694
`
`J. Opt. Soc. Am. A/Vol. 3, No. 5/May 1986
`
`J.P. Mills and B. J. Thompson
`
`=1==~
`
`b~
`
`Effect of aberrations and apodization on the performance of
`coherent optical systems. I. The amplitude impulse
`response
`
`j. P. Mills* and B. J. Thompson
`
`Institute of Optics, University of Rochester, Rochester, New York 14627
`
`Received July 18, 1985; accepted December 5,1985
`
`A systematic study was performed in which the effects of aberrations and apodization on the performance of a
`coherent optical system were investigated. The system performance was characterized by the modulus and the
`phase of the amplitude impulse response. The aberrations considered were defocus, spherical aberration, coma,
`and astigmatism. The apodizer was Gaussian in amplitude transmittance. The results of the study indicate that,
`within certain limits, the apodizer was effective in removing the sidelobes from the aberrated amplitude impulse
`response. This has significant implications for the performance of coherent imaging and beam-propagation
`systems.
`
`INTRODUCTION
`
`The importance of coherent optical systems has grown dra-
`matically since the inception of the laser. At present a great
`variety of coherent optical systems exist in research, indus-
`trial, government, and business applications. For the pur-
`pose of this paper, these coherent systems can be divided
`into two broad categories: (1) image-forming systems and
`(2) beam-propagation and -focusing systems. Examples of
`image-forming systems include optical data storage and re-
`trieval, optical image processing, and some microscopy. La-
`ser welding, eye surgery, and high-power lasers for fusion
`research are examples of beam-focusing systems.
`A common feature of all these systems is the presence of
`optical aberrations. Even in the most highly corrected sys-
`tems, such as those used in photomicrolithography, there are
`some residual aberrations; and most systems are not so well
`corrected. Those systems that were relatively well correct-
`ed in the design phase can have additional aberrations intro-
`duced by the manufacturing process and environmental
`stresses. Aberrations result in phase errors in the imaging
`(or beam-propagation) wave front as it traverses the optical
`system. These errors have weU-known effects on the perfor-
`mance of these systems.TM The usual response from an
`optical designer when an aberration has resulted in an unac-
`ceptable degradation of performance is to alter the surface
`contours of the optical elements in the optical system to
`decrease the overall amount of aberrations. It will be shown
`that the use of apodization is also useful in controlling the
`effects of aberrations in coherent systems.
`Apodization is the deliberate modification of the trans-
`mittance of the optical system. This modification results in
`a significantly altered system impulse response, which in
`turn affects the imaging or beam-propagation characteris-
`tics of the optical system. For aberrated incoherent optical
`systems, the use of apodization has theoretically been shown
`to moderate the deleterious effects of the aberrations on
`system performance.2-n However, the use ofapodization to
`improve the performance of aberrated coherent optical sys-
`tems has apparently not been studied.
`
`0740-3232/86/050694-10502.00
`
`The work reported here seeks to fill that void. A system-
`~÷~r ~t,,,t,, wa~ n~f, rm~ investigating the effects of aberra-
`tions and apodization (separately and in combination) on
`selected aspects of the performance of coherent optical sys-
`tems. The present paper reports the results of the portion
`of the study related to the amplitude impulse response,
`which is the fundamental function of importance in coherent
`and partially coherent systems. (For incoherent systems it
`is the irradiance impulse response that is the fundamental
`function; the irradiance impulse response is the product of
`the amplitude impulse response with its complex conjugate.)
`Future papers will address the analogous aspects of coherent
`imaging systems and coherent beam-propagation systems.
`
`THEORY
`
`The amplitude impulse response is defined as the complex
`optical field amplitude in the image plane of an optical
`system when the object is a point source of unit amplitude
`and zero phase. In the presence of aberrations other than
`defocus, the image plane is the one in which the best Strehl
`ratio is obtained. In the case of defocus, the image plane is
`the one determined by geometrical optics.
