SURVEY OF OPHTHALMOLOGY VOLUME 54 � NUMBER 6 � NOVEMBER–DECEMBER 2009

HISTORY OF OPHTHALMOLOGYMICHAEL F. MARMOR, EDITOR

The Rainbow: From Ancient Greece to Modern OpticsIvan R. Schwab, MD, FACS,1 W. Barry Lee, MD, FACS,2 and David Bisno, MD3

1Department of Ophthalmology and Vision Sciences, University of California, Davis Medical Center, Sacramento,California, USA; 2Piedmont Hospital, Eye Consultants of Atlanta, Atlanta, Georgia, USA; and 3Atlanta, Georgia, USA

� 2009 byAll rights

Abstract. We provide a historical perspective on the rainbow, with a review of early research andobservations regarding the rainbow’s origin and a discussion of some of the major contributors to ourcurrent understanding of what the rainbow represents. An overview of the various types of rainbows isundertaken. We conclude with a discussion of the rainbow’s link to refraction, light, the visual system,and perception. (Surv Ophthalmol 54:714--720, 2009. � 2009 Elsevier Inc. All rights reserved.)

Key words. light theory � rainbow � reflection � refraction

A late afternoon summer shower moves across thelandscape leaving a stunning rainbow in its wake.Who has not been awestruck by its mystique? Theenigmatic rainbow represents a link between magicand science, right and left brain, science and art,and reality and imagination.

The rainbow has been observed, feared, orrevered throughout history. Of the surprisingly fewbiblical references to the rainbow, the best known isGod’s gift of a rainbow to Noah as a covenant thatthe world would never again be destroyed by water.34

Although Noah and his people saw the rainbow asa providential promise, the rainbow is not alwaysportrayed as a provider of mercy and good fortune.Cultures throughout Australia, Africa, and SouthAmerica saw the rainbow as a destructive mythicalgiant snake known as the rainbow serpent.10 Therainbow also held special meaning in mythology.Homer in the Iliad described Iris as the goddess ofthe rainbow, acting as a messenger or bridgebetween the gods.4 In some references, Iris wasregarded as evil, and in other, a source of promise.4

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Whether for good or evil, we have always sought anexplanation of this ephemeral sight.

Historical Perspectives of the Rainbow

Although we have no record of the precise yearwhen the rainbow, or bow, was first noticed, theTalmud discusses its creation at twilight on the 6th

day. Depictions of bows on cuneiform tablets of theSumerian and Babylonian cultures from 5,000 yearsago represent the first physical evidence of humanrecognition of bows. In 211 CE, Alexander ofAphrodisias was the first to document a dark stripbetween the double bow, an area between theprimary and secondary bow that is devoid of color,still known as Alexander’s phenomenon (Fig. 1).5

Aristotle, in about 322 CE, theorized that tinymirrors in the clouds were responsible for reflectionof light to create the bow, producing colors bya weakening of the light.3,33 He was the first to

0039-6257/09/$--see front matterdoi:10.1016/j.survophthal.2009.03.003

Fig. 1. Alexander’s phenomenon is the dark strip devoidof bow colors seen between the double bow as seen here.Note that the colors of the second bow are reversed whencompared to the primary bow. This reversal is caused bythe second reflection within the raindrops.

Fig. 2. Descartes recognized that there would be tworeflections and two refractions with the droplet, and whencombined with the law of refraction, explained the bow.The illustration is similar to Descartes’ original with hisexplanation.

THE RAINBOW 715

explain correctly the geometric shape of the arc andspeculate on the origin of the second bow.

Aristotle’s reflection theory persisted for nearly1,500 years. In the 13th century, Robert Grosseteste(Robert of Lincoln) first suggested refraction asa necessary component for the bow.5 In 1292, JohnPeckham, Archbishop of Canterbury, summarizedthe concepts developed in the 13th century, theimportance of refraction within a bow (although notyet completely understood), the establishment of itsradius at 42 degrees, and the importance of in-dividual raindrops rather than the cloud as a whole.

