cially available translation stages for such a setup[11]. In Fig. 7(b) the graph of the minimum sensitiv-ity Smin Sx 0 is obtained with regard to l xmax for the angle prisms previously considered. A va-

lue of Smin 0:01mm1 has been obtained for l 15mm, whereas with Eq. (23), Smax 2:4Smin be-cause k 2:3 for l 15mm, as discussed in Figs. 6and 7. This sensitivity allows for use of the devicein certain applications when finite diameter beamsare necessary for which the attenuation range ob-tained is satisfactory and when (especially in porta-ble instruments) the compact mechanical setup is ofprime importance, as in other apparatus with double

wedges [7].A detailed analysis of the sources of error that canappear in rotary Risley prism applications is re-ported in [17]. For our device, the accuracy in deter-mination of the level of attenuation depends on threeparameters from Eqs. (15) and (16): (i) the prism an-gle, which for commercially available prisms has tol-erances of the order of seconds of arc; (ii) index ofrefraction n; and (iii) absorption coefficient . Factors(ii) and (iii) depend on the quality of the glass; nowa-days materials with a high degree of homogeneity,e.g., ultrapure synthetic fused silica, are commer-

Fig. 6. Graphs of (a) absorption coefficient l xmax and (b) at-tenuation range kl xmax for min 0:1; D 10mm; n 1:517and for three prism angles of 1 10, 2 20, and 3 30.

Fig. 7. Graphs of (a) minimum transmission coefficient minl xmax and (b) minimum sensitivity Sminl xmax for D 10mmand n 1:517 and for three pairs of values of 1 0:4 and1 10; 2 0:2 and 2 20; and 3 0:1 and 3 30.

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cially available. All the surfaces of the wedges mustbe of high quality; surface accuracy of=4 is also com-mercially available nowadays for Risley prisms [18].

The mechanical tolerances of the mounting systemcan have an influence on the practical achievement ofthe calculated sensitivity. However, currently avail-able translation stages provide [18] a good straightline accuracy of0:2m=in:, with a positional repeat-ability of0:13m. This is a translation advantage ofRisley prisms over rotary prisms for which servomo-tors with controllers had to be developed [17] toachieve high precision, closed-loop positioning de-

vices. For an optical attenuator, error (d=dx) x ofthe transmission coefficient obtained from the equa-

tion of x and produced by x 0:26m an accu-mulated positioning error of the two prisms for theparameters and for the three pairs of the previouslyconsidered characteristic values of the prisms arepresented in Fig. 8. This error is smaller and almostconstant with regard to maximum displacement l ofthe prisms if double wedges are used, so the sameconclusion as in the previous discussion can bedrawn, with regard to the advantage of wedges ver-sus prisms. In particular, from Fig. 8, for a 10prism angle and l xmax 15mm, a maximum errorof 0.002% in the transmission of the attenuator isproduced, which demonstrates that the sensitivity

of the device is only slightly affected by the toler-ances of the mounting system that are due to thehigh precision of commercially available translationstages.

5. Conclusions

The complete, rigorous mathematical study per-formed for the neutral density filters with Risleyprisms has allowed for selection of the best of thethree possible solutions of this optical attenuator,both from an optical point of view and from a me-chanical point of view. We have demonstrated that

the device with two identical prisms with symmetri-cal movement has the highest attenuation range,interval, and sensitivity, with the best possible mini-mum transmission coefficient. The characteristicparameters defined and obtained to perform thiscomparison also allowed for the development of thedesign calculus of these devices. Although we haveconsidered double prisms, the double-wedge solution(with a prism angle of some 10) provides the max-imum attenuation range of 22

:5x, which we have

demonstrated as the best this type of attenuatorcan offer. The attenuator has a good, however, notconstant, sensitivity and a proper minimum trans-mission coefficient. This study is part of our researchconcerning different types of optical attenuator [14]in connection with several domains in which they areused, e.g., radiometry and colorimetry [15]. Our fu-ture work comprises experiments and applicationsof this optimal attenuator with double prismsdeveloped in apparatus for which compact deviceswith finite diameter beams are required, particularlyin a colorimeter with improved characteristics thatwe are currently developing [19].

The research was supported by the RomanianEducation and Research Ministry through NationalUniversity Research Council (NURC) grant 1896/2008. Special thanks to the reviewers, whose valu-able comments were included entirely in the final

version of the paper.

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Fig. 8. Graphs of the error of the transmission coefficient pro-duced by a x 0:26m accumulated positioning error of thetwo prisms for min 0:1; D 10mm; n 1:517, and for threepairs of values: 1 0:4 and 1 10; 2 0:2 and 2 20;and 3 0:1 and 3 30.

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14. V. F. Duma, Theoretical approach on optical choppers fortop-hat light beam distributions, J. Opt. A Pure Appl. Opt.10, 064008 (2008).

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2009 ao nd filters with risley prisms

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