`
`Theoretical Development
`The effects of aberrations and apodization were investigated
`theoretically by considering the idealized optical system
`shown in Fig. 1. In the object plane, an on-axis point source
`radiates a diverging spherical wave. The lens L1, a distance
`from the object plane equal to its focal length fi, collimates
`the diverging spherical wave. The lens L1 is assumed to
`produce a perfectly collimated wave front. This assumption
`is the usual one in which al! the diffraction effects are associ-
`ated with the limiting aperture of the system. The (xl, Y0
`plane is a distance fl + f2 away from lens L1 and a distance f2
`away from image plane, where f2 is the focal length of lens
`L2. The apodizer, the lens L2, and the limiting aperture
`were all assumed to be coincident with the (xl, yz) plane,
`which is also the exit pupil of the system. The transmit-
`tance of the exit pupil can then be described by
`LG Electronics, Inc. et al.
`© 1986 Opti ai Society of America EXHIBIT 1008
`IPR Petition for
`U.S. Patent No. RE43,106
`
`

`
`J. P. Mills and B. J. Thompson
`
`Vol. 3, No. 5/May 1986/J. Opt. Soc. Am. A 695
`
`yo
`
`APODIZER
`
`Po,.T 1x°
`
`soURCEJ
`
`h
`\L
`
`v -
`
`)(-x~
`
`n
`IIj ,.
`
`. ’.L2
`
`fl+ f2
`
`"-LIMITING
`= = fz’’~ APERTURE
`
`I IVlRvr=.
`IpLANE
`
`Fig. 1.
`
`The geometry of the analyzed coherent optical system.
`
`-i21r
`X exp[~2 @(xz, yl)]B(xl, yl),
`
`(1)
`
`where A(xi, Yl) is the amplitude transmittance of the apo-
`dizer, f2 is the focal length of lens L2, and ;k is the illumina-
`tion wavelength. The first exponential term accounts for
`the modification of the wave front by a perfect thin lens, the
`second exponential term accounts for the aberration intro-
`duced by a real lens, and B(xl, yl) represents the finite
`extent of the limiting aperture.
`The apodization used in this study had an amplitude
`transmittance that was Gaussian in form:
`
`A(Xl, Yl) = exp[-3(xl2 + Yz2)] = exp(-3r2), (2)
`
`where the constant in the exponent was chosen so that the
`value of A(xl, Yz) at the edge of the limiting aperture was
`equal to 0.050. This function is displayed in Fig. 2 as a solid
`curve. The dashed curve in this figure represents the ampli-
`tude transmittance of the unapodized limiting aperture.
`The reason for selecting a Gaussian apodization was that
`it was desirable to have an apodizer that produced a real and
`positive amplitude impulse response.12 Many of the delete-
`rious effects seen in coherent imaging (edge ringing, for
`example) occur because the amplitude impulse response has
`negative regions. By contrast, an incoherent system has an
`impulse response that is always real and positive. This
`observation led to the conclusion that it would be appropri-
`ate to make the amplitude impulse response for the coherent
`case real and positive as well. The amplitude impulse re-
`sponse of an optical system is proportional to the Fourier
`transform of the optical field in the exit pupil of the system.
`So, if the transmittance of the apodizer has the functional
`form of a Gaussian truncated far from its center, then the
`amplitude impulse response will be nearly real and positive.
`The aberrations considered were those described by the
`Siedel wave front representationla
`
`@(r, 0) = a20r2 + a40r4 + a31r3 cos e + a22r2 cos2 e, (3)
`
`where a2o, a40, a3z, and a22 are the amounts of defocus, spher-
`ical aberration, coma, and astigmatism, respectively. The
`quantities r and ~ are defined by
`
`r2 = xl2 + yl2, 0 ---- tan-l(yl/Xl).
`
`The image-plane optical field arising from the point
`
`0~
`
`=
`
`source is proportional to the two-dimensional Fourier trans-
`form of the exit pupil field distribution14
`
`K(x2, Y2)
`
`= I--L-II:. T(xpyl)
`X2f22
`
`yly~)]dx,dy,. (4)
`
`x e=pI (X!X2 + .......
`_r-i2~xf2
`K(x2, Y2) then is the expression for the amplitude impulse
`response of the idealized optical system of Fig. 1.
`There are several assumptions implicit in Eq. (4). It was
`assumed that a scalar diffraction treatment of this problem
`was sufficient. Also, the usual paraxial assumptions were
`made, namely, the diameter of the exit pupil was much
`greater than the wavelength of illumination and the maxi-
`mum linear distance in the region of interest in the image
`plane was much less than the distance from the exit pupil to
`the image plane.
