OptikOptikOptik 121 (2010) 113

pa

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ing, O

Max

rorsusins ha

r 2008 Elsevier GmbH. All rights reserved.

esting for imaging in the THz spectral range becausethey are achromatic and nearly lossless (i.e. they neither

tion of the imaging performance of different arrange-ments of the OAPMs. Up to now the object was scanned

that astigmatism and coma are inherent aberrations dueto the off-axis geometry. But these analytic expressionsare not very demonstrative and the resulting aberrations

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Corresponding author at: Fraunhofer Institute of AppliedOptics and Precision Engineering, Optical Systems Department,

Albert-Einstein-Str. 7, 07745 Jena, Germany. Tel.: +493641 807251;

fax: +493641 807602.

of different arrangements of OAPMs are not straight-forward to estimate. Furthermore in [4] the optimal

0030-4026/$ - see front matter r 2008 Elsevier GmbH. All rights reserved.

doi:10.1016/j.ijleo.2008.05.024

E-mail address: claudia.brueckner@iof.fraunhofer.de

(C. Bruckner).have the inherent Fresnel losses of lenses nor absorptionof the bulk material). In THz time domain imagingsystems the Fresnel losses do not only lead to a reduceddetected electric eld amplitude but also cause pulseechoes. As lens material in the THz spectral range water-free plastics are preferred. Their absorption increasesstrongly above 1THz (300 mm). The off-axis geometryhas additionally the advantage to have no centralobscuration that damps the middle transferred spatial

and for the image quality only the object point on theoptical axis was important. For an object point in thefocus of the OAPM, the image is always an ideal pointimage. The imaging properties for off-axis eld pointsbecame important with the development of line and areadetectors. As shown in this paper, the arrangement ofthe OAPMs has a high inuence on the resultinggeometrical optical aberrations and the image is notdiffraction limited even for the large THz wavelengths.Analytical expressions for the wavefront error of

OAPMs can be found in [2,3]. These publications showKeywords: THz; Off-axis parabolic mirrors

1. Introduction

Off-axis parabolic mirrors (OAPMs) are used asoptical elements for imaging and focusing in THz-measurement systems [1]. OAPMs are especially inter-

frequencies. Within the off-axis apertures the 901deviation enables the highest possible focal length at acertain aperture diameter.The arrangement of the OAPMs is mostly realized

based on spatial requirements. There was no investiga-Optimal arrangement of 901 off-axis

Claudia Brucknera,b,, Gunther Notnia, An

aFraunhofer Institute for Applied Optics and Precision Engineer

Albert-Einstein-Str. 7, 07745 Jena, GermanybInstitute of Applied Physics, Friedrich-Schiller-University Jena,

Received 22 January 2008; accepted 25 May 2008

Abstract

The inherent aberrations of 901 off-axis parabolic mirdifferent combinations of OAPMs for collimating and focarrangement with coinciding axes of the parent parabolaOptics119

rabolic mirrors in THz setups

as Tunnermanna,b

ptical Systems Department,

-Wien-Platz 1, 07743 Jena, Germany

(OAPMs) are investigated. The resulting aberrations ofg the THz radiation are investigated. It is found that thes a minimal geometrical optical spot of confusion.

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arrangement for off-axis ellipsoidal mirrors was derivedfor a quasi-optical setup, i.e. for Gaussian beampropagation in the millimeter and submillimeter wave-length range. It is shown how the distortion of theGaussian beam by off-axis ellipsoidal and parabolicmirrors leads to mode conversion by straying intensityinto higher order modes. A conguration with minimaldistortion is derived. But this analysis gives noappreciation of the geometrical optical aberrations andthe imaging quality. The imaging performance of asingle lens in the THz spectral range was analyzed andoptimized with the optical design program ZEMAXin [5,6].In this paper, the analysis tools of ZEMAX are used

to demonstrate the inherent aberrations of OAPMs.First a detailed analysis of the geometrical opticalaberrations of a single OAPM is given. Then, aberra-tions of different arrangements are demonstrated.Within this analysis the optimal arrangement of twoOAPMs is found, resulting in mostly cancelled geome-trical optical aberrations.

Fig. 1 shows the setup and the imaging performanceof a conguration of one of these OAPMs and aparaxial lens. The coordinate system in the object planeis shown in Fig. 1, upper left): the positive y-axis of theright-handed coordinate system shows to the left, thepositive x-axis shows into the drawing plane and thez-axis shows into the direction of the optical axis. Forthe object the positive y-axis shows to the right and thepositive x-axis shows into the drawing plane. The eldpoints investigated are the point on the optical axis (0, 0)and the off-axis eld points (x, y) (4mm, 0); (4mm,0); (0, 4mm); (0, 4mm). The spot diagram (Fig. 1,upper right) serves for a rst appreciation if the system isdiffraction limited. It shows beneath the circle ofconfusion the Airy disk (black circle). A system can beconsidered as diffraction limited if the circle of confu-sion is smaller than the Airy disk. For the axis point theimage is ideal point shaped. For the off-axis eld pointsthe circle of confusion is in the same order of magnitudeas the Airy disk. Thus, even for the off-axis eld pointsthe optical system is diffraction limited. But forwavelengths smaller than 750 mm or frequencies above0.4 THz the image of the off-axis eld points is not

