JOSA COMMUNICATIONSCommunications are short papers. Appropriate material for this section includes reports of incidental research results,comments on papers previously published, and short descriptions of theoretical and experimental techniques.Communications are handled much the same as regular papers. Proofs are provided.

Rayleigh wave-front criterion: comment

Adriaan van den Bos

Department of Physics, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands

Received February 16, 1999; revised manuscript received May 3, 1999; accepted May 13, 1999

A correction is presented to a quantitative study [J. Opt. Soc. Am. 55, 572 (1965)] of Rayleigh’s criterion forallowable wave-front amplitude aberration. Also, for comparison, Marechal’s alternative root-mean-squarecriterion is evaluated under the same conditions. © 1999 Optical Society of America [S0740-3232(99)01909-2]

OCIS codes: 080.1010, 110.3000.

1. INTRODUCTIONReference 1 discusses the influence of a wavy wave-frontaberration on the optical transfer function of a slit. Theaberration nowhere exceeds a quarter of the wavelengthand therefore meets the classical Rayleigh criterion guar-anteeing a Strehl ratio larger than 0.8. The transferfunction of the slit with the chosen aberration is com-puted and compared with the ideal triangular transferfunction corresponding to absence of aberrations. Theconclusion of Ref. 1 is that, although the aberration no-where exceeds a quarter-wavelength, the transfer func-tion seriously deviates from the ideal one, and the Strehlratio is substantially lower than 0.8.

The wave-front aberration chosen in Ref. 1 is theLegendre polynomial of order 20, shown in Fig. 1. Thisgraph differs from the graph of the same polynomial pre-sented in Ref. 1, where the peaks at either end of the in-terval are absent. This was probably caused by thechoice of grid points at which the polynomial was com-puted. In Section 2 of this Communication the opticaltransfer function of a slit is computed for the wave-frontaberration function shown in Fig. 1. Comparison of thistransfer function with that computed in Ref. 1 then showsthat the transfer function of Ref. 1 and the conclusionsdrawn from it are not correct.

As an alternative to the Rayleigh criterion, Marechal2,3

proposed a root-mean-square (rms) aberration smallerthan one fourteenth of the wavelength as a criterion guar-anteeing a Strehl ratio larger than 0.8. This criterion isinvestigated in Section 3 of this Communication; theStrehl ratio is computed for two wave-front aberrationswith the same Fourier amplitude content and thereforethe same rms value but a highly differing amplituderange.

2. AMPLITUDE ABERRATIONLet w(x) be the wave-front aberration as a function of thespatial coordinate x across the slit, and suppose that w(x)

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is expressed in wavelength units. Then the pupil func-tion is equal to exp@ j2pw(x)# across the slit and to zeroelsewhere. The transfer function is the correlation func-tion of the pupil function. The amplitude range (differ-ence of maximum and minimum) of the Legendre polyno-mial of order 20, shown in Fig. 1, is equal to 1.40.Therefore, to produce a wave-front aberration meetingRayleigh’s criterion, the polynomial has to be multipliedby 0.25/1.40. Figure 2 shows the corresponding transferfunction.

The computed transfer function’s deviations from theideal triangular transfer function are almost an ordersmaller than those in Ref. 1. Furthermore, the Strehl ra-tio for the computed transfer function is equal to 0.97.This is well above 0.8, the minimum value guaranteed byRayleigh’s quarter-wavelength criterion. The conclusionis that these results differ substantially from those of Ref.1, where for the twentieth-order Legendre polynomial aStrehl ratio of 0.70 is computed and Rayleigh’s quarter-wavelength criterion is questioned.

3. ROOT-MEAN-SQUARE ABERRATIONAs an alternative to Rayleigh’s criterion, Marechal2,3 pro-poses to judge the influence of wave-front aberrations bytheir rms value instead of by their amplitude range. Hiscriterion to guarantee a minimum Strehl ratio of 0.8 isthat the rms value of the aberration not exceed one four-teenth the wavelength.

