ENGINEERING AND LABORATORY NOTES

Tolerance on defocus precisely locates the farfield �exactly where is that far field anyway?�

James E. Harvey, Andrey Krywonos, and Dijana Bogunovic

The Fraunhofer criterion defines the location of the boundary between the Fresnel and the Fraunhoferdiffraction regions and thus determines the location of that region commonly referred to as the far field.The Fraunhofer criterion is usually given as an axial distance much greater than some amount relativeto the maximum dimension of the aperture. By recognizing that Fresnel diffraction patterns are merelydefocused Fraunhofer diffraction patterns, we show that the Fraunhofer criterion can be written preciselyin terms of an allowable tolerance on defocus. This new criterion provides insight that is useful to opticaldesigners and engineers who routinely deal with such tolerances. © 2002 Optical Society of America

OCIS codes: 050.0050, 080.0080, 260.0260, 050.1940, 080.1010.

1. Introduction

We obtain the Fresnel and the Fraunhofer diffractionintegrals by making explicit approximations to themore-general Rayliegh–Sommerfeld diffraction inte-gral. These approximations restrict the region ofspace over which the respective integrals are valid, asshown in Fig. 1.1,2 Note that the Fraunhofer regionis contained in the Fresnel region and that both theFraunhofer and the Fresnel regions are contained inthe Rayliegh–Sommerfeld region. The far field iswidely understood to be synonymous with the Fraun-hofer region; however, there is less agreement in theliterature upon the definition of the near field. Wedefine the near field to be that region of space thatdoes not satisfy even the Fresnel criterion. Hencethe near field and the far field are mutually exclusivebut not all-inclusive.

The Fraunhofer criterion specifies the minimumdistance of the observation plane from a diffractingaperture to ensure the validity of the Fraunhoferdiffraction integral,

z �� �k�2�� x12 � y1

2�max. (1)

That the Fraunhofer criterion is usually given as anaxial distance much greater than some amount rela-

The authors are with The Center for Research and Education inOptics and Lasers, P.O. Box 162700, 4000 Central Florida Boule-vard, University of Central Florida, Orlando, Florida 32816.

Received 5 November 2001; revised manuscript received 14 De-cember 2001.

0003-6935�02�132586-03$15.00�0© 2002 Optical Society of America

2586 APPLIED OPTICS � Vol. 41, No. 13 � 1 May 2002

tive to the maximum dimension of the aperture israther vague and unsatisfying to the optical engineerwanting to know exactly how far he must be from agiven diffracting aperture for safely using the Fraun-hofer diffraction integral.

2. Fraunhofer Criterion in Terms of a Tolerance onDefocus

It has been shown that the Rayleigh–Sommerfelddiffraction integral can be written �without use of anyexplicit approximations� as a Fourier transform inte-gral of a generalized pupil function containing phasevariations that are identified with conventionalwave-front aberrations.3,4 Near-field diffraction pat-terns are thus aberrated Fraunhofer diffraction pat-terns. Fraunhofer diffraction is therefore aberrationfree, because it is precisely the reference from whichour aberrations are defined. And Fresnel diffraction

Fig. 1. Illustration of the regions of validity of the Fraunhofer,Fresnel, and Rayliegh–Sommerfeld diffraction integrals.

can be described as merely a defocused Fraunhoferdiffraction pattern.3–6

Recall that the wave-front aberration function isexpressed as a power-series expansion of the appro-priate field and pupil parameters7,8:

where � is the normalized field position of the obser-vation point and a is the normalized pupil height.The hats �ˆ� merely indicate that those quantities arealso normalized by the wavelength of light; i.e., z �z��, a � a��, and W020 � W020��.

