diffractive optical elements based on fourier optical

$: Diffractive optical elements based on Fourier optical$
Diffractive optical elements based on Fourieroptical techniques: a new class of optics forextreme ultraviolet and soft x-ray wavelengths

Chang Chang, Patrick Naulleau, Erik Anderson, Kristine Rosfjord, and David Attwood

A diffractive optical element, based on Fourier optics techniques, for use in extreme ultraviolet�soft x-rayexperiments has been fabricated and demonstrated. This diffractive optical element, when illuminatedby a uniform plane wave, will produce two symmetric off-axis first-order foci suitable for interferometricexperiments. The efficiency of this optical element and its use in direct interferometric determinationof optical constants are also discussed. Its use in direct interferometric determination of optical con-stants is also referenced. Its use opens a new era in the use of sophisticated optical techniques at shortwavelengths. © 2002 Optical Society of America

OCIS codes: 070.2580, 050.1970.

1. Introduction

Coherent extreme ultraviolet �EUV� and soft x-rayradiation1 facilitates phase-sensitive techniques thatprovide new opportunities in various fields, e.g., bio-logical imaging, material characterization, and nano-technology. However, challenges are presented inthat very limited optical elements are available atthese wavelengths. No appropriate materials existfor lenses and prisms due to high absorption. Mostexperiments either utilize low efficiency diffractiveoptics such as Fresnel zoneplates, or glancing inci-dence reflection mirrors and normal incidence multi-layer mirrors that result in restrictive off-axis opticalsystems and a limited spectral region, respectively.Therefore, devising novel optical elements that caneffectively and efficiently achieve wave-front shapingis of crucial importance for researches conducted atEUV�SXR wavelengths. Here, Fourier optical tech-niques are introduced to accomplish the desiredwave-front manipulation.

In our first example of these techniques, which are

The authors are with the Center for X-Ray Optics, LawrenceBerkeley National Laboratory, Berkeley, California 94720. C.Chang, K. Rosfjord, and D. Attwood are also with the Departmentof Electrical Engineering & Computer Sciences, University of Cal-ifornia, Berkeley, California 94720. E-mail address for C. Changis [email protected].

Received 17 March 2002; revised manuscript received 28 June2002.

0003-6935�02�357384-06$15.00�0© 2002 Optical Society of America

7384 APPLIED OPTICS � Vol. 41, No. 35 � 10 December 2002

new to the best of our knowledge, we have designedand fabricated, based on Fourier optics techniques, adiffractive optical element that combines the func-tions of the grating and zone plate through a bit-wiseeXclusive OR �XOR� operation. By use of this com-pound diffractive optical element allows the efficiencyand the contrast of the interferometer to be greatlyincreased. This optical element has been used in anEUV interferometer to directly determine the indexof refraction at EUV wavelengths.2 Similar activi-ties are underway at soft x-ray wavelengths.

2. XOR Pattern

This XOR diffractive optical element is obtained bycombining a 50% duty-cycle binary intensity gratingand a 50% duty-cycle intensity zoneplate. The bi-nary grating and zoneplate are first pixelized, witheach pixel being either 1 or 0 for transmission andabsorption, respectively. The two pixelized patternsare then overlapped and compared pixel by pixel toproduce the resulting XOR pattern, i.e., at each pixelposition, if the pixel values of the grating and zon-eplate are the same �both 0’s or both 1’s�, the value ofthe corresponding pixel on the XOR pattern is 0.Otherwise, the value of the corresponding pixel onthe XOR pattern is 1.

For a 50% duty-cycle grating of period d, the trans-mitted intensity function is

G� x, y� �12

�1 � sgn�cos �x��, (1)

where � � 2��d.Similarly, for a 50% duty-cycle zoneplate of diam-

eter D and outermost zone width �r, the transmittedintensity function is3

ZP� x, y� �12

�1 � sgn�cos r2��, (2)

where r � x2 � y2 and � ��r�D � �r��.Expand these two patterns in their Fourier series,

G� x, y� � m��

� sin�m��2�

m�exp��jm�x�, (3)

ZP� x, y� � n��

� sin�n��2�

n�exp��jnr2�. (4)

Note that by comparing the Fourier series of a zon-eplate with a lens, one finds that the zoneplate func-tions as multiple lenses with nth order focal lengthfn � ��n��.

