Ray-tracing and physical-optics analysis of the apertureefficiency in a radio telescope
Luca Olmi1,2,* and Pietro Bolli3,4
1INAF, Istituto di Radioastronomia, sezione di Firenze, Largo E. Fermi 5, I-50125 Firenze, Italy2Department of Physics, University of Puerto Rico, Rio Piedras Campus, Box 23343, UPR Station, San Juan,
Puerto Rico 00931-3343, USA3INAF, Osservatorio Astronomico di Cagliari, Loc. Poggio dei Pini, Strada 54, I-09012 Cagliari, Italy4INAF, Istituto di Radioastronomia, sezione di Bologna, Via P. Gobetti 101, I-40129 Bologna, Italy
Performance evaluation is a critical step in the designof any optical system, either at microwave or visible–IR wavelengths. The image quality criteria morecommonly used, however, are quite different in thesetwo regions of the electromagnetic spectrum. In fact,in the analysis of microwave antennas and radio tele-scopes the two fundamental figures-of-merit used bydesigners and users are the aperture efficiency andthe beam efficiency, whereas in optical systems theStrehl ratio and ray aberrations are often quoted.This is because of the coherent nature of most micro-wave antennas, where single-mode receivers are
generally used (exceptions may be millimeter andsubmillimeter bolometers used in radio astronomy),making the phase distribution in the image as im-portant as the amplitude distribution in determiningthe performance of the optics. In fact, the apertureefficiency is intrinsically dependent on the phase dis-tributions since it is calculated as a correlation inte-gral between the focal region field produced by anincident plane wave and the horn aperture field.
The difference between the microwave and thevisible–IR wavelength regimes, in terms of the imagequality criteria applied to astronomical telescopes, hasbeen reduced over the past 10–15 years thanks to thedevelopment of focal plane arrays (FPA, hereafter).In fact, the noise performance of receivers used inradio astronomy has improved dramatically duringthis time, especially at millimeter and submillimeter
wavelengths. As a consequence, it has become clearthat the best means of increasing observing efficiencyfor mapping extended sources or to conduct blindsearches is to use imaging arrays located at the focalplane of the telescope. This implies the need of alarger field of view (FOV) with few aberrations in therange of frequencies used by the array(s) of receivers.Very often these FPAs require some relay optics toconvert the telescope focal ratio (which, in somecases, may be quite large, i.e. �10) to the smallerfocal ratios of the individual feed-horns. As a conse-quence, the overall image quality of the total system,telescope and reimaging optics, must be evaluatedover a wide FOV, thus effectively contributing tobridging the gap between the microwave and the vis-ible wavelength regimes.
A number of commercial ray tracing packages existthat are being used to analyze the performance ofFPAs for use with existing or planned (sub)millime-ter telescopes. However, many of these packageshave been specifically designed for use with optical(i.e., visible and IR) systems and thus, although veryflexible and sophisticated, they do not provide theappropriate parameters to fully describe microwaveantennas, and thus to compare with specifications.The possibility to easily convert an optical-based de-sign parameter, such as the Strehl ratio, to a funda-mental antenna-based design parameter, such as thephase efficiency, gives the designer�user of the tele-scope the opportunity to use the faster commercialray-tracing software to optimize the design. Once thedesign is optimized, a full Physical Optics softwarecan be used to analyse more thoroughly all the crit-ical performance parameters of the antenna (e.g.,spillover, antenna noise temperature, etc.). Anotheradvantage offered by this conversion consists of thepossibility to study the degrading effects on the wave-front caused by obstructions to the beam (e.g., sec-ondary reflector and its support struts) which arenotoriously difficult to simulate in Physical Opticssoftware.
In this paper we review the main design parame-ters generally used in evaluating the performance ofoptical designs at both microwave and visible wave-lengths. Based on this review we find a simple rela-tionship between the (antenna-based) apertureefficiency and the Strehl ratio. We also show the re-sults of several tests performed to check the validityof this relationship that we carried out using a ray-tracing software, ZEMAX, and a full Physical Opticssoftware, GRASP9.3, applied to three different tele-scope designs.
In Section 2 we review and discuss the definitionsof antenna gain and aperture efficiency. In Section 3we analyze the definition of Strehl ratio and derive asimple relationship between the aperture efficiencyand the Strehl ratio. In Section 4 we show the resultsof a comparison obtained using a Physical Optics anda ray-tracing program and, finally, we draw our con-clusions in Section 5.