`Noting that these assumptions are valid in most optical
`systems, Eq. (4) was used to calculate the amplitude impulse
`response for the system in the presence of various amounts
`of defocus and the third-order aberrations, both with and
`without the Gaussian apodization.
`
`Theoretical Data
`Equation (4) was computed on a VAX 11/750 computer
`using a fast-Fourier-transform subroutine from the IMSL14
`package.
`The quantity K(x2, Y.2) in Eq. (4) is, in general, complex.
`Consequently, the output of the program is in terms of real
`and imaginary coefficients, i.e.,
`
`K(x2, Y2) ffi a(x2, Y2) + ib(x2, Y2)
`
`= re(x2, y2)exp[ip(x2, Y2)],
`
`(5)
`
`where a(x2, Y2) and b(x2, Y2) are the coefficients and re(x2, y2)
`
`TIr)
`
`=
`
`0
`
`i
`I ....... "1
`1
`
`i
`
`,
`
`-1
`
`i
`0
`r’
`Fig. 2. The amplitude transmittance T(r) of the Gaussian filter
`used in this study (solid curve) shown relative to the unapodized
`amplitude transmittance of the exit pupil (dashed curve).
`
`

`
`696
`
`J. Opt. Soc. Am. A/Vol. 3, No. 5/May 1986
`
`J. P. Mills and B. J. Thompson
`
`Q~
`
`o
`
`o i
`
`(0}
`
`(b)
`
`(c}
`
`Fig. 3. The impulse response of an unaberrated, unapodised, cir-
`cularly symmetric optical system in terms of (a) modulus, (b) phase,
`and (e) irradiance. The u and v axes are in terms of canonical
`distance coordinates. All plots have the same distance scales.
`
`and p(x2, Y2) are the modulus and the phase, respectively, of
`the amplitude impulse response.
`Coherently illuminated systems are linear in complex am-
`plitude. However, if measurements are to be made in an
`experiment, it is the irradiance 1(x2, y2) that is usually mea-
`sured. The irradiance of the amplitude impulse response is
`related to its complex amplitude by
`
`(6)
`
`I(Xs, Y2) ffi K(xs, Ys)K*(xs, Ys) = IK(x2, Ys)I 2,
`where the asterisk denotes the complex conjugate.
`A typical output of this program is displayed in Fig. 3.
`The system in this case was unaberrated and unapodized,
`and the pupil function was a circular aperture. The dis-
`tance coordinates used in Fig. 3, as well as in many other
`figures in this paper, are the canonical distance coordinates
`u and v, defined by
`
`2~a
`2~a
`U = d--T xi, v = --di y,,.
`
`(7)
`
`where a is the radius of the exit pupil, di is the distance from
`the exit pupil to the image plane, and xi and Yl are the spatial
`coordinates in the image plane.
`A coherent optical system is linear in field amplitude,
`which is a complex quantity. So, unlike in an incoherent
`system, the phase in the optical field is critically important
`in determining the f’mal image irradiance distribution. For
`this reason, the phase was calculated for the unaberrated
`case and is displayed in Fig. 3(b). The phase was also calcu-
`lated for many of the aberrated cases considered later.
`The phase distribution in the amplitude impulse response
`of an unaberrated system [see Fig. 3(b)] is uniform every-
`where except along concentric circles, where the phase
`jumps discontinuously by an amount equal to v rad. These
`jumps occur at the same spatial locations as do the zero
`
`values in the modulus distribution of Fig. 3(a). This implies
`that in every other ring the amplitude impulse response is
`composed of negative values. It is these negative regions
`that cause the ringing phenomenon seen in the image of an
`edge.
`The irradiance distribution of this impulse response is
`shown in Fig. 3(c). It is obtained by squaring the modulus
`distribution. Since the optical system in this case is unaber-
`rated and has a circular exit pupil, it is expected that the
`irradiance distribution would approximate an Airy15 pat-
`tern. A careful comparison of Fig. 3(c) with a theoretical
`Airy distribution reveals the error between the two patterns
`to be less than 1% at any point. This error arises mainly
`from the problem of adequately representing a circular aper-
`ture with a rectangular array of samples.