The XOR pattern of the combined grating and thezoneplate is obtained by forming

XOR� x, y� � G� x, y� � ZP� x, y� � 2G� x, y�ZP� x, y�

� m��

� sin�m��2�

m�exp��jm�x�

� n��

� sin�n��2�

n�exp��jnr2�

� 2�12

� m��m�0

� sin�m��2�

m�exp��jm�x��

� �12

� n��n�0

� sin�n��2�

n�exp��jnr2��

�12

� 2� m��m�0

� sin�m��2�

m�exp��jm�x��

� � n��n�0

� sin�n��2�

n�exp��jnr2�� . (5)

This combined diffractive element, when illuminatedby a uniform wavefront, has the interesting propertythat it produces two symmetric off-axis focal spots,�m, n� � ��1, 1�, at the back focal plane of the zon-eplate. Note that both the grating and the zoneplatehave to be of 50% duty cycle for the on-axis focal spotto disappear, i.e., m � 0 and n � 0 in the summation.The separation of these two beam spots �x can bedetermined by multiplying the two exponentials inEq. �5�, completing the square for x-terms, thus re-sulting in �x � 2�r�D � �r��d � 2�rD�d. Note thatthis separation is independent of wavelength �.Thus as the wavelength is varied for spectral deter-mination of the index of refraction, the focal length�distance from the XOR pattern to the sample mask�

varies, but the lateral separation of the two beamspots remains fixed. The invariance of the spot sep-aration over the wavelength allows the EUV inter-ferometer to operate at different wavelengthswithout the need of changing the image-plane samplemask. This is a desirable property for EUV inter-ferometers because the scale of the sample mask forEUV applications requires it to be micro or nanofabricated, and thus immutable after being made.

Simulation of the XOR Pattern. A computer simula-tion is performed to see if these patterns produce theexpected results. An XOR pattern of a grating �pe-riod d � 16 �m� and a zoneplate �outermost zone-width �r � 0.2 �m, diameter D � 400 �m� isproduced, as shown in Fig. 1�a�. This pattern is thenFresnel-propagated to the first order focal plane ofthe zoneplate and the resulting intensity distributionis shown in Fig. 1�b�. As expected, only two off-axisspots exist in this focal plane.

3. Efficiency of the XOR Pattern

The XOR pattern, as expressed in Eq. �5� gives theefficiencies of the individual orders. First, we needto determine the overall transparent area of this XORpattern. Because we know that the percent of thetransparent area on the grating and the zoneplate is1�2, we find that the overall transparent area of theXOR pattern to be 1�2 � 1�2 � 2�1�2��1�2� � 1�2from Eq. �5�.

Next, we calculate the efficiency of individual or-ders from their relative strength. From Eq. �5�, wehave, for m, n � 0,

�m,n � �4

m2n2�4

0

if m, n are both odd,

if m or n is even,

(6)

where ¥k�0� �1��2k � 1�2� � ��2�8� is used in the

calculation.Another way to look at this is that we can think of

this XOR pattern as a binary amplitude zoneplate,multiplied by a �-phase-shift grating that does nothave any absorption. Therefore the overall absorp-tion of this XOR pattern is the same as that of abinary amplitude zoneplate, i.e., 1�2 and the effi-ciency of its individual orders is given by multiplyingthe corresponding orders of the binary amplitudezoneplate and the �-phase-shift grating. The effi-ciency �m of a 50% duty-cycle �-phase-shift grating is

�m � � 4m2�2 for m � �1, �3, · · · ,

0 for m is even.(7)

The efficiency �n of a binary amplitude zoneplate is

�n � � 1n2�2 for n � �1, �3, · · · ,

0 for n is even.(8)

10 December 2002 � Vol. 41, No. 35 � APPLIED OPTICS 7385

By comparing Eq. �6� with Eq. �7� and Eq. �8�, weindeed see that the efficiency of the individual ordersof the XOR pattern, �m,n, is given by �m � �n, i.e., themultiplication of the corresponding orders of thephase grating and amplitude zoneplate.