2. Antenna Gain and Aperture Efficiency
A. Definitions
The gain of an antenna is a measure of the couplingof the antenna to a plane wave field, and it can bewritten in terms of the effective area (we assume thatohmic losses are negligible):
G��, �� �4�
�2 Aeff��, ��. (1)
For an aperture type antenna the gain is express-ible in terms of the illumination by the feed. We canassume that the illumination is linearly polarized,and that the aperture lies on an infinite plane. In thiscase the gain is expressible in terms of Ea�r��, themagnitude of the (in-phase) illuminating electric fieldin the aperture plane. If almost all the energy in thefield is contained in a small angular region about thez� axis, and if we use the scalar-field approximation,then G��, �� can be written as [1]
G��, �� �4�
�2
��AP�
�a�r�, R�dS��2
�
Ea2�r��dS�
, (2)
with
�a�r�, R� � Ea�r��ej�r��ejkR·r�
R · r� � r� sin � cos�� � ���
dS� � r�dr�d�� (3)
where we have introduced the complex electric fieldin the aperture, �a�r�, R�. We have also indicated withk � 2��� the wavenumber, and the field point Q atposition r� on the aperture plane (see Fig. 1) has polarcoordinates �r�, ���. R is the unit vector along thedirection to the observation point, with � represent-
Fig. 1. Coordinate systems used to calculate the antenna gain.
ing the angle formed by the direction to the observa-tion point and the optical axis and � being the anglemeasured in the plane of scan, i.e., perpendicular tothe optical axis �z��, as shown in Fig. 1. The integralin the numerator is calculated over the antenna ap-erture, whereas the integral in the denominator mustextend over the entire plane if there is any spilloverillumination in the case of reflector antennas.
The phase aberration function, �r��, in Eq. (3)defines the phase at point r� in the aperture plane,which accounts for any change in the optical pathlength resulting from the structural deformation ofthe primary reflector, the displacements of the sec-ondary reflector and the feed. Thus, it is in �r�� thatone can take into account the positions of differentfeed-horns in a FPA.
For aperture type antennas, the effective aperturecan be related directly to the antenna geometric area,Ag, by means of the aperture efficiency, �A��, �� (e.g.,see Ref. 2),
Aeff��, �� � Ag�A��, ��. (4)
Therefore,
G��, �� �4�Ag
�2 �A��, ��, (5)
�A��, �� �
��AP�
�a�r�, R�dS��2
Ag�
Ea2�r��dS�
. (6)
The on-axis gain, Go, is obtained by setting R · r�� 0, then we obtain
Go �4�Ag
�2 �o, (7)
�o �
��AP�
Ea�r��ej�r��dS��2
Ag�
Ea2�r��dS�
. (8)
If the phase is constant over the aperture the on-axisgain attains its maximum value, GM:
GM �4�Ag
�2 �M, (9)
�M �
��AP�
Ea�r��dS��2
Ag�
Ea2�r��dS�
. (10)
A case of special interest is that of uniform illumina-tion over the aperture, i.e., Ea�r�� � const over theantenna aperture and zero outside. Hence, we obtain�M � 1 and the ideal gain, Gideal, is then defined as
Gideal �4�Ag
�2 GM. (11)
Thus, we obtain the well-known result that the uni-form field distribution over the aperture gives thehighest gain of all constant-phase distributions overthe aperture [1].
B. Phase-Error Effects
In the previous section we showed that if the phasedistribution is constant over the aperture, the maxi-mum gain, GM, is obtained in the direction of theoptical axis, i.e. R · r� � 0. However, if a phase-errordistribution is present over the aperture, this may nolonger be the case. A phase-error over the aperture,i.e., deviations from uniform phase, may arise fromvarious causes, such as a displacement of the feed-horn from the on-axis focus (e.g., in FPAs), or distor-tion of the optical surfaces, or it may be caused byphase-error in the field of the feed-horn.
