January 1, 2000 / Vol. 25, No. 1 / OPTICS LETTERS 1

Diffractive Alvarez lens

Ian M. Barton, Sham N. Dixit, Leslie J. Summers, Charles A. Thompson, Kenneth Avicola, and Julia Wilhelmsen*

Mailstop L-438, Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, California 94550

Received September 14, 1999

A diffractive Alvarez lens is demonstrated that consists of two separate phase plates, each havingcomplementary 16-level surface-relief profiles that contain cubic phase delays. Translation of these twocomponents in the plane of the phase plates is shown to produce a variable astigmatic focus. Both sphericaland cylindrical phase profiles are demonstrated with good accuracy, and the discrete surface-relief features areshown to cause less than l�10 wave-front aberration in the transmitted wave front over a 40 mm 3 80 mmregion. 2000 Optical Society of America

OCIS codes: 050.1970, 220.3630, 220.3620, 220.1000, 010.1080.

The concept of the Alvarez lens was first developed in1967 by Louis Alvarez for ophthalmic applications.1 – 3

It consists of two phase plates that each have a cubicphase profile, one the inverse of the other. Whenthese plates are placed in registration, the resultingphase profile is a null, owing to the cancellation of thetwo phase profiles. However, if one of the plates istranslated in the plane of the phase profile, there isno longer perfect cancellation of the two phase profiles.The residual phase variation is the differential of thecubic profiles, resulting in a quadratic phase variation.This quadratic phase variation is equivalent to a lens ofa certain focal length, as determined by the steepnessof the individual cubic phase profiles and the transla-tion distance. By varying the relative translation inthe x and y directions, one can generate an astigmaticfocus (or defocus), as illustrated in Fig. 1. The versa-tility of the Alvarez lens allows for dynamic correctionof arbitrary astigmatic aberrations, e.g., thermal lens-ing in high-power laser systems. Other versions alsoexist that correct for other distortions such as coma.4 – 6

The Alvarez lens was invented more than threedecades ago; however, it is not widely used because offabrication difficulties. The profile of each componentof the lens can be fabricated either as a continuouscubic profile or as a quantized diffractive structure.Traditional continuous fabrication techniques such assmall-tool polishing can be prohibitively expensive, es-pecially for large-aperture optics. An additional limi-tation is that if the desired correction requires that thecomponents have surface profiles with large excursion,then the fabrication difficulty (and hence the cost) in-creases considerably.

The Alvarez lens can also be fabricated as a diffrac-tive structure, in which, as with a Fresnel lens, thephase profile of each component is reduced to a modulo2p version before further quantization to a number ofdiscrete phase levels. The diffractive equivalent offersan advantage over a standard Alvarez lens in ease offabrication, since standard photolithography7 can beused. Also, the thin nature of the components makesthe diffractive lens ideal for high-power laser applica-tions and allows for the size of each component to bemade arbitrarily large without increasing the depthof the surface-relief profile. This latter fact is espe-cially important for the Alvarez lens, since the aperturesize of each component determines the dynamic range

0146-9592/00/010001-03$15.00/0

of astigmatic correction that can be achieved with thelens for a specific design. The fabrication errors thata diffractive Alvarez lens incurs will lead to mostlysmall-scale length aberrations (depending on the pixel-lation size), in contrast with the long-scale length er-rors that are usually incurred during surface-polishingmethods. Depending on the optical system that is inuse, the unwanted small-scale modulation could be spa-tially filtered out.

We describe the fabrication and the optical perfor-mance of what we believe to be the first diffractive Al-varez lens. The original Alvarez lens was designed forthe generation of a variable spherical focus by transla-tion in a single axis. In this original design, variableastigmatism is possible by translations along a diago-nal. We use a modified version of Alvarez’s formula,1

where variable astimatism is generated along the axis

Fig. 1. Schematic showing the operation of the Alvarezlens. Translation of the two complementary cubic phaseplates with respect to each other along the x axis �y axis�generates focus (defocus) in that axis. A variable astig-matic lens is formed by control of the relative displacementin both directions.

2000 Optical Society of America

2 OPTICS LETTERS / Vol. 25, No. 1 / January 1, 2000

of translation. The surface-relief profile of the firstof the Alvarez lens components (phase plate A) is de-scribed by

z � ax3 2 bx 2 cy3 1 dy , (1)

where z is the surface-relief height. We chose theparameters to be a � 1.634 3 1027 mm22, b � 2.593 3

1024, c � 4.085 3 1028 mm22, and d � 1.335 3 1024.The second phase plate (B) has the inverse of thisprofile. Both phase plates were fabricated to havedimensions of 92 mm 3 132 mm.

We fabricated the two components of the diffractiveAlvarez lens in fused-silica substrates, using standardmultistep photolithography. Each element was com-posed of 16 phase levels, which meant that eight maskswere required in total (four for each element). Theetching step was performed with a chemical etch of abuffered HF acid solution. The surface-relief heightsof the elements (denoted A and B) were measured witha white-light interferometer and are detailed in Fig. 2.The expected heights are also included for comparison.