4. Visible Light Experiment

A first XOR pattern, designed for proof-of-principletesting at visible wavelengths, is fabricated usinge-beam lithography4 to directly observe the intensitydistribution at the back focal plane. The pattern is

Fig. 1. Computer simulation of the XOR pattern: The parameters used in this simulation are set equal to the actual fabricated element.The pattern in �a� is obtained by taking the XOR of the binary grating and zoneplate. 4096 � 4096 pixels are used to generated thispattern. This pattern is then Fresnel propagated in a computer by one focal length, and the resulting intensity distribution is shown in�b�. A horizontal cross-section through the focal spots is also shown. The two symmetric off-axis first-order foci is clearly visible in thissimulation. The other two outer spots are caused by the third orders �m � �3� of the grating, with nine times lower intensity.

Fig. 2. Visible experiment is performed to directly verify the intensity distribution at the back focal plane of the XOR pattern. Forcomparison an OR pattern obtained by taking the bit-wise OR of a grating and a zoneplate is also fabricated. The effect of this OR patternis equivalent to that of a grating and a zoneplate placed in tandem, which is the conventional setup for interferometric experiments. �a�Shows that the intensity distribution at the back focal plane of the XOR pattern consists of only two symmetric off-axis foci, as predictedby the theory. As a comparison, the focal plane intensity distribution of the OR pattern is shown in �b�, which has three foci, with onestrong on-axis focus and two weaker off-axis symmetric foci. The grating used by the XOR and OR patterns in this visible experimenthas a period of 5 �m, and the diameter and the outermost zone width of the zoneplate is D � 5 mm and 2 �m, respectively. A He–Nelaser �� 633 nm� is used for illuminating the XOR and OR patterns.


defined by electroplating a 100-nm thickness ofnickel, which is highly absorptive in both EUV andvisible wavelengths. The grating used in this visibleversion has a period of 5 �m, the zoneplate diameteris 5 mm, and the outermost zone width is 2 �m. Ascreen is put at its back plane, which is 15.8 mm awayfrom this visible XOR pattern. A collimated He–Nelaser beam �� 633 nm� is then used to illuminatethis visible version of the XOR pattern and the re-sulting intensity distribution at the back focal planeis shown in Fig. 2�a�. As expected, the two symmet-ric off-axis foci are directly observable and there is noon-axis focus presented. The separation betweenthese two off-axis spots are measured to be 4 mm,which agrees with the designed value. As a compar-ison, an OR pattern made from the same grating andzoneplate is also fabricated and tested, as shown inFig. 3. Combining the grating and zoneplatethrough an bit-wise OR operation is equivalent toplacing them in tandem. Therefore this OR patterndemonstrates the back focal plane intensity distribu-tion of a traditional separate grating and zoneplatesetup. Figure 2�b� shows the resulting intensity dis-tribution at the back focal plane of this OR pattern.Three foci are clearly observed, with the strongestfocus on-axis and two weaker symmetric off-axis foci.The separation between the on-axis and the off-axisspots are measured to be 2 mm, which again agreeswith the designed value.

5. First use in Extreme Ultraviolet Interferometry

The XOR pattern employed in our first application toEUV interferometry is fabricated using the samee-beam lithography tool,4 and a scanning electron

microscopy image of the actual pattern is shown inFig. 4. The grating has a period d of 16 �m andcovers a 400 �m � 400 �m square area. The zon-eplate has a diameter D � 400 �m and a outermostzone width �r � 0.2 �m. Undulator beam line 12 atthe advanced light source provides the EUV radiationfor this measurement.5 The wavelength at whichthis measurement was performed is � � 16.53 nm �75eV� and the monochromator at the beam line is set at�� 1100.