If the phase distribution is a linear function of theaperture coordinates, then it can be shown that thefar field is the same as that of the constant-phasedistribution but displaced with respect to the z� axis,i.e., the direction of peak-gain is no longer in thedirection of the system optical axis [1]. In the case ofarbitrary phase ditributions over the aperture, if thephase error does not deviate too widely from constantphase over the aperture, and if it can be decomposedinto a linear phase distribution and higher-orderterms, then we may write
�r�� � 1�r�� � ab�r��, (12)
where 1�r�� is linear in the coordinates over theaperture and causes an undistorted beam shift, i.e., achange in direction of the peak gain (now correspond-ing to � � �pk), whereas ab�r�� accounts for the truewavefront distortion. The shifted far-field beam canthen be considered to have arisen from a tilted aper-ture plane, i.e., from the aperture projected onto aplane normal to the direction of the peak gain, Rpk, asshown in Fig. 2. In the projected aperture the linearphase distribution term cancels out, leaving onlyhigher-order phase errors, i.e.,
and where r is the position of a point in the projectedaperture plane, indicated with AP, such that R · r� 0 for R � Rpk. If ab�r� � 0 then the field distribu-tion has constant phase over the projected apertureand the antenna gain in this aperture will be givenby [1]
GMP � GM cos �pk, (15)
where cos �pk � 1 for most radio astronomical appli-cations. Therefore, in the following sections we willrefer to the antenna gain and aperture efficiency asthe gain and aperture efficiency in the projected ap-erture plane, unless noted otherwise.
C. Main Contributions to the Aperture Efficiency
The aperture efficiency of an antenna is determinedby a number of phenomena and hence it can be writ-ten as the product of a number of individual contri-butions (e.g., see Ref. 3):
�A��, �� � �spill�taper��, ���phase��, ��, (16)
where �spill is the spillover efficiency, �taper is the taperefficiency, and �phase takes into account all phase-erroreffects causing a distortion of the wavefront. We havealso assumed that ohmic losses are negligible andthat the aperture is unblocked. The spillover effi-ciency includes all spillover contributions from thefeed, subreflector, diffraction, etc.,
�spill �
�AP
Ea2�r�dS
�
Ea2�r�dS
. (17)
�taper accounts for the aperture illumination taper dueto the feed and the reflector geometry,
�taper��, �� �
��AP
Ea�r�ejkR·rdS�2
Ag�AP
Ea2�r�dS
, (18)
and finally, �phase accounts for the residual high-orderphase distortions of the wavefront at the apertureplane due to optical aberrations, surface errors, ormisalignments, etc.,
�phase��, �� �
��AP
�a�r, R�dS�2
��AP
Ea�r�ejkR·rdS�2 . (19)
In the direction of the peak gain R · r � 0, as weearlier mentioned, and thus the R � ��, �� depen-dence can be dropped from �taper and �phase.
In the case of on-axis, dual-reflector systems thecentral subreflector and its support structure cause apartial shadowing of the aperture, which leads to aloss of efficiency. To take this effect into account theintegral at the numerator of Eq. (13) can be written inthe case of a partially blocked aperture:
�APblock
�a�r, R�dS ��AP
�a�r, R�dS ��subr
�a�r, R�dS
(20)
where APblock represents the area of the apertureplane subtracted of the blocked part, AP indicates asusual the full area of the aperture plane and subrindicates the integration area over the subreflector,assuming this is the main source of blockage. Bysubstituting Eq. (20) into Eq. (13) we thus obtain
�A��, �� �
��AP
�a�r, R�dS�2
Ag�
Ea2�r�dS
��1 �
�subr
�a�r, R�dS
�AP
�a�r, R�dS �2
,
(21)
where the first term at the right can once again bewritten as in Eq. (16) and thus the second term can beinterpreted as the blocking efficiency due to the sub-reflector,
�block��, �� ��1 �
�subr
�a�r, R�dS
�AP
�a�r, R�dS �2
. (22)
We note that in the direction of the peak-gain�Rpk · r � 0�, for a uniform, unaberrated �ab�r� � 0�field we find the well-known result
Fig. 2. Off-axis feed and tilted aperture plane geometry.
where Aprim and Asubr are the surface areas of theprimary and secondary reflectors, respectively. Ingeneral, the geometric blockage caused by the sup-port struts can be up to several times larger than theblockage caused by the secondary mirror, especiallyin open-air antennas. Therefore, the blockage effi-ciency given by Eq. (23) usually overestimates thereal efficiency and should be corrected including thestrip blockage of the plane wave and the blockagefrom the converging spherical wave between the pri-mary mirror and the subreflector (e.g., see Ref. 4).