We tested the Alvarez lens by setting up the twoelements in an interferometer and measuring thetransmission wave front with them in various configu-rations. The operational aperture was designed to be40 mm 3 80 mm and was used for all measurements.Note that these measurements were performed withHe–Ne light �l � 633 nm�, as opposed to the designwavelength of 600 nm, which causes a slight phaseoffset from the optimal in the measured wave fronts.With the two elements in perfect registration, thewave-front aberration was found to be approximately0.14l (peak-to-valley variation for 633-nm light). Thetransmission wave front contains small scale-lengthaberrations caused by the mismatch of the surface-relief heights in the two elements; however, the aber-rations are very small in magnitude, contributing lessthan 0.1l modulation. The long scale-length varia-tion is predominantly from the two substrates, whichwere each measured to have a 0.05l transmission wavefront before fabrication.

When plate B is translated by 10 mm and plate Aby 210 mm in the y direction, a cylindrical phaseprofile can be seen (Fig. 3). This profile is quadraticin the y direction, with a power of 2.95 waves, andmaintains the 0.17l wave front along the x axis. Thisprofile is as expected from theoretical calculations.An elliptical phase profile is produced when plate Bis translated diagonally by 210 mm and plate A by10 mm along both the x and the y axes. This profile isshown in Fig. 4, and the power of measured astigmaticfocus is in close agreement with the theoretical value.It follows that any astigmatic focus can be producedwith different translations, and phase profiles showingnearly eight waves of correction are observed withthis setup. The residual wave-front error (when tilt,power, and astigmatism are removed) was found tobe ,0.04l rms beyond the desired relative translationrange in each direction. These values are detailed inFig. 5 for y-axis translation, along with a comparisonof the measured and theoretical peak-to-valley wave-front excursion, also shown as functions of y-axis

translation. The efficiency of the elements was notmeasured, but from the measured phase-level heightsand previous analysis of similar elements we expect

Fig. 2. Comparison of the measured depth of each phaselevel with the expected value for each component of theAlvarez lens (A and B). The deviation of the measureddepths from the theoretical values is ,15 mm.

Fig. 3. Transmission wave front of a 40 mm 3 80 mm re-gion of the Alvarez lens when the two components are dis-placed along the y axis by 20 mm and registered inthe x axis. The resulting phase profile is a cylindricallens with some small fine-scale structure as a result ofquantization and fabrication errors in the diffractive ele-ments and of the test wavelength’s being different from thedesign wavelength.

Fig. 4. Transmission wave front of a 40 mm 3 80 mmregion of the Alvarez lens when the two components are dis-placed in each axis by 20 mm. The resulting phase profilecorresponds to an astigmatic lens with a 1:4 power ratio.

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Fig. 5. Performance of the diffractive Alvarez lens asa function of the relative translation (shear) of the twocomponents in the y axis is detailed. The measured peak-to-valley (P-V) wave-front excursion is compared with thatpredicted from theory over a 50-mm translation range.The residual rms wave-front error is also shown.

each component to have a diffraction efficiency of.95% (excluding Fresnel losses).

In conclusion, the diffractive version of the Alvarezlens has been shown to be a feasible optical devicethat can produce arbitrary astigmatic focus with goodaccuracy. Aberrations owing to mismatches in theheights of the diffractive structures were shown tocontribute less than 0.1l modulation to the resultingtransmitted wave front. Applications of the diffractive

Alvarez lens to high-power laser systems are currentlyunder investigation.

This work was performed under the auspices of theU.S. Department of Energy by Lawrence LivermoreNational Laboratory under contract W-7405-Eng-48.The authors gratefully acknowledge contributionsfrom Jerry Britten, Curly Hoaglan, Mike Johnson,Thad Salmon, and Tilman Stuhlinger through-out this project. I. M. Barton’s e-mail address isbarton12@llnl.gov.

*Permanent address, Stevens Institute of Tech-nology, Hoboken, New Jersey 07030.

References

1. L. W. Alvarez, ‘‘Two-element variable-power sphericallens,’’ U.S. patent 3,305,294 (February 21, 1967).

2. L. W. Alvarez and W. E. Humphrey, ‘‘Variable-powerlens and system,’’ U.S. patent 3,507,565 (April 21, 1970).

3. L. W. Alvarez, J. Am. Optom. Assoc. 49, 24 (1978).4. N. Lopez-Gil, H. C. Howland, B. Howland, N. Charman,

and R. Applegate, J. Opt. Soc. Am. A 15, 2563 (1998).5. M. Oka, N. Eguchi, and H. Suganuma, ‘‘Apparatus and

method for compensating coma aberration,’’ U.S. patent5,726,436 (March 10, 1998).

6. I. A. Palusinski, J. M. Sasian, and J. E. Greivenkamp,Proc. SPIE 3482, 96 (1998).

7. See, for example, H.-P. Herzig, ed., Micro-Optics:Elements, Systems, and Applications (Taylor & Francis,London, 1997).