This interferometer utilizes the strongest nonze-roth order, i.e., �m, n� � ��1, 1�, which has a theoret-ical efficiency of 4��2 � 1��2 � 4��4 � 4.1% as givenby Eq. �6�. Experimentally, the efficiency of thisXOR pattern is measured by recording the totalcounts on the CCD while scanning a knife-like beamstop transversely across the back focal plane. Start-ing with the beam stop placed at the back focal planesuch that the entire beam is blocked, as the beam stopslowly moves aside, allowing a fraction of light topass, the total count on the CCD increases. Theresult of this efficiency measurement is shown in Fig.5. The two abrupt steps at the center is caused bythe two symmetric off-axis first order foci, �m, n� ��1, 1�, being released one at a time by the scanning

Fig. 3. Microscope image of the OR pattern used in the experi-ment with visible light.

Fig. 4. Center part of the XOR pattern is shown. This diffractiveoptical element is obtained by taking the bit-wise XOR of a binaryamplitude grating and a binary amplitude zoneplate. The func-tionality of this XOR pattern is equivalent to that of a binary phasegrating overlapping a binary amplitude zoneplate, as discussed inthe text. The grating used here has a 16 �m period and thezoneplate has a 400 �m diameter and a 0.2 �m outermost zonewidth.


beam stop. However, when determining the effi-ciency of the �m, n� � ��1, 1� order, the effect ofundiffracted straight-through light needs to be re-moved. Because the position of the transverselyscanning beam stop is directly proportional to thefraction of the straight through light that passes it,

the effect of straight through light can be determinedby the constant slope of the two straight sections.After removing the effect of the straight through lightby least-square fitting the slope of the two straightsections, the individual strength of the �m, n� � ��1,1� order is shown to be around 4.0%, which agrees

Fig. 5. Efficiency of this XOR pattern is measured by scanning a knife-like beam stop across the focal plane. Starting with the beamstop placed at the back focal plane such that the entire beam is blocked, as the beam stop slowly moves aside, the total counts on the CCDincreases, allowing fractions of light to pass. The constant slope of the two straight sections results from the effect of zeroth order �straightthrough� light. The two abrupt steps at the center is caused by the two symmetric off-axis first-order foci being released one at a timeby the beam stop. Their strength is shown to be around 4.0%, which agrees with the theoretical value.

Fig. 6. Object wave, which consists of two converging spherical wavefronts, interferes with a reference plane wave, and the resultingintensity interference pattern is usually referred to as a CGH. This CGH is then binarized for nanofabrication by e-beam lithography.�a� Shows its binarized form. When illuminated by a uniform plane wave, this optical element reconstructs the object wave �twoconverging spherical waves� as shown in �b�. Note that the two spots are symmetrically off-axis.


with the theoretical value. Note that the definitionof diffraction efficiency for this element is the sum ofthe flux in the two desired orders divided by the totalincident flux on the pattern. We measured the dif-fracted flux to the two desired orders and the totalflux through the XOR pattern. The latter is as-sumed to be half of the total flux incident on the XORpattern, as half the pattern is transparent. There-fore the diffraction efficiency is obtained by dividingthe diffracted flux in the two orders by twice the totalflux through the XOR pattern.

In comparison with the traditional separate binarygrating and zoneplate setup, in which the �1st ordersof the grating are being focused by the first order ofthe zone plate with an overall efficiency of 1��4 �1.0%, this XOR pattern provides a 4-times improve-ment in theory. In practice, the required exposuretime is actually reduced by approximately 10 timesbecause of the fact that the substrates on which theseoptical elements are fabricated have finite absorp-tion, and only one substrate is needed in this case.This improvement in efficiency enables the first di-rect measurement of the refractive index at EUVwavelengths, where the two symmetric first-orderfoci are used as two arms of an interferometer, andtherefore enables a direct phase measurement for thedispersive part of the index of refraction.2

6. Comparison with the Computer GeneratedHologram

A computer generated hologram �CGH� having sim-ilar functions can be constructed by encoding the ob-ject wave, which consists of two converging sphericalwavefronts by a reference plane wave to form aninterference pattern �hologram�. This CGH, whenilluminated by a reference plane wave, would pro-duce two converging spherical wavefronts that can beused for interferometric experiments. These twospherical wavefronts would be identical and symmet-rically distributed with respect to the optical axis.