3. Strehl Ratio
A. Strehl Ratio On Axis
While the main antenna-based figures of merit areusually, though not necessarily, defined in the farfield of the aperture, the Strehl ratio of an opticalimaging system is defined as the ratio of the aber-rated to unaberrated incoherent point spread func-tion (PSF, hereafter [5]).
When considering the optical system in receivingmode, the PSF refers to the instantaneous field dis-tribution in the focal plane of an optical imaging sys-tem produced by a far-field point source. For simplicitywe assume that the fields are emerging from the exitpupil of the optical system with a system focal lengthf, and converging toward the image plane. Let us sup-pose that the exit pupil is on an infinite plane locatedat z � 0, and with the normal unit vector in the direc-tion of the z axis, n � z� (see Fig. 3)1. The focal planeand the observation point in the far field defined by Rin the previous sections (where the optical system wasconsidered in the transmission mode) are in oppositedirections with respect to the x�y� plane. This will betaken into account in Subsection 3.C. Then, followingRefs. 5 and 6, the scalar field at point P at position ��in the paraxial focal plane (see Fig. 3) is given by
Ef���� ��AP�
Eex�r��e�jkf
��·r�dS�, (24)
where Eex is the field amplitude at a point Q at posi-tion r� on the system’s exit pupil and f is also equal tothe radius of curvature of the reference sphere cen-tered at point O in the focal plane. In the case of apoint source in the far field of the system Eex is uni-form over the pupil.
In Eq. (24) the substitution of the exit pupil for theantenna aperture plane, and the consequent use of r�in both cases, is justified by using the equivalent pa-rabola (e.g., in a dual-reflector system) and by the factthat when the point source object is at infinity, thenthe diameter (assuming a circular aperture) of theexit pupil can be substituted with the diameter of theentrance pupil, or main dish in a dual-reflector sys-tem (see Ref. 6, p. 184), and the system focal lengthwould be in this case the focal length of the equivalentparabola [7]. In other words, the spherical (i.e.,aberration-free) wavefront leaving the equivalent pa-rabola and converging to the focus is identified herewith the Gaussian reference sphere centered on theexit pupil. Then, we can state that the (unaberrated)incoherent PSF is simply the square modulus ofEf����, i.e., PSF � I���� � |Ef����|2.
Equation (24) is strictly valid in the absence ofphase errors that may modify the perfectly sphericalconvergent wave that was assumed earlier in thespecial case of an aberration-free wavefront. In themore general case of a distorted wavefront Eq. (24)should be rewritten as
Ef���� ��AP�
Eex�r��e�jkf
��·r�ej�r��dS�, (25)
where �r�� is the phase error term. The Strehl ratio,S, of the imaging system is then given by the ratio ofthe central (i.e., �� � 0) irradiance of its aberrated andunaberrated PSFs. From Eq. (25) S can be written inthe form [5,8]
So �I�0�
I�0�|�0�
��AP�
Eex�r��ej�r��dS��2
��AP�
Eex�r��dS��2 , (26)
where So � S��� � 0�. The Strehl ratio can also beused as a measure of the on-axis PSF away from itscentral irradiance peak, and thus we can write
S���� �I����
I�0�|�0�
��AP�
Eex�r��e�jkf
��·r�ej�r��dS��2
��AP�
Eex�r��dS��2 .
(27)B. Strehl Ratio Off Axis
In Eqs. (24)–(27) the position in the paraxial focalplane of the central irradiance peak of the PSF was
Fig. 3. Coordinate frame at the exit pupil �x�, y�, z�� and position�� of point P at the focal plane. The field amplitude at point r� onthe system’s exit pupil is Eex�r��.
taken as the origin of a Cartesian system of axes andalso as the center of the (unaberrated) Gaussian ref-erence sphere [8]. The observation of an object pointoff-axis, which is equivalent to having the feed later-ally displaced in a microwave antenna, introducesboth a change in the position of the PSF peak (ordirection of peak gain in an antenna) and wave-frontaberration. The quasi-spherical (i.e., aberrated) wavewill be thus converging to a point displaced with re-spect to point O in Fig. 3. If �pk� represents the posi-tion of the off-axis PSF peak in the local plane, thenEq. (27) can be rewritten as
S���� �I����
I��pk��|ab�0�
��AP�
Eex�r��e�jkf
��·r�ej�r��dS��2
��AP�
Eex�r��e�jkf
�pk�·r�ej1�r��dS��2,
(28)
where ab and 1�r�� have been defined in Eq. (12)and Subsection 2.B. Thus, I��pk��|ab�0 represents thepeak irradiance of the unaberrated, off-axis PSF.