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: lay

C. Bruckner et al. / Optik 121 (2010) 113119114Fig. 1. Imaging properties of a single off-axis parabolic mirror2. Aberrations of a single OAPMs

A typical OAPM is used having an on axis focallength of 59.7mm. The angle of reection is 901. Thusthe off-axis focal length is 119.4mm. The mirror has afree aperture of 63.5mm. The wavelength for theinvestigation is 750 mm corresponding to 0.4 THz.grid distortion (lower right).diffraction limited and geometrical optical aberrationsdominate the image formation. For the classication ofthe wavefront errors the optical path difference (OPD)map is used (Fig. 1, lower left). The OPD map shows theintersections in x- and y-direction through the wave-front in the exit pupil. The maximum scale of the OPD isa quarter wavelength. A system is also considered to bediffraction limited if the OPD is smaller than a quarter

out (upper left), spot diagram (upper right), OPD (lower left),

wavelength. The OPD analysis shows clearly astigma-tism for the eld points (0, 4mm) and (0, 4mm). Forthe eld point (0, 4mm) the curvature of the wavefrontin y-direction (tangential) is too strong, whereas thecurvature of the wavefront in x-direction (sagittal) is tooweak. Thus, the focus of the tangential rays is in front ofthe ideal focus, whereas the focus of the sagittal rays isbehind the ideal focus. For the eld point (0, 4mm)the relation is reversed. For the eld points (4mm, 0)and (4mm, 0) the apparent wave front errors are verylow. But the spot diagram shows that the aberrations inx-direction must be in the same order of magnitude asthe aberrations in y-direction. The reason for this is thatthe astigmatism for eld points in x-direction is rotatedabout 451. This aberration is only visible from thecomplete wavefront map. Fig. 2 shows the wavefronts ofthe off-axis eld points in the exit pupil. The uppergures show the astigmatism rotated about 451. Fig. 1(lower right) shows the grid distortion. The grid width inx- and y-direction is 8mm. The system has a bowingdistortion as it can be present in off-axis systems. Thismeans the distortion is only pronounced in y-direction.The maximum distortion is 5.28% for the eld point

astigmatism under 451. The other aberrations are oneorder of magnitude smaller than the maximum astigma-tism and are not given in the table for the sake of clarity.

3. Correction possibilities

Replacing the paraxial lens in Fig. 1, upper left, by asingle lens only the spherical aberration can be correctedby using an aspherical lens with higher-order asphericalcoefcients. Then the axis point can be imaged as anideal point. But a single lens cannot correct the inherentastigmatism of the outer eld points. Also, the inherentastigmatism cannot be corrected by using different conicconstants for the OAPM in x- and y-direction. Forcorrection every eld point needs an own conic constantin x- and y-direction. Thus, an improvement in theimaging quality for one eld point will yield degradationfor another eld point. But an adept arrangement of theOAPMs could lead to the introduction of the sameaberrations with invert sign, such that a big part of theaberrations is cancelled. The following sections willoutline this possibility in detail.

The layout of this arrangement is shown in Fig. 3. Thespot diagram shows that the image is diffraction limited

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for t

C. Bruckner et al. / Optik 121 (2010) 113119 115(x, y) (4mm, 4mm).For a quantitative analysis of the wavefront errors the

Zernike fringe coefcients are used. The relationshipbetween Zernike polynomials and third-order aberrationsis derived in [7]. The Zernike fringe coefcients with thehighest contribution to the aberration polynomial arethat for astigmatism Z5 and Z6. Table 1 shows thecoefcients for the astigmatism under 01 and the

Fig. 2. Wavefront analysis for a single off-axis parabolic mirror(0, 4mm) (lower left), and (0, 4mm) (lower right).

he eld points (4mm, 0) (upper left), (4mm, 0) (upper right),4. Two OAPMs with coinciding axes of theparent parabolas

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at

ptikTable 1. Zernike fringe coefcients Z5 and Z6 for astigmatism

Coefcient Designation 4mm, 0

Z5 Astig. @01 0.011Z6 Astig. @451 0.178

C. Bruckner et al. / O116for all eld points. The OPD diagram shows that thewavefront aberrations are only a very small fraction ofthe wavelength. The analysis of the Zernike fringecoefcients (Table 2) constitutes that result: the 451astigmatism for the x-direction and the 01 astigmatismfor the y-direction cancel for the most part. A smallamount of 01 astigmatism remains for all eld pointsleading to a liform point image in y-direction for alloff-axis eld points. To demonstrate the spot gure andwavefront aberration in Fig. 4 the spot diagram andOPD map are shown for a wavelength of 100 mm. Itshows that this system is diffraction limited up tol 100 mm or n 3THz. In y-direction the distortionadds and is also of the bowing type. The maximum

Fig. 3. Imaging properties of two OAPMs with coinciding axes

(upper right), OPD diagram (lower left), grid distortion (lower righ

Table 2. Zernike fringe coefcients Z5 and Z6 for astigmatism at

parabolas

Coefcient Designation Field (4mm, 0) Fi

Z5 Astig. @ 01 0.012 Z6 Astig. @ 451 0.00001 and 451 for a single off-axis parabolic mirror

4mm, 0 0, 4mm 0, 4mm

0.011 0.167 0.1900.178 0.000 0.000

121 (2010) 113119distortion amounts 10.09% at the eld points(x, y) (4mm, 4mm) and (x, y) (4mm, 4mm).