To nanofabricate this CGH, it is necessary to bina-rize the smooth areal interference pattern into 0’sand 1’s. This binarized pattern, shown in Fig. 6�a�,will then be used to produced the computer-aideddesign �CAD� file that nanofabricates the holographicoptical element. To see the effect of binarization onthe reconstructed wavefront, this binarized holo-graphic optical element is Fresnel propagated to theplane where the object wave converges to two pointsand the intensity distribution is shown in Fig. 6�b�.

The CGH can be optimized for optical flux through-put, while the XOR pattern is not specifically de-signed for maximum efficiency. However, it is verydifficult for the CAD program of an electron-beamcolumn to generated a CGH data file due to the largememory requirement imposed by the large amount ofvery small and irregularly-shaped structures partic-ularly at the outer edge of the CGH. In addition, thefiner details required by the CGH also make it moredifficult to nanofabricate. The XOR pattern pro-

vides a more practical solution in that it requiresmuch less computer memory and relatively less strin-gency in nanofabrication. For the XOR pattern thedigital data files of the grating and the zoneplate arealready accurately calculated and taking the bit-wiseXOR operation of the two data files is trivial in com-puters.

7. Conclusion

To the best of our knowledge, this paper demon-strates, for the first time, a novel diffractive opticalelement based on Fourier optics techniques. It isshown, both in theory and in experiment, that bycombining two diffractive elements, a grating and azoneplate, through a bit-wise XOR operation, the re-sultant optical element produced a new functionality:two symmetric off-axis foci with a higher efficiency.The two symmetric off-axis foci at the back focalplane are ideal for interferometric experiments.Specifically, it is shown that interferometric experi-ments that require better contrast and higher coher-ent power benefit from this XOR design due to thesymmetricalness of the intensity distribution at theback focal plane and the improved overall efficiency,respectively. Although useful at all wavelengths,this pattern has particular value at the short wave-lengths of interest here. This group of optical ele-ments shown in this paper brings sophisticatedFourier optical techniques to open new experimentalfrontiers in an area rich with opportunities on nano-meter scales and with element-specific identificationsand applications.

The authors thank Bruce Harteneck for help withscanning-electron microscopy and Phil Batson’s greatengineering team: Brian Hoef, Paul Denham, SenoRekawa, for excellent engineering support. Thiswork was supported by the Director, Office of Science,Office of Basic Energy Sciences, Division of MaterialsSciences and Engineering, of the U.S. Department ofEnergy under Contract No. DE-AC03-76SF00098.

References1. D. T. Attwood, Soft X-rays and Extreme Ultraviolet Radiation:

Principles and Applications �Cambridge University, Cam-bridge, U.K., 1999�.

2. C. Chang, P. Naulleau, E. H. Anderson, E. M. Gullikson, K. A.Goldberg, and D. Attwood, “Direct index of refraction measure-ment at extreme ultraviolet wavelength region with a novelinterferometer,” Opt. Lett. 27, 1028–1030 �2002�.

3. J. W. Goodman, Introduction to Fourier Optics, 2nd ed.,�McGraw-Hill, New York, 1996�, Chap. 5, Problem 14.

4. E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G.Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood,“Nanofabrication and diffractive optics for high-resolutionX-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2985�2000�.

5. D. T. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang,R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike,H. Medecki, and J. H. Underwood, “Tunable coherent radiationin the soft X-ray and extreme ultraviolet spectral regions,”IEEE J. Quantum Electron. 35, 709–720 �1999�.


diffractive optical elements based on fourier optical

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