In Subsection 2.B we saw that by tilting the aper-ture plane so that it becomes perpendicular to thedirection of the peak gain, it is possible to write theaperture efficiency in terms of ab only. Likewise, inthe definition of the PSF it is possible to align the zaxis along the direction from the center of the exitpupil to the off-axis Gaussian image point, which canalso be taken as the origin of a new Cartesian systemof axes. The Gaussian image point is also the centerof curvature of the (tilted) wavefront, and for thispoint all path lengths from the spherical wavefrontwould be equal, in the absence of higher-order phasedistortions. Then, Eq. (28) takes the same form as Eq.(27), i.e.,
S��� �I���
I�0�|ab�0�
��AP
Eex�r�e�jkf
�·rejab�r�dS�2
��AP
Eex�r�dS�2 ,
(29)
where the peak of the PSF is now at point � � 0 in thenew system of axes, centered on the Gaussian imagepoint in the focal plane, and r now lies on a tiltedplane, AP, perpendicular to the direction of the off-axis PSF peak. Thus we have in the projected plane,
So �I�0�
I�0�|ab�0�
��AP
Eex�r�ejab�r�dS�2
��AP
Eex�r�dS�2 . (30)
C. Strehl Ratio and Aperture Efficiency
In this section we use the previous results to derive arelationship between aperture efficiency and Strehlratio. First, we use Eq. (13) to form the ratio of theaberrated and unaberrated aperture efficiency (in theprojected aperture plane), i.e.,
�A��, ���MP
�
��AP
�a�r, R�dS�2
��AP
Ea�r�dS�2 , (31)
with
�MP �
��AP
Ea�r�dS�2
Ag�
Ea2�r�dS
, (32)
where �A�R� � �A��, �� is the aberrated apertureefficiency measured in the generic direction R ���, �� (i.e., not coincident with the direction of thepeak gain, Rpk), for the general case in which thedirection of peak gain is not along the main opticalaxis of the system, as explained in Subsection 2.B.�MP � �A�Rpk�|ab�0 is the unaberrated aperture effi-ciency measured in the direction of the (off-axis) peakgain, i.e., �MP represents the peak aperture efficiencyas measured in the projected aperture plane. Recallingthat in the direction of the peak-gain Rpk · r � 0 (seeSubsection 2.C) the R-dependence can be droppedfrom �MP. From Eqs. (10) and (15) it also follows that
�MP � �M cos �pk � �M (33)
if �pk �� 1, where �M is the maximum aperture effi-ciency as defined in Subsection 2.A. From Eq. (32)and Eqs. (17) and (18) we also see that �M ��spill�taper.
Then, we note that Eqs. (29) and (31) have thesame form and, for small angles close to the opticalaxis it holds that
� · r � �R · r,
where we have defined � � ��f (see the discussion inRef. 9). However, since Eex represents the field pro-duced by a point source in the far field of the system,in order to conclude that Eqs. (29) and (31) are fullyequivalent one must assume that the incident field onthe optical system from a distant source has an apo-dization equivalent to that produced by the feed illu-mination on the antenna aperture (see Subsection3.A). In this case we can write Eex�r� � Ea�r�, and thus
Then, by comparing Eqs. (16) to (19) with Eq. (34) onecan see that, in general,
�MS��� � �spill�taper�R��phase�R�. (35)
Usually, however, one is interested in the apertureefficiency at the nominal position of the peak gain(i.e., at the center of the far-field beam), or equiva-lently at the center of the PSF, then it also holds that
�o � �MSo, �M � �spill�taper, (36)
So � �phase, (37)
with Rpk · r � 0 and �o � �A�R � Rpk� is the apertureefficiency in the direction of the peak-gain, corre-sponding to Eq. (8) in the projected aperture plane,i.e.,
�o �
��AP
Ea�r�ejab�r�dS�2
Ag�
Ea2�r�dS
, (38)
where we have not used the subscript “p” (for “pro-jected parameter”) in �o because of the approximationin Eq. (33). Therefore, Eq. (37) finally shows theequivalence between the Strehl ratio and phase effi-ciency.
