Sensitivity of the fractional Fourier transformto parameters and its application inoptical measurement

Zhiping Jiang, Qisheng Lu, and Yijun Zhao

The fractional Fourier transform ~FRT! is becoming important in optics and can be used as a new tool toanalyze many optical problems. However, we point out that the FRT might be much more sensitive toparameters than the conventional Fourier transform. This sensitivity leads to higher requirements onthe optical implementation. On the other hand, high parametric sensitivity can be used in opticaldiffraction measurements. We give the first proposal, to our knowledge, of the FRT’s applications inoptical measurement. © 1997 Optical Society of America

Key words: Fourier transforms, optical standards and testing.

The fractional Fourier transform ~FRT! was first in-troduced in 1980 by Namias1 but did not garner muchattention until its reintroduction to optics in 1993 byMendlovic and Ozaktas.2 Since then a lot of workhas been done on its properties, optical implementa-tions, and applications.2–22 So-called fractional Fou-rier optics have also been proposed.17 The concept ofthe fractional transform is not limited to the Fouriertransform; the fractional Radon transform,23 frac-tional Wigner distribution function,24 and fractionalWiener filter25 have also been proposed. Many otherfractional-operation concepts might be proposed inthe future.

The FRT is mathematically well defined. If it isused as only a new way of describing some opticalsystems, then there is no problem. However, if onewishes to do the optical FRT, attention must be paidto its feasibility. In this paper we show that the FRTis much more sensitive to parameters than is theconventional Fourier transform, which may give riseto difficulties in precision optical implementation.On the other hand, this high parametric sensitivitycan find applications in optical diffraction measure-ments. One example is given. To our knowledge, it

The authors are with the Department of Applied Physics, Na-tional University of Defense Technology, Hunan, Changsha410073, China.

Received 21 February 1997; revised manuscript received 7 May1997.

0003-6935y97y328455-04$10.00y0© 1997 Optical Society of America

is the first proposal of the FRT’s application in opticalmeasurements.

Figure 1 shows the optical setup of the ~1yQ!th-order FRT, where l is the input-to-lens and lens-to-FRT plane distances, f1yQ is the focal length of lens,and the following relations hold:

l 5 f tan~py4Q!, f1yQ 5 fysin~py2Q!, (1)

where f is a constant. The propagation matrix is

FAC

BDG 5 F cos~py2Q!

2sin~py2Q!yff sin~py2Q!cos~py2Q! G . (2)

For simplicity here we consider only the one-dimensional case, but the conclusions are also validfor the two-dimensional case. Let E1~x1! and E2~x2!be the fields at the input and output planes, respec-tively. According to the Collins relation,26 we have

E2~x2! 51

ÎlB *2`

`

E1~x1!

3 expF iplB

~Ax12 2 2x1x2 1 Dx2

2!Gdx1

51

~lf sin f!1y2 *2`

`

E1~x1!expFiplf

~x12 cot f

2 2x1x2 csc f 1 x22 cot f!Gdx1, (3)

where l is the wavelength and f 5 py2Q.

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By introducing the dimensionless coordinates x19and x29,

x1 5 x19ys, x2 5 x29ys, (4)

where s 5 1y=lf is a proportional constant, we have,from Eqs. ~3! and ~4!,

E2~x29ys! 51

sin1y2 f *2`

`

E1~x19ys!exp@ip~x192 cot f

2 2x19x29 csc f 1 x292 cot f!#dx19. (5)

Equation ~5! is essentially the same as Eq. ~14! of Ref.6, except for the scale factor s and the constant phaseterm.

Equation ~2! is the precision FRT matrix. How-ever, if we assume that the input-to-lens and lens-to-FRT plane distances are not exactly equal to l buthave errors of lD1 and lD2, respectively, then thepropagation matrix is

FAC

BDG

5 F10 l~1 1 D2!1 GF 1

21yf1yQ

01GF10 l~1 1 D1!

1 G5 Fcos f 2 2D2 sin2~fy2!

2sin fyff sin f 1 ~D1 1 D2! f cos f

cos f 2 2D1 sin2~fy2! G .

(6)

The terms that contain D1D2 are omitted. Now, theFRT of Eq. ~5! should be replaced with Eq. ~7! ~theintensity distribution instead of the amplitude distri-bution!:

I2~x29ys! 5 uE2~x29ys!u2 51

sin f 1 ~D1 1 D2!cos f

3 U*2`

`

E1~x19ys!exp( ipsin f 1 ~D1 1 D2!cos f

3 $x192@cos f 2 2D1 sin2~fy2!#

2 2x19x29%)dx19U2

. (7)

Fig. 1. Optical setup for a ~1yQ!th-order two-dimensional FRT.For the one-dimensional FRT, the lens should be replaced by acylindrical lens. Parameters: l 5 f tan~py4Q! and f1yQ 5 fysin~py2Q!.

8456 APPLIED OPTICS y Vol. 36, No. 32 y 10 November 1997

To carry out a computer simulation of the sensitiv-ity of the FRT to the errors D1 and D2, we consider theFRT of a slit. Let the slit have a width 2w with anerror 2wD; then the dimensionless width and errorare w9 5 ws and w9D 5 wsD, respectively. Hence

I2~x29ys! 5 uE2~x29ys!u2 51

sin f 1 ~D1 1 D2!cos f

3 U*2~11D!w9

~11D!w9

E1~x19ys!