`
`Defocus
`The aberration of defocus has the functional form ~(r, 0) ffi
`a2or2, where the coefficient a20 is the amount of aberration.
`Defocus is the simplest type of aberration in that the real
`wave front differs from the spherical reference wave front
`only in its radius of curvature. A calculation of the ampli-
`tude impulse response, for the case when a20 = 0.5 wave,
`:,~elded the results shown L-~ Fig. 4. in this figure the top two
`plots show the modulus (on the left) and the phase of the
`amplitude impulse response when the exit pupil of the opti-
`cal system has a uniform transmittance, i.e., when there is no
`apodization. For ready comparison, the amplitude impulse
`response (modulus and phase) for the same system with a
`Gaussian apodizer is shown in the bottom two plots of the
`same figure.
`The modulus and the phase of the unapodized amplitude
`impulse response (top two plots) should be compared with
`the analogous plots of Fig. 3 where there are no aberrations
`in the system. The peak value of the modulus in the aber-
`rated case decreased relative to the unaberrated case. The
`zero values in the modulus pattern for the unaberrated ease
`
`Fig. 4. The amplitude impulse response (modulus and phase) in
`the presence of 0.5-wave defocus and for the ease Of an unapodized
`and a Gaussian apodized aperture. The top two plots are for the
`unapodized case, and the bottom two are for the case of a Gaussian
`apodizer. The vertical scales for the modulus plots (left-hand col-
`umn), and the phase plots (right-hand column) are indicated by the
`top two plots. The same scaling is used in Figs. 7, 10, and 13.
`
`

`
`J. P. Mills and B. J. Thompson
`
`Vol. 3, No. 5/May 1986/J. Opt. Soe. Am. A 697
`
`AMPLITUDE IMPULSE
`RESPONSE
`
`1.O
`
`O.6
`
`DEFOCUS
`
`@
`
`0,1
`
`Fig. 5. The amplitude impulse response (modulus and phase) with
`varying amounts of defocus for the case of an unapodized and a
`Gaussian apodized exit pupil. The amount of aberration for each
`column is indicated at the bottom of that column.
`
`3
`
`e~
`
`’"
`
`~
`
`OJ’:,~
`-20
`
`"’,-/
`0
`v
`
`~ ~ ,~
`20
`
`0
`
`-20
`
`0
`v
`
`20
`
`u~ / ~
`
`.--- ...... ~.~
`
`-
`
`-20
`
`0
`v
`
`20
`
`-311"
`
`¯
`-20
`
`i
`0
`v
`
`~ ’
`20
`
`Fig. 6. Central slices through the plots of modulus and phase in the
`presence of varying amounts of defocus and for the unapodized
`(left-hand column) as well as the Gaussian apodized case:
`0.1 wave; -- -- --, 0.5 wave; ..... ,1.0 wave.
`
`evolved to relative minimums that do not go to zero. The
`phase of the aberrated amplitude impulse response (upper-
`right-hand plot of Fig. 4) no longer has the discontinuities
`evident in the unaberrated case.
`When the apodizer described by Eq. (2) and plotted in Fig.
`2 is applied to this aberrated system, the modulus of the
`amplitude impulse response (lower-left-hand plot of Fig. 4)
`is considerably smoothed, as is the phase. The phase varies
`by less than 7r rad over the region of the modulus plot where
`
`(cid:128)
`
`)mae
`
`t~
`
`the modulus is greater than 10% of its peak value. So this
`amplitude impulse response does not change sign until the
`absolute value of the amplitude is quite small. Thus the
`impulse response is almost real and positive.
`This has important implications for the imaging perfor-
`mance of optical systems. For instance, the ringing in the
`coherent image of an edge is caused by the negative regions
`of the impulse response. In this case the apodizer has
`smoothed the amplitude impulse response such that it has
`very little amplitude in the regions where there are negative
`values of amplitude. It can be expected, then, that the
`image of an edge through this system would be free from
`ringing.
`The amplitude impulse response, both unapodized and
`apodized, for other values of defocus is shown in Fig. 5.
`Here the amount of aberration is different for each column
`of plots, varying from 0.1 wave on the left to 1.0 wave on the
`right. From these plots the evolution of the modulus and
`the phase can be seen as more defocus is added to the system.