Clearly, �M takes into account both taper and spill-over effects, whereas So is a measure of the phaseaberrations. Therefore, in the case of an unaberratedwave front, i.e. So � �phase � 1, the aperture efficiencyis �o � �M and depends only on the spatial distribu-tion of the field over the antenna aperture. Further-more, by explicitly writing the aberration function,ab�r�, in terms of the primary aberrations (e.g., seeRef. 5) it would be possible to derive the individualcontributions to the aperture efficiency by, e.g., coma,astigmatism, and curvature of field, which are usu-ally the most relevant aberrations in radiotelescopes.However, this is beyond the scope of this work andwill not be done here.
4. Comparison of Strehl Ratio and Aperture Efficiency
In this section we want to compare the values of theStrehl ratio, obtained from a ray-tracing optical soft-ware, ZEMAX (Focus Software [10]), and the associ-ated value of �phase, obtained through the numericalintegration of Eq. (19) and using the aperture fieldvalues computed by a Physical Optics program,GRASP9.3 (TICRA Engineering Consultants [11]).Several configurations have been analyzed and arediscussed below.
A. Description of Software Packages
The analysis has been conducted using theGRASP9.3 package, which is a commercial tool forcalculating the electromagnetic radiation from sys-tems consisting of multiple reflectors with severalfeeds and feed arrays. This package can use severalhigh-frequency techniques for the analysis of largereflector antennas, such as physical optics (PO) sup-plemented with the physical theory of diffraction(PTD), geometrical optics (GO) and uniform geomet-ric theory of diffraction (GTD), which require a mod-erate computational effort.
The PO technique is an accurate method that givesan approximation to the surface currents valid forperfectly conducting scatterers which are large interms of wavelengths. The PO approximation as-sumes that the current in a specific point on a curvedbut perfectly conducting scatterer is the same as thecurrent on an infinite planar surface, tangent to thescattering surface. For a curved surface, the PO cur-rent is a good approximation to the actual one if thedimensions of the scattering surface and its radius ofcurvature are sufficiently large measured in wave-lengths. The well-known GO method uses ray-tracingtechniques for describing wave propagation. SinceGO gives discontinuities in the total electromagneticfield, GTD is often applied in addition to GO, sinceGTD methods may account for diffraction effects.
On the other hand, ZEMAX is a classical opticaldesign tool based on ray-tracing methods, which com-bines three major categories of analysis in one pack-age: lens design, physical optics, and non-sequentialillumination�stray light analysis.
B. Calculation of the Aperture Efficiency with GRASP9.3
As described in Subsection 4.A, GRASP9.3 allowsseveral methods for electromagnetic analysis of re-flecting surfaces. An interesting tool of GRASP9.3,based on the ray-tracing, for calculating the aperturefield is the so-called “Surface Grid” [12]. This methodreturns the reflected magnetic field on the surfaceaccording to the formula: Hr � Hi � 2n�n · Hi�, whereHi is the magnetic incident field and n is the normalto the surface. The magnetic reflected field on thesurface, Hr, is then projected, with a phase adjust-ment, on the aperture plane. As described in Subsec-tion 2.B, when the feed is placed off-axis the apertureplane is tilted according to the direction of the peakgain. For a dual reflector configuration, the scatteringfrom the secondary and primary mirrors has beenanalyzed through the GTD technique and the “Sur-face Grid”, respectively.
This approach is particularly appropriate when thediameter-to-wavelength ratio of the primary reflectoris very large and when the observation point is in thenear field (such as the aperture plane case). Underthese conditions the PO method would be very time-consuming: in fact, it would require a huge number ofpoints on the reflector where currents need to beevaluated. Using the method described here to ana-lyze the primary reflector the diffracted field from the
edge of the reflector is not considered. However, thenumerical results obtained with this “hybrid” tech-nique have been compared with those obtained byapplying the PO method to both the primary andsecondary mirrors, resulting in a very good agree-ment between the two methods.
In order to calculate the aperture efficiency fromEq. (31) we use the complex electric field in the ap-erture plane, i.e. �a�r, R�, produced by GRASP9.3,which is tabulated through its real and imaginarycomponents. These can then be used to calculate theamplitude and the phase function of the field. Thecomplex electric field is finally read by a proprietarycode which evaluates Eq. (19) in order to determinethe phase efficiency.
