Optomechanical integration method for finite integrals

Jenny Magnes,1,* Gregory Schwarz,2 John Hartke,2 Derek Burt,3 and Noureddine Melikechi4

1Department of Physics and Astronomy, Vassar College, 124 Raymond Avenue, Poughkeepsie, New York 12604, USA2Department of Physics, U.S. Military Academy, West Point, New York 10996-1790, USA

3Department of Mathematical Sciences, U.S. Military Academy, West Point, New York 10996-1790, USA4Department of Physics and Pre-Engineering and Center for Research and Education in Optical Sciences and Applications,

Delaware State University, Dover, Delaware 19901, USA

*Corresponding author: jemagnes@vassar.edu

Received 6 July 2007; revised 15 August 2007; accepted 17 August 2007;posted 17 August 2007 (Doc. ID 84858); published 21 September 2007

We present, for the first time to our knowledge, an optomechanical integration method for finite functions.This technique allows for the integration of any finite function by combining optical and mechanicalprincipals. The integrated function can then be determined using curve fitting methods. Furthermore, theoriginal function can be reproduced through numerical or analytical integration. © 2007 Optical Societyof America

OCIS codes: 120.4820, 200.4560, 200.4740, 220.4480.

1. Introduction

The pioneers of calculus realized the significance ofexpressing and calculating the properties of mathe-matical functions [1]. Mathematicians, scientists,and engineers depend on these to predict volume,speed, fields, and countless other physical properties[2]. Numerical methods became unavoidable to solvechallenging or time consuming problems [3]. Further-more, numerical methods became more attractiveand less tedious with the arrival of the computer age[3]. Computations have entered a phase that allowsone to find solutions using genetic algorithms [4] andto give results almost instantly through experimentaloptimizations using these genetic algorithms [5]. Inshort, there has been an effort to meet mathematicalchallenges using experimental methods.

In this paper we present an experimental methodthat allows for the integration of functions for bothcases, when these functions can be expressed ana-lytically or not. In fact, the optomechanical integra-tion method allows for the integration of a functioneven if its mathematical expression is unknown. Itshould be noted, however, that this paper contains amere introduction to the optomechanical integration

method. Even though we are not claiming that thismethod can integrate all functions, this paper dem-onstrates that this method can handle a wide varietyof functions.

Some existing physical integration methods in-clude a planimeter, which compares to the function-ality of the optomechanical integrator in that it canhelp determine the total area of a function or object.However, a planimeter does not integrate the func-tion itself. This tedious mechanical integrator wascommonly used by engineers and has become obsoletesince it is restricted to linear or polar use. Miller et al.[6] developed an optical planimeter that was de-signed to determine the total area of a leaf. Anothertraditional method to evaluate integrals involves thegraphing of a function on paper and then weighingthe cut-out of the function. Most of these traditionaltools have been replaced by computer methods.

The optomechanical integration technique lends it-self to potential applications in the area of integra-tion, shadow techniques [7,8], and education. Thistechnique seems to be particularly useful when inter-nalizing the concept of integration [9] since the stu-dent becomes part of the integration process itself.

For a given function f�x� an integral is generallydescribed as the area on an interval �a, b� betweenthe curve and the interval �a, b� of the horizontal axis[10], as shown in Fig. 1.

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Integrations for which analytical solutions are in-accessible can be performed using numerical tech-niques. Numerical integrals typically rely on thedefinition of a finite integral [10]:

lim�xi→0

�i�1

n

f�xi��xi ��a

b

f�x�dx, (1)

where the integer n → � as �xi → 0 on the interval�a, b�, and xi is the midpoint of each segment �xi.Numerical integration can be achieved by adding fi-nite yet small segments of the areas contained in thesegments bounded by �xi.

The function F�x�, shown in Fig. 1(b), gives the areabounded by a curve and the horizontal axis. F�x� canthen be recorded as a function of displacement if werecord the area bounded by point a and each point xk

as indicated in Fig. 1.In this paper, we use three functions to demon-

strate the optomechanical integrator: a straight line,a parabola, and a house. The first two functions arewell known and verify the validity of our method. Thelast function is of a more complex mathematical na-ture and demonstrates the numerical benefits of thistype of integration. Uncertainties are discussed inSection 4.