`The modulus and the phase along slices through the cen-
`ter of some of these impulse responses are shown in Fig. 6.
`The relationship of phase to modulus is clearly seen in this
`figure.
`Apou,za~lou, in each case of ,L^ last *~’- ~ r ....
`smoothes both the modulus and the phase. In each case, the
`amplitude impulse response becomes almost real and posi-
`tive when the apodizer is applied. There are, however, lim-
`its to this process. As the amount of aberration increases,
`the apodizer becomes less effective in making the amplitude
`impulse response almost real and positive. For the case of
`one wave of defocus, it appears that the phase has changed
`by more than ~r/2 rad over the region where the modulus is
`still relatively large.
`
`Spherical Aberration
`Spherical aberration has the functional form ~(r, 0) = a4or4.
`Spherical aberration, like defocus, is a radially symmetric
`aberration. Owing to its fourth-power dependence on the
`radial distance parameter r, spherical aberration describes a
`
`Fig. 7. The amplitude impulse response (modulus and phase) in
`the presence of 0.5-wave spherical aberration and for the case of an
`unapodized and a Gaussian apodized aperture. The top two plots
`are for the unapodized case, and the bottom two are for the case of a
`Gaussian apodizer. See Fig. 4 for the scaling.
`
`

`
`698
`
`J. Opt. Soc. Am. A/Vol. 3, No. 5/May 1986
`
`J. P. Mills and B. J. Thompson
`
`,Fq
`
`o
`
`AMPLITUDE IMPULSE
`RESPONSE
`
`Fig. 8. The amplitude impulse response (modulus and phase) with
`varying amounts of spherical aberration for the case of an unapo-
`dized and a Gaussian apodized exit pupil. The amount of aberra-
`tion for each column is indicated at the bottom of that column.
`
`01 A1
`
`C
`
`-20
`
`,
`0
`v
`
`20
`
`-~
`-20
`
`~
`v
`
`20
`
`particularly the first sidelobe. The position of the ring of
`minimum values between the central lobe and the first side-
`lobe changes little for this value of aberration.
`The impulse response when the apodizer is employed is
`shown in the bottom two plots of Fig. 7. As in the case of
`defocus, the use of the apodizer has resulted in a much
`smoother impulse response. An examination of the phase
`distribution shows that the phase is nearly uniform over the
`region of the impulse response having significant amounts of
`energy. This impulse response can also be described as
`being almost real and positive.
`The evolution of the unapodized and apodized amplitude
`impulse responses as more spherical aberration is added to
`the system is shown in Fig. 8. Central slices through some of
`these plots are shown in Fig. 9. The same general phenome-
`na seen in the case of defocus are seen here as well. The use
`of the apodizer results in an impulse response that is free
`from sidelobes in the modulus pattern and that has a rela-
`tively fiat phase over the region where there is a Significant
`amount of energy.
`Again there are limits to this process. When the amount
`of spherical aberration is about one wave, the impulse re-
`sponse has significant amounts of energy in regions where
`the phase has changed by about 4/2o So the apo~zer 1~ nnt
`totally effective, although the apodized impulse response is
`still much smoother than the unapodized one.
`
`Coma
`The aberration of coma can be described by ¢(r, 0) = aslr~
`cos 0. Coma is the first aberration considered for which the
`wave front in the exit pupil depends on the polar angle as
`well as on the radial distance r. This aberration therefore
`produces the unsymmetrical amplitude impulse response
`seen in Fig. 10 for the case a31 = -0.5 wave. The use of the
`apodizer in this case appears to be less effective than in the
`previous cases, because the first sidelobe is still evident in
`the modulus of the apodized impulse response (lower-left-
`
`-31T ,
`-20
`
`Fig. 9. Central slices through the plots of modulus and phase in the
`presence of varying amounts of spherical aberration and for the
`unapodized (left-hand column) as well as the Ganssian apodized
`case: ,0.1 wave; .... ,0.5 wave; ..... ,1.0 wave.