C. Comparison of Results
The values of the Strehl ratio and phase efficiencyobtained with ZEMAX and GRASP9.3, respectively,have been compared using three different optical sys-tems. These systems have been selected to representstandard telescope designs, and the frequencies usedin the simulations cover the mm- and submm-wavelength regimes. For the electromagnetic analy-sis with GRASP9.3, we have always used a linearlypolarized Gaussian feed. Although more realistic feedmodels to describe circular horns could be adopted,for the sake of comparison with ZEMAX and to avoidintroducing any systematic error due to different feedillumination, we report the results obtained with aGaussian model only. The level of apodization inZEMAX has then been chosen to be consistent withthat produced by the Gaussian feed-horn.
1. Single-Dish AntennaFirst, we have carried out the comparison in the sim-plest possible case, i.e., an unblocked spherical reflec-tor antenna. This choice eliminates or minimizespotential discrepancies due to different handling inZEMAX and GRASP9.3 of effects such as multiplereflections, aperture blocking, and diffraction at sec-ondary surfaces.
The surface chosen for this simulation is sphericalbecause it ensures that spherical aberration will limitthe overall FOV to small ��1°� angles near the opticalaxis. This is required in order to avoid introducingfurther variables in the comparison between ZEMAXand GRASP9.3 due to the incidence angle of radiationover the aperture of the feed-horn in the focal plane,which may affect the coupling between the PSF andthe electric fields on the horn aperture. The selectedaperture was 105 cm in diameter with a f�# � 2 andthe simulations have been carried out at a wave-length of 500 �m. For the electromagnetic analysiswith GRASP9.3, a linearly polarized Gaussian feedhas been used with a taper level of �12 dB at 14°.
The results are shown in Fig. 4: the comparison hasbeen extended up to a maximum offset angle of �1.4°,or about 44 beams at 500 �m, and the maximummeasured difference between the Strehl ratio calcu-lated by ZEMAX and the phase efficiency calculatedby GRASP9.3 is 0.38% at the maximum offset angle.
We also note, however, a 0.25% discrepancy on bore-sight, which will be discussed in the next section.
2. Dual-Reflector Antenna: CassegrainConfigurationWe then analyzed the most common radio telescopedesign, consisting of a dual-reflector antenna. Wefirst consider the classical Cassegrain configura-tion, which we have derived from the design of the“Balloonborne Large Aperture Submillimeter Tele-scope” (BLAST) [13]. Compared to the original de-sign with a spherical primary mirror [14] and to thenewer telescope design with a Ritchey-Chretien op-tical configuration, the system analyzed here has aparabolic primary and a hyperbolic secondary. Thediameters of primary and secondary mirrors are181.61 and 42.76 cm, respectively, and the systemfocal ratio is 5. As in the single-reflector case, a lin-early polarized Gaussian feed has been used, butwith a taper level of �9 dB at 6°.
The results are shown in Fig. 5: the comparison hasbeen extended up to a maximum offset angle of�0.79°, or about 42 beams at 500 �m, thus quiteequivalent to the previous simulation. The maximummeasured difference between the Strehl ratio calcu-lated by ZEMAX and the phase efficiency calculatedby GRASP9.3 is about 0.59% at an offset angle ofabout 0.5°. We observe that the discrepancy between
Fig. 4. Plot of the Strehl ratio and of the phase efficiency at awavelength of 500 �m for the case of a spherical reflector 105 cmin diameter with a f�# � 2.
Fig. 5. Plot of the Strehl ratio and of the phase efficiency at awavelength of 500 �m for the case of a classical Cassegrain tele-scope. The diameters of primary and secondary mirrors are 181.61and 42.76 cm, respectively, and the system focal ratio is 5.
the two methods is also relevant �0.2 � 0.3%� foroffset angles near boresight and it is possibly moresystematic in this case than in the single-reflectordesign analysed in the previous section.
This on-axis difference is likely due to the rela-tively small secondary diameter to wavelength ratio,Dsec��, which may cause an on-axis decrease of theantenna gain due to diffraction effects from the edgeof the secondary. To test this hypothesis, we havescaled-up the BLAST telescope, while keeping con-stant the wavelength, in order to obtain an opticaldesign with a much larger Dsec�� ratio, comparable tothat used in the next section for the “Sardinia RadioTelescope”. We have thus obtained a telescope withthe same focal ratio at the Cassegrain focus but witha primary and secondary reflector diameter equal to12.2 m and 2.6 m, respectively.