5. Two OAPMs rotated 901 about theircommon optical axis

The layout and imaging properties of an opticalsystem with two OAPMs rotated 901 about theircommon optical axis are shown in Fig. 5. The imagingquality is a little bit worse than that of the single OAPMwith the paraxial lens. The analysis of the Strehl ratio independence of the eld height delivered that the image is

of the parent parabolas: layout (upper left), spot diagram

t).

01 and 451 for two OAPMs with coinciding axes of the parent

eld (4mm, 0) Field (0, 4mm) Field (0, 4mm)

0.012 0.008 0.0090.000 0.000 0.000

ARTICLE IN PRESSptikC. Bruckner et al. / Odiffraction limited up to an object height of 2.7mm inx- and y-direction. The Zernike analysis (Table 3) showsthat 01 and 451 astigmatism are now present for eacheld point. The grid distortion diagram (Fig. 5, lowerright) shows that bowing distortion is now present inx- and y-direction. The maximum distortion amounts8.13% for the eld point (x, y) (4mm, 4mm).

Fig. 4. Spot diagram and OPD for two OAPMs with coincidin

Fig. 5. Imaging properties of two OAPMs rotated 901 about th(upper right), OPD (lower left), grid distortion (lower right).

Table 3. Zernike fringe coefcients Z5 and Z6 for astigmatism at

optical axis

Coefcient Designation Field (4mm, 0) Fi

Z5 Astig. @ 01 0.150 0Z6 Astig. @ 451 0.163 0121 (2010) 113119 1176. Two OAPMs rotated 1801 about theircommon optical axis

Layout and imaging properties are shown in Fig. 6.The spot diagram shows that the circle of confusion isthe largest for this arrangement. The same feature showsthe OPD diagram. The Zernike fringe coefcient

g axes of the parent parabolas at a wavelength of 100 mm.

eir common optical axis: layout (upper left), spot diagram

01 and 451 for two OAPMs rotated 901 about their common

eld (4mm, 0) Field (0, 4mm) Field (0, 4mm)

.170 0.152 0.153

.157 0.152 0.154

ARTICLE IN PRESSptikC. Bruckner et al. / O118analysis (Table 4) shows that the 01 and 451 astigmatismadd, respectively. The grid distortion diagram (Fig. 6,lower right) shows that the distortion cancels to zero.The eld curvature analysis delivered that the tangentialand sagittal eld points drift about 16mm away fromeach other for the outermost eld points. The Strehlratio analysis yielded that the image is only diffractionlimited for eld points up to 2mm.

7. Conclusion

It is shown that a 901 OAPMs introduces astigmatism,which cannot be corrected by a single lens. The astigmatismcan be reduced if a second identical OAPM is used forfocusing. If the OAPMs are arranged with coinciding axisof the parent parabolas the astigmatism cancels for the mostpart but distortion adds. If the OAPMs are arranged in away that they are rotated by 1801 about their commonoptical axis, the astigmatism introduced by both mirrorsadds such that the highest amount of astigmatism appears.

Fig. 6. Imaging properties of two OAPMs rotated 1801 about th(upper right), OPD (lower left), grid distortion (lower right).

Table 4. Zernike fringe coefcients Z5 and Z6 for astigmatism at

optical axis

Coefcient Designation Field (4mm, 0) Fi

Z5 Astig. @ 01 0.023 Z6 Astig. @ 451 0.320121 (2010) 113119In this case the image is not diffraction limited even for thelarge THz wavelengths. Distortion cancels for this arrange-ment. The amount of astigmatism arises with the eld size.Thus, the larger the object, the worse the image quality. Insummary, the ideal arrangement is the one where the twooff-axis parabolic mirrors have a coinciding axis of theirparent parabolas. But never the less, image distortionscannot be avoided at all.

Acknowledgment

This research was supported by the FraunhoferGesellschaft Internal Programs under Grant no. MAVO813907.

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ARTICLE IN PRESSC. Bruckner et al. / Optik 121 (2010) 113119 119

Optimal arrangement of 90deg off-axis parabolic mirrors in THz setupsIntroductionAberrations of a single OAPMsCorrection possibilitiesTwo OAPMs with coinciding axes of the parent parabolasTwo OAPMs rotated 90deg about their common optical axisTwo OAPMs rotated 180deg about their common optical axisConclusionAcknowledgmentReferences