`
`wave front having the largest deviation from the spherical
`reference wave front of any of the aberrations considered. A
`calculation of the impulse response (modulus and phase)
`when a4o = 0.5 wave is shown in Fig. 7. The top two plots
`(the unapodized case) should be compared with the unaber-
`rated impulse response of Fig. 3. The presence of spherical
`aberration causes a decrease in the value of the central peak
`(Strehl ratio) and an increase in the energy in the sidelobes,
`
`Fig. 10. The amplitude impulse response (modulus and phase) in
`the presence of -0.5-wave coma and for the case of an unapodized
`and a Gaussian apodized aperture. The top two plots are for the
`unapodized case, and the bottom two are for the case of a Gaussian
`apodizer. See Fig. 4 for scaling.
`
`

`
`J. P. Mills and B. J. Thompson
`
`Vol. 3, No. 5/May 1986/J. Opt. Soc. Am. A 699
`
`AMPLITUDE IMPULSE
`RESPONSE
`
`Y COMA
`
`Fig. 11. The amplitude impulse response (modulus and phase)
`with varying amounts of coma for the case of an unapodized and a
`Gaussian apodized exit pupil. The amount of aberration for each
`culunm is indicated at the bottom of that column.
`
`u}
`
`O
`=E
`O"
`
`-20 0 20
`
`,
`o i S-Y
`-20
`0
`V v
`
`-31T~ ,
`-20
`
`,
`0
`v
`
`,
`20
`
`-31T-~--~
`-20
`
`:
`0
`v
`
`--
`20
`
`:
`20
`
`Fig. 12. Central slice through the plots of modulus and phase in the
`presence of varying amounts of coma and for the unapodized (left-
`hand column) as well as the Gaussian apodized case: ,0.1 wave;
`.... ,0.5 wave; ..... ,1.0 wave.
`
`hand plot of Fig. 10). Also, there is a ~r phase change in the
`region of this first sidelobe.
`The evolution of these impulse responses with increasing
`amounts of aberration is shown in Fig. 11. Central slices of
`some of these data are shown in Fig. 12. The slices are along
`the axis showing the minimum amount of symmetry. The
`same general conclusions that were drawn for the cases of
`defocus and spherical aberration can be drawn for this case
`as well. First, the apodizer is effective in transforming the
`
`aberrated impulse response into a much smoother function.
`Second, there are limits to the effectiveness of the apodizer.
`The limit in this case appears at a lower value of aberration
`than in the other cases. For as little as half a wavelength
`there is a ~ phase change in a region of the impulse response
`where the modulus shows a significant sidelobe.
`
`Astigmatism
`The aberration of astigmatism is described by ~(r, 0) = a22r2
`cos2 0. Like coma, astigmatism is an unsymmetric aberra-
`tion. The appearance of the unapodized and apodized im-
`pulse responses when a~2 = 0.5 wave is shown in Fig. 13.
`Both phase plots show a saddle shape, which is characteristic
`of astigmatism. Astigmatism results in a wave front that
`has different radii of curvature along two orthogonal direc-
`tions in the plane of the exit pupil. This behavior is evident
`also in the paraxial focal plane as seen in Fig. 13.
`The effect of apodization is again to smooth both the
`amplitude and the phase of the impulse response. This
`behavior holds as the amount of aberration is increased from
`0.1 wave to 1.0 wave, as seen in Figs. 14 and 15. The general
`shapes of the functions in Fig. 15 differ little from the analo-
`gous plots for the case of defocus (Fig. 6). This is because, in
`^-~ .=" ^- : .... .: --~." ...... 1 ^ ~- ^
`uii~ ulm~iiSiu*1, ~wgui~b~*a ,vSu~t~ ~, a wave ,,~n~
`*-^ + that is
`spherical but has a radius of curvature different from the
`reference wave front.
`Again, the apodizer smooths the aberrated impulse re-
`sponse for a limited amount of aberration. The limit in the
`case of astigmatism appears to be about one wave.
`
`On-Axis Calculations
`A common feature in all the impulse data is that the addition
`of aberrations to an unaberrated optical system always re-
`sults in a decrease in the peak value of the central lobe of the
`modulus. This behavior is plotted in Fig. 16 for various
`values of the four aberrations and the two cases of apodiza-
`tion. The peak values have been normalized to unity for the
`
`Fig. 13. The amplitude impulse response (modulus and phase) in
`the presence of 0.5-wave astigmatism and for the case of an unapo-
`dized and a Ganssian apodized aperture. The top two plots are for
`the unapodized case, and the bottom two are for the case of a
`Gaussian apodizer. See Fig. 4 for scaling.