That a Fresnel diffraction pattern is merely adefocused Fraunhofer diffraction pattern is evi-denced by the fact that the two diffraction integralsdiffer only by a quadratic phase factor.1,2 For aplane wave incident upon a circular aperture ofdiameter d, the aberration coefficient for defocus isgiven by3,4

W020 �z2 � d

2z�2

. (3)

Solving for z, we obtain

z �d2

8W020. (4)

Equation �4� serves as a new expression for theFraunhofer criterion, based on a specified tolerancefor the amount of defocus allowed for a given appli-cation. For a circular aperture 4.0 cm in diameterand for a wavelength of 0.5 �m we obtain from Eq. �1�that an observation distance of much greater than 2.5km is required for the Fraunhofer diffraction integralto be valid. We now see from Table 1 that, if we arewilling to tolerate 1�2-wave of defocus, an observa-

tion distance of only 800 m is required. However, ifwe tighten the allowable tolerance on defocus to ��20,8 km is required!

The Fraunhofer criterion, as usually stated, saysthat the amount of defocus allowed must be much less

than ��2�. Clearly there are many practical appli-cations in which this tolerance can be relaxed. Also,by multiplying Eq. �3� by 2�, we obtain the followingequation for the maximum phase error across theaperture,

��

4�zd2 �rad�. (5)

This measure of quality is used in many areas ofphysical optics and antenna measurement. Finally,we would comment that the above analysis approachis applicable to far-field measurements made by useof Fourier transform lenses to bring the measure-ment plane to the focal plane of the transform lens.In this case, lens aberrations and positioning errorsrelative to the focal plane also affect the far-fieldquality.

3. Summary and Conclusions

We have discussed the Fraunhofer, Fresnel, andRayliegh–Sommerfeld regions behind a diffractingaperture illuminated by a plane wave. After es-tablishing that Fresnel diffraction patterns aremerely defocused Fraunhofer diffraction patterns,we expressed the rather vague Fraunhofer criterionthat defines the axial location of the boundary be-tween the Fresnel region and the Fraunhofer regionin terms of a precise axial distance that depends onan allowable tolerance on the aberration defocus.This new expression for the Fraunhofer criterionshould be particularly useful to optical designersand engineers that routinely deal with such toler-ances.

References1. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics

�Wiley, New York, 1978�.2. J. W. Goodman, Introduction to Fourier Optics, 2nd ed.

�McGraw-Hill, New York, 1996�.

Table 1. Fraunhofer Criterion for Different Tolerances upon Defocus

W020 z �km�

��2 0.80��4 1.60��8 3.20��10 4.00��20 8.00

W � W000 (Piston Error) Zero-

� W200�4

Piston� W020a

2

Defocus� W111�a cos�� � �

Lateral Magnification Error � Second-

� W400�4

Piston� W040a

4

Spherical Aberration� W131�a3 cos�� � �

Coma

� W222�2a2 cos2�� � �

Astigmatism� W220�

2a2

Field Curvature� W311�

3a cos�� � �Distortion

� Fourth-

� Higher-Order Terms, (2)

1 May 2002 � Vol. 41, No. 13 � APPLIED OPTICS 2587

3. J. E. Harvey and R. V. Shack, “Aberrations of diffracted wavefields,” Appl. Opt. 17, 3003–3009 �1978�.

4. J. E. Harvey, “Fourier treatment of near-field scalar diffractiontheory,” Am. J. Phys. 47, 974–980 �1979�.

5. V. N. Mahajan, “Aberrations of diffracted wave fields. I. Op-tical imaging,” J. Opt. Soc. Am. A 17, 2216–2222 �2000�.

6. V. N. Mahajan, Optical Imaging and Aberrations: Part II�SPIE, Bellingham, Wash., 2001�.

7. V. N. Mahajan, Optical Imaging and Aberrations: Part I�SPIE, Bellingham, Wash., 1998�.

8. H. H. Hopkins, Wave Theory of Aberrations �Clarendon, Oxford,UK, 1950�.

2588 APPLIED OPTICS � Vol. 41, No. 13 � 1 May 2002