2. Experimental Setup and Method

This method follows the principle of knife-edge beamprofiling that allows for quick, inexpensive, and ac-curate determination of the cross-sectional laserbeam parameters [11–13]. The optomechanical inte-

grator requires a sharp edge (typically a razor blade),a translation stage with a micrometer, a mask out-lining the shape of the function of interest, either apowermeter for continuous light (coherent or incoher-ent) or an energy meter when working with laserpulses, and a stable light source configured as shownin Fig. 2.

The light source can be any stable light source,even a desk lamp can work reasonably well. The lightsource is arranged in such a way that the intensitydistribution across the mask is fairly uniform, e.g.,the distance between the light source and the maskmust be much larger than the dimensions of the maskto achieve a relatively uniform intensity distribution.

Fig. 3. (a), (b) Masks are used to demonstrate the integration ofknown functions. In this case a straight line and an inverted pa-rabola. (c) A mask in the shape of a house is used to demonstratethe integration of an unknown function.

Fig. 1. (a) Inverted parabola and (b) integrated parabola on theinterval �a, b�.

Fig. 2. Top view of principle experimental setup. The light travelsfrom the source past the razor blade through a collection lens intoa detector.

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The masks are taped to microscope slides for stabilityand are about 1 cm2 in dimension. The razor blade isplaced as close as possible after the mask to avoiddistortions of the image. The collection lens collectsthe light from the image and directs the light onto thedetector. Initially, the razor blade covers the imagecompletely to prevent any light from reaching thedetector. The razor blade is then moved incremen-tally to allow the light to reach the detector as thepower, P�x�, is recorded as a function of the displace-ment, x. The graph P�x� then represents the integral,F�x�, of the function, f�x�, that is outlined by themask.

The data presented in this section were obtainedusing a Coherent Verdi-V5 continuous wave laser at532 nm as a light source. A PM 100 powermeter fromThorLabs, Inc., was used as a detector. A lens, L1,with a focal length of 7 cm was used to increase thespot size of the laser beam. Another lens, L2, with a3.5 cm focal length was used to direct the light ontothe detector. We chose two masks, shown in Figs. 3(a)and 3(b), to demonstrate our method. The two masksin Figs. 3(a) and 3(b) represent well known mathe-matical functions, a straight line with a negativeslope and an inverted parabola. The mask shown inFig. 3(c) has the shape of a house and represents anunknown function.

3. Experimental Results and Discussion

The integral of the straight line contained in themask shown in Fig. 3(a) is computed optomechani-cally [Fig. 4(a)]. As expected, the fitted function is asecond order polynomial representing the integral ofa straight line. The validity of our method can beverified by differentiating the data points numeri-cally and then fitting a function. Alternatively, dif-ferentiating the function fitted to the optomechanicalintegral in Fig. 4(a) results in an analytical confir-mation of the results. It should be noted that thenumerical differentiation tends to amplify any noiseintroduced during measurements as shown in Fig.4(d). Nevertheless, the two methods do not differ sig-nificantly, as can be seen qualitatively in Fig. 4. Forexample, the straight lines obtained to verify the datain Fig. 4(d) show a less than 1% difference in slopeand a 2% difference in y intercept. The differentiatedfunction can then easily be scaled to the originalmask.

The numerical integration of the data points wascompleted by taking the derivative of the data usingan algorithm that allows for smoothing of the deriv-ative. In this case, the derivative at any data pointwas taken by averaging the derivative of two adjacentdata points:

Fig. 4. (a)–(c) Optomechanically integrated representations of a straight line, a parabola, and a house, respectively. The data points in (a)and (b) allow for a second and third order polynomial fit obtaining the integral of the function in the mask. Error bars are too small to be shownon this scale. (d)–(f) The differentiated data corresponding to (a)–(c) and Fig. 3, verifying the validity of our method. The solid lines in (d) and(e) represent the fit to the numerical differentiated points, whereas the dashed lines are the analytically differentiated function to the fit in(a) and (b).