`
`

`
`700
`
`J. Opt. Soc. Am. A/Voi. 3, No. 5/May 1986
`
`J. P. Mills and B. J. Thompson
`
`AMPLITUDE IMPULSE
`RESPONSE
`
`¢J
`rn
`
`UNAPODIZED
`
`APODIZED
`
`0.30]
`
`OSFOCUS
`X SPHERICAL
`0 ASTIGMATISM
`
`i
`oA-
`
`o.i.
`
`’.,.
`
`o.lo
`
`o,s
`WAVES OF ABERRATION
`WAVES OF ABERRATION
`Fig. 16. The central value of the modulus as a function of the type
`and amount of aberration for cases of unapodized and Gaussian
`apodized apertures. For each plot, the abscissa is the amount of
`aberration in waves and the ordinate is the peak of the modulus.
`
`,s
`
`UNAPODIZED
`
`APODIZED
`
`"V 0.2
`
`O" ASTIGMATISM
`
`0.1
`
`Fig. 14.
`The amplitude impulse response (modulus and phase)
`with varying amounts of astigmatism for the case of an unapodized
`and a Gaussian apodized exit pupil. The amount of aberration for
`each column is indicated at the bottom of that column.
`
`-20
`
`0
`v
`
`20
`
`-20
`
`0
`v
`
`20
`
`1
`
`WAVES OF ABERRATION
`WAVES OF ABERRATION
`The central value of the irradiance as a function of the type
`and amount of aberration for cases of unapodized and Ganssian
`apodized apertures. For each plot, the abscissa is the amount of
`aberration in waves and the ordinate is the peak value of the irradi-
`anee.
`
`UNAPODIZED
`
`APODIZED
`6 DEFOCUS
`x SPHERICAL
`0 0 ASTIGMA11SM
`v CO~A
`
`-20
`
`20
`
`-20
`
`0
`v
`Fig. 15. Central slices through the plots of modulus and phase in
`the presence of varying amounts of astigmatism and for the unapo-
`dized (left-hand column) as well as the Gaussian apodized case:
`--, 0.1 wave; .... ,0.5 wave; ..... ,1.0 wave.
`
`0
`v
`
`20
`
`UJ -t.s
`U)
`<
`-r
`
`U)1~1I~.,,r"~-3-;t’ -l’
`
`unaberrated system. From this plot, it can be seen that the
`peak value of modulus occurs when there are no aberrations.
`The addition of any amount of the four aberrations to an
`unaberrated system results in a decrease in the peak value.
`Notice that in the case of defocus, the central peak complete-
`ly disappears for 1.0 wave of aberration. This is so-called
`fringe of defocus.
`Figure 17 shows the behavior of the central value of irradi-
`ance as the same aberrations were added. It was obtained
`
`~.
`
`WAVES OFABERRATION
`WAVES OF ABERRATION
`Fig. 18. The cez~tral value of the phase as a function of the type and
`amount of aberration for cases of unapodized and Gaussian apo-
`dized apertures. For each plot, the abscissa is the amount of aber-
`ration in waves and the ordinate is the value of the phase in radians
`at the peak of the modulus.
`
`

`
`J. P. Mills and B. J. Thompson
`
`Vol. 3, No. 5/May 1986/J. Opt. Soc. Am. A
`
`by squaring each value in Fig. 16. When the irradiance is
`used as in Fig. 17, the plots could also be labeled as the Strehl
`ratio. Note that the use of the apodizer immediately de-
`creases the Strehl ratio by a factor of 10 for the unaberrated
`case.
`Figure 18 shows the behavior of the phase at the central,
`peak of the impulse response as a function of the amount of
`aberration and apodization present. For most ofthe aberra-
`tions, the phase is relatively unchanged as aberrations are
`added. For the case of defocus, however, there is a ~r phase
`
`gles are data points from individual elements in the CCD
`array. Both sets of data conform closely to the expected
`Airy pattern.