The results are shown in Fig. 6: in this case thecomparison has been extended up to a maximum off-set angle of �0.33°, or about 116 beams at 500 �m.As expected, the discrepancy near the optical axis hasdecreased compared to both the single-dish and theoriginal BLAST cases. The maximum difference isabout 0.61%, thus still quite similar to that observedin the original BLAST design despite the much largeroffset angle in beam units used in the scaled-up tele-scope. These results indicate that diffraction effectsare calculated differently in GRASP9.3 and ZEMAX.
3. Dual-Reflector Antenna: GregorianConfigurationThe third system analyzed during this comparison isanother dual-reflector antenna, though in a Grego-rian configuration. In this case we have changed thewavelength to a larger value of 3 mm and have alsochosen a telescope with a much higher D�� ratio. Thebaseline design is in this case the “Sardinia RadioTelescope” (SRT [15]); however, we have convertedthe original shaped design of the SRT to a more stan-dard Gregorian configuration, keeping the same ap-erture (64 m) and system focal ratio (2.34) of the SRT.As in the previous two cases, a linearly polarizedGaussian feed has been used with a taper level of�12 dB at 12°.
The results are shown in Fig. 7: the comparison hasbeen extended up to a maximum offset angle of�0.136°, or about 42 beams at � � 3 mm, thus con-sistent with the simulations used for the single-dishand the BLAST configurations. The maximum mea-sured difference between the Strehl ratio calculatedby ZEMAX and the phase efficiency calculated byGRASP9.3 is about 1.9%, thus larger than in theoptical systems discussed above. However, in therange of offset angles where the Strehl ratio (or equiv-alently the phase efficiency) is �0.95, i.e. the rangewhich is normally targeted by the optical design ofdiffraction-limited telescopes, the difference betweenStrehl ratio and phase efficiency is �0.5%, consistentwith that observed in the BLAST telescope.
5. Conclusions
We have reviewed the main design parameters gen-erally used in evaluating the performance of opticaldesigns at both microwave and visible wavelengths.In particular, we have reviewed the classical conceptof antenna gain and the main contributions to theaperture efficiency, with special attention to phase-error effects. We then described the formalism withwhich to compare the aperture efficiency and its com-ponents with the Strehl ratio, which is the standardparameter used to evaluate the image quality of dif-fraction-limited telescopes at visible–IR wavelengths.We have shown that a simple relationship can befound between Strehl ratio and aperture efficiency:the Strehl ratio is equal to the phase efficiency whenthe apodization factor is taken into account.
We then compared these two parameters by run-ning ray-tracing software, ZEMAX and full PhysicalOptics software, GRASP9.3, on three different tele-scope designs: a single spherical reflector, a Casseg-rain telescope, and finally a Gregorian telescope.These three configurations span a factor of �10 interms of D��. The simple spherical reflector allowsthe most direct comparison between Strehl ratio andphase efficiency, as it is only marginally affected byedge diffraction effects. In this case we find that thesetwo parameters differ by less than 0.4% in ourZEMAX and GRASP9.3 simulations, up to an angle of
Fig. 6. Same as Fig. 5 for the scaled-up version of the BLASTtelescope. The primary and secondary reflector diameters equal to12.2 m and 2.6 m, respectively.
Fig. 7. Plot of the Strehl ratio and of the phase efficiency at awavelength of 3 mm for the case of a classical Gregorian telescope,with a primary reflector diameter of 64 m and a system focal ratioof 2.34.
about 44 beams off-axis. The other two configurationsare more prone to diffraction effects caused by thesecondary reflector, especially in the case of thesmaller Cassegrain telescope.
The phase-efficiency is the most critical contribu-tion to the aperture efficiency of the antenna and themost difficult parameter to optimize during the tele-scope design. The equivalence between the Strehlratio and the phase efficiency gives the designer�userof the telescope the opportunity to use the faster (andless expensive) commercial ray-tracing software tooptimize the design using their built-in optimizationroutines.
This work was partly sponsored by the Puerto RicoNASA Space Grant Consortium.
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