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dPT

dx �12�yi�1 � yi

xi�1 � xi�

yi � yi�1

xi � xi�1 (2)

gives the power at a particular point x. This type ofderivative may be computed quickly by using spread-sheets. It should be noted that depending on the func-tion, other numerical differentiation methods may bemore appropriate.

We also demonstrate the power of the optome-chanical technique by using an inverted parabola[Fig. 3(b)] as shown in Figs. 4(b) and 4(e), which showthat a function with a varying derivative may rendermore convincing results in a classroom setting. Inthis case, the varying derivative tends to hide fluctu-ations in the signal. The function can be verifiedthrough differentiation as shown in Fig. 4(e).

Finally, we can use this method to integrate anyunknown function and determine the mathematicalexpression of the integral and�or the function itselfthrough a fitting routine. In this case, we used theoutline of a house [Fig. 3(c)] to demonstrate our op-tomechanical integration method. We differentiatedthe data numerically to demonstrate to the reader thevalidity of our method.

4. Uncertainties

In this section we demonstrate the reliability of theintegration technique and its sensitivity to opticalalignment. We integrated the following function us-ing our optomechanical integration method:

y�x� �a sin2�bx � d�

x � c, (3)

where a, b, c, and d are constants. In this example,a � b � 1, d � 0, and c � 0.025. The plot in Fig. 5shows the integrated function. The curve fit was ac-complished using the Levenberg–Marquardt [14] al-gorithm. After scaling the data to the dimensions ofthe original curve, the agreement of the originalcurve and the fitted curve is good as indicated by the

fitted values for the constants a, b, c, and d in thegraph.

Each data point was the average of 1000 datapoints taken over a 2 minute period using the Coher-ent LabMaster Ultima (model LM Ultima) powerme-ter. The uncertainty bars reflect temporal and spatialfluctuations in power. The intensity of the light pass-ing through the mask varied less than 3% throughoutthe area of the mask. Nevertheless, the inconsisten-cies in the derivative from intensity variations arenegligible since the change in light intensity from onearea segment to the next is ��3%. Larger contribu-tions to the uncertainty include inconsistencies in themask and relative rotation of mask and razor blade.

A relative rotation of the mask and the razorblade distort the integration process in such a waythat the x axis and y axis form a nonorthogonalcoordinate system. The integral appears then dis-torted as shown in Fig. 6.

5. Summary

The differentiated data in Figs. 4(d)–4(f) are in goodagreement with the original functions in Fig. 3. Itshould be kept in mind that this method works wellwhen using a stable light source. Battery operatedlight sources distort the image as the decreasing bat-tery power is reflected in the data points. Further-more, the features that are being investigated mustbe large enough not to display any diffraction effects,i.e., the wavelength ��� features. Final considerationmust be given to the slope of the feature itself. Thelarger the magnitude of the slope, the better thesignal-to-noise ratio. For example, a horizontal linewith a slope equal to zero will reveal any irregulari-ties in the spatial intensity distribution of the lightsource and the mask. This would certainly be amethod to apply in quality control.

The optomechanical integrator serves also as apedagogical tool when teaching the principles of in-tegration. Most calculus laboratories highlight themathematical relationship between physical phe-nomena such as speed and acceleration. This labo-

Fig. 5. Function as defined in Eq. (3) is used to demonstrate theagreement with the original mask.

Fig. 6. Razor blade is rotated 10° with respect to the mask. Alateral shift in the peak is apparent. The relative height of apexeschange as well.

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ratory exercise makes the student part of theintegration process by conveying the meaning of theintegration process itself.

While our technique assumes a uniform intensitydistribution, it should be noted that there are manyapplications to knife-edge techniques designed to de-tect changes in intensity distributions that could bemodified to improve and expand our technique. Dif-fraction effects could be reduced [15], and a modifiedversion of the two dimensional intensity measure-ment technique could be applied here to integrateshapes that are not functions [16]. The literaturereveals more techniques that may also help to refinethe optical integration method. Karabacak et al. [17],e.g., work with a reflected beam.

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