3 exp( ipsin f 1 ~D1 1 D2!cos f

3 $x192@cos f 2 2D1 sin2~fy2!#

2 2x19x29%)dx19U2

. (8)

We use Eq. ~8! to calculate the dependence of theintensity distribution on the parametric errors D, D1,and D2. Figures 2 and 3 show the normalized inten-sities. The only difference between Figs. 2 and 3 isthe dimensionless width w9 of the slit. However,since there exists a scaling law, the change of thewidth of the slit can also be considered to be thechange of the FRT order.8 The cases of the conven-tional Fourier transform ~order 1! are shown in Fig. 4for comparison. It is seen that only 5% of all threeerrors leads to a large change of the output intensitiesfor the 0.5-order FRT, whereas for the conventionalFourier transform only the slit-width change D leadsto output-intensity changes, and the changes aremuch smaller.

Figure 5 presents the differences of curves a and b~Figs. 2–4!. Figure 5 shows us that the FRT is muchmore sensitive to changes of the parameters than is

Fig. 2. Dependence of the FRT on the parametric errors: w9 51.0, Q 5 2.0. Curve a: D 5 D1 5 D2 5 0; curve b: D 5 0.05, D1

5 D2 5 0; curve c: D1 5 0.05, D 5 D2 5 0; curve d: D2 5 0.05, D5 D1 5 0. The transversal coordinate is x29; the vertical axis is thenormalized intensity.

the conventional Fourier transform. Consequently,higher requirements must be satisfied to obtain aprecise optical implementation of the FRT. In opticsthe FRT is essentially a special Fresnel diffraction, sothis strong dependence of the FRT on parameters isnot surprising.

The dependences of the FRT on the wavelength land the constant f can also be calculated readily.The influences of l and f are through only the pro-portional factor s @Eq. ~8!#. However, l and f areusually fixed in the experiment, and we do not discussthem in this study.

It is well known that Fraunhofer diffraction ~whichis essentially a Fourier transform! can be used tomeasure the dimension of a small hole or slit whenthe size is too small for direct measurement. Thedimension of the diffraction pattern is inversely pro-portional to the dimensions of the hole or slit, a smallhole or slit resulting in a large diffraction pattern,and can easily be measured. Besides, diffraction

Fig. 3. Same as for Fig. 2 but with w9 5 1.6. According to thescaling law, the change of w9 can also be considered to be thechange of the FRT order.8

Fig. 4. Same as for Fig. 2 but with Q 5 1.0 ~i.e., the conventionalFourier transform!. Curves a, c, and d are practically indistin-guishable.

measurements are noncontact measurements andtherefore can be used in large-scale, fast, automaticmeasurements.

Because of the higher sensitivity of the FRT toparameters, it may have advantages in optical dif-fraction measurements over the Fourier transform.Because array detectors ~e.g., CCD detectors! areavailable, the whole experimental curve can be mea-sured and fitted to a theoretical curve to give thewanted parameter values ~e.g., the width of the slit!.In this application, higher sensitivity to some param-eters indicates smaller detectable changes of theseparameters. Roughly speaking, from Fig. 5 we cansee that the average error of the 0.5-order FRT ismore than 4 times higher than that of a conventionalFourier transform ~we do not give a very quantitativecomparison because the average errors are depen-dent on the range of the transversal coordinate!.Therefore we propose here that the FRT could beused to increase the sensitivity and precision of themeasurement.

Figures 2 and 3 show the results for Q 5 0.5. Wedo not give the results for other fractional orders.When the slit width w9 decreases or the order Q goesto 1 the FRT becomes more and more like the Fouriertransform, whereas when the slit width w9 increasesor the order Q goes to 0 the FRT becomes more andmore like the slit itself. One should also note that itis the dimensionless width w9, not the width w itself,that appears in Eq. ~8!. To take advantage of theFRT’s high parametric sensitivity, w9 should be of theorder of 1. If w9 ,, 1, then the pattern of the FRT ispractically the same as that of Fruanhofer diffrac-tion.

We need to know if the parameters shown in Figs.2 and 3 can be satisfied experimentally. Here weconsider an example. Let Q 5 2.0, l 5 0.6328 mm~He–Ne laser!, and f 5 300 mm; then s 5 1y=lf 5 2.3mm21. Assume that w 5 0.5 mm; then w9 5 ws 51.15 mm. In addition, from Eq. ~1! the distance l hasa value of 124 mm and the focal length f1yQ has avalue of 424 mm. The size of the FRT pattern is ofthe order of 1 mm and can be detected easily by a

Fig. 5. Differences between curves a and b in Figs. 2–4.

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CCD camera. It is seen that all these conditions caneasily be satisfied experimentally.

Unlike the Fourier transform, the FRT istransverse-shift variant. That is to say, if the inputfield is shifted the FRT pattern will be shifted too.Although this feature is used in shift-variant informa-tion processing,20 it may increase the difficulty of datatreatment. However, like the parametric sensitivity,the disadvantage of the transverse-shift variance mayalso find application in optical measurement. Thisquestion is under consideration.

In this paper we have considered the sensitivity ofthe FRT to parameters. Higher sensitivity may leadto higher requirements for optical implementation,but it can also confer advantages in optical diffractionmeasurements. We believe that this is the first pro-posal of the FRT’s application to optical measure-ment and hope that it will inspire other applicationpossibilities. Similar problems and possibilitiesmay also exist in other fractional operations.

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