`When the apodizer was placed in the same system, the
`data displayed in Fig. 21 were collected. The solid curve in
`the figure represents the expected irradiance distribution
`based on the measured aberrations and the presence of the
`apodizer. The dashed curves represent data from the detec-
`tor array in two orthogonal orientations.
`The data in Figs. 20 and 21 indicate that (1) the experi-
`
`t~
`[ml.
`
`corresponds to the location of the zero in the modulus of the
`unaberrated impulse response. This is consistent with the
`earlier observation that zeros in the modulus distribution of
`an unaberrated amplitude impulse response are accompa-
`nied by ~ phase jumps.
`
`the expected diffraction patterns, (2) the theoretical model
`was effective in predicting experimental results, and (3) the
`constructed f’llter was indeed Gaussian in amplitude trans-
`mittance.
`
`EXPERIMENT
`
`Experiments were conducted to test the theoretical predic-
`tions of the previous section. The experimental arrange-
`ment is illustrated schematically in Fig. 19. This configura-
`tion was designed to measure the irradiance impulse re-
`sponse of the optical system formed by lens LI, the apodizer,
`the iris, and lens L2.
`In this system, coherent illumination originated from an
`0.5-roW, linearly polarized He-Ne laser. The absorbing
`filter was used to adjust the irradiance level of the impulse
`response at the detector plane. Mirrors M1 and M2 direct-
`ed the beam into the spatial filter, which served as the point
`source for the optical system. Lens L1 was a high-f-number
`telescope objective, which provided an essentially aberra-
`tion-free, collimated beam to the remainder of the optical
`system. The lens L2 was one of several lenses chosen and
`oriented to provide aberrations.
`Beyond the image plane, two configurations were used.
`In the one shown on the optical axis in Fig. 19, lens L3
`magnified the imaged point source, which is of course the
`irradiance impulse response. A Fairchild 256-element lin-
`ear charge-coupled-device (CCD) array was placed in the
`magnified image plane to measure the irradiance along a line
`through the center of the impulse response.
`The configuration, shown shifted to the right of the opti-
`cal axis in Fig. 19, sometimes replaced lens L3 and the detec-
`tor array. It was used to measure the aberrations in the exit
`pupil. This was done by centering the point diffraction
`interferometer (PDI) on the impulse response. Lens L4 was
`adjusted to image the iris onto the film plane in the camera.
`The photographed interference patterns were digitized and
`analyzed by the fringe analysis program wisP.16
`The Gaussian apodizing filter was made using a technique
`developed by Welter17 and others.18-21
`
`Data from an Unaberrated System
`As an initial test of the system, an aberration-free configura-
`tion was constructed. A good-quality, high-f-number tele-
`scope objective was selected for use as lens L2. The absolute
`value of each of the measured third-order aberrations was
`less than 0.1 wave. The result of a measurement of the
`irradiance impulse response for this case is shown in Fig. 20,
`where the solid curve is the theoretically expected impulse
`response based on the measured aberrations and the trian-
`
`~ M2
`
`I~ ~-I Absorbing C
`Filter
`
`p Spatial
`Filter
`
`Image
`Plane
`
`Detector Acray
`Fig. 19. The experiments] configuration used to measure the irra-
`diance impulse response of the optical system formed by L1, L2, the
`iris, and the apodizer.
`
`Camera
`
`,
`
`0.1
`
`!
`
`tu
`
`~ 0.01
`
`rr
`0.001
`I0
`0
`-10
`NORMALIZED DISTANCE V
`Fig. 20. A comparison of theoretical (solid curve) and experimen-
`tal (triangles) results. The vertical axis is the relative irradiance
`plotted on a logarithmic scale. The horizontal axis for each plot is
`the normalized distance V.
`
`" I
`
`

`
`702
`
`J. Opt. Soc. Am. A]Vol. 3, No. 5]May 1986
`
`J. P. Mills and B. J. Thompson
`
`1’o
`
`b
`
`10
`
`0.001
`
`NORMALIZED DISTANCE V
`Fig. 22. Theoretical (solid lines) mid experimental (dashed lines)
`results are compared for two systems. In the first system, a, astig-
`matism is 0.1 wave, coma is 0.1 wave, and spherical is -0.2 wave; and
`in the second system, b, astigmatism is 0.1 wave, coma is 0