As an illustration, a wave-front aberration is chosenconsisting of 1 period of the sum of the first 31 harmoni-cally related cosine functions with equal amplitudes andwith phase angles equal to either 0 or p rad. If all phaseangles are chosen equal to zero, the aberration has a rela-tively large peak factor. This is the ratio of the ampli-tude range to 2A2 times the rms value. The normalizingfactor 2A2 is introduced to make the peak factor of a sinewave equal to 1. The aberration concerned is shown inFig. 3. Its peak factor is equal to 3.44.

1999 Optical Society of America

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A more deliberate assigning of the value 0 or p to thephase angles of the harmonics may substantially reducethe peak factor.4,5 For example, the particular choice of

Fig. 1. Legendre polynomial of order 20.

Fig. 2. Optical transfer function for a slit aperture as a functionof the normalized spatial frequency. The aperture wave-frontaberration is shaped like the Legendre polynomial of Fig. 1 andhas an amplitude range of a quarter-wavelength. The triangu-lar aberration-free optical transfer function is also shown.

Fig. 3. High peak-factor function.

phase angles proposed in Ref. 4 produces the aberrationshown in Fig. 4. The peak factor of this aberration isequal to 1.30. However, in spite of the substantially dif-fering peak factors, the rms values and the Fourier am-plitude spectra of the functions shown in Figs. 3 and 4 areexactly equal. Therefore, if the rms instead of the ampli-tude criterion is used, then for both aberration functionsthe Strehl ratios should be larger than or equal to 0.8 ifthe rms value is 1/14, in spite of the substantial differ-ences in amplitude range.

To test this assertion, the aberrations are scaled suchthat their rms value is 1/14, and the optical transfer func-tions and corresponding Strehl ratios are computed. Forthe aberrations of Figs. 3 and 4, the Strehl ratios arefound to be equal to 0.90 and 0.82, respectively. It is con-cluded that these values agree with Marechal’s criterionbut are quite different.

4. CONCLUSIONSFor completeness, the aberrations of Figs. 1, 3, and 4 arescaled to a quarter-wavelength amplitude range (Ray-leigh) or to an rms value equal to one fourteenth of awavelength (Marechal), and all pertinent Strehl ratiosare computed. The results are presented in Table 1.

The main conclusion is that, in all cases considered, theStrehl ratio exceeds 0.8. Therefore, on the basis of theseresults, there is no reason to question either criterion.On the other hand, it is seen that for both the Rayleighand the Marechal criterion the Strehl ratios differ sub-

Fig. 4. Low peak-factor function with the same rms value andFourier amplitude content as the function of Fig. 3.

Table 1. Strehl Ratios

Aberration

Strehl Ratio

Peak FactorRayleigh Marechal

Fig. 1 0.97 0.82 3.15Fig. 3 0.98 0.90 3.44Fig. 4 0.83 0.82 1.30

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stantially from one aberration to another. Therefore, thesuggestion2 that in this respect the Marechal criterionwould be an improvement is not substantiated by theseexamples.

A. van den Bos may be reached at a.vandenbos@tn.tudelft.nl.

REFERENCES1. R. Barakat, ‘‘Rayleigh wavefront criterion,’’ J. Opt. Soc. Am.

55, 572–573 (1965).

2. M. Born and E. Wolf, Principles of Optics (Cambridge U.Press, Cambridge, UK, 1997).

3. A. Marechal, ‘‘Etude des effets combines de la diffraction etdes aberrations geometriques sur l’image d’un point lu-mineux,’’ Rev. Opt. Theor. Instrum. 26, 257–277 (1947).

4. M. R. Schroeder, ‘‘Synthesis of low-peak-factor signals andbinary sequences with low autocorrelation,’’ IEEE Trans.Inf. Theory IT-16, 85–89 (1970).

5. A. van den Bos, ‘‘A new method for synthesis of low-peak-factor signals,’’ IEEE Trans. Acoustics, Speech, Signal Pro-cess. ASSP-35, 120–122 (1987).