The Rainbow and Refraction

The law of refraction is essential to the under-standing of the bow. This law governs the behavior oflight rays as they propagate through an interfacebetween two transparent media by defining howstrongly the rays will be bent. Stated simply, it relatesthe angle of incidence (angle between the incidentray and normal ray) to the angle of refraction (anglebetween the refracted ray and normal ray), statingthat light rays are bent towards the normal ray whenpassing from a medium with a lower index ofrefraction to one with a higher. While the principleof refraction was widely recognized, it was not fullyunderstood until the 17th century. Kepler in 1604tried unsuccessfully to reduce the law of refraction totrigonometric functions. Between 1606 and 1609, hecorresponded with Thomas Harriot, who may havebeen the first to understand the law of sines as the lawof refraction. Harriot, in letters to Huygens, correctlydescribed the law of refraction (at least according toHuygens).2,8 His description was never published,

which is why we do not call the law of refraction,Harriot’s law. Instead, we call it ‘‘Snell’s law’’ becauseWillebrond Snell, in 1621, presented the correctexplanation of the law of sines and refraction ata scientific meeting and in an unpublished manu-script. It is believed by some that his audienceincluded a young Descartes, who, in his publicationof 1637 entitled Dioptics, would be the first to publishthis law of refraction, including his explanation of thebow with the correct geometry and shape.7

Descartes made the laborious calculations for thefirst- and second-order bows but declined to gofurther probably because the calculations wouldhave been formidable when done by simple raytracing and individual ray calculation.31 We will seethat he neglected the important issue of a wave frontand the infinite number of rays that would bestriking the droplet, making his explanationincomplete at best. Nevertheless, this is the theorythat is taught in most high school classes even today.Descartes’, or Cartesian, theory is illustrated in Fig. 2with the representation of individual rays.7

Descartes recognized that as his experimental raysmoved superiorly on the drop they would come toa critical point. At this point on the droplet edgeabove the axial ray, a light ray would enter the drop,be refracted and then proceed to the posteriorsurface of the drop (concave mirror) and bereflected at the steepest or greatest possible angle.This is known as Descartes’ ray, and all rays strikingthe surface of the water drop above the Descartes’ray would be refracted, strike the back of the drop

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(concave mirror), and be reflected at a less steepangle. He realized that this would explain theconcentration of light at the arc of 40--42 degrees,directly above the antisolar point, an imaginarypoint exactly opposite the sun along the celestialsphere. Descartes also realized that at this criticalangle, rays became trapped, creating a concentrationof light.7

Descartes also correctly calculated the position ofthe secondary bow at 50--52 degrees and recognizedthat light would be trapped above this bow(Fig. 3).7,31 Essentially, according to his theory,there would be no light distributed within the first-and second-order bows. Thereafter, Descartesexclaimed that the problem of the explanation ofthe bow ‘‘had been solved.’’ Unfortunately this wasnot the entire picture. To understand the colors ofthe bow, we must review the history of theunderstanding of light.

Light

Early observers did not appear to have a conceptof light, though they certainly realized when itbecame dark. Aristotle recognized that there wassuch an entity as light, but he considered itinstantaneous. In 1666, Newton conducted simpleexperiments, perhaps the greatest ever on color. Heused a prism to break light into its componentcolors and a second prism to reunite these colors.He proved that white light could be separated intoseven basic colors. He also took a second prism andshowed that the individual colors could not bebroken or divided further into additional colors.Nevertheless, when he presented this brillianttheory at the Royal Society in 1672, he was roundlycriticized. Sensitive to this criticism, he did not

Fig. 3. A diagrammatic representation of the singlereflection and dual refraction that produces the firstbow as well as the representation of the dual reflectionsand dual refractions that creates the secondary bow.

publish this work until 1704, by which time othershad already accepted and published his idea.11,24,25

The Rainbow Story (Continued)

In a time when the position of higher order bowsremained unknown, Sir Edmund Halley (who alsodescribed his namesake comet) calculated theposition of the third- and fourth-order bows in1700 using Newton’s calculus. To the surprise ofeveryone, these bows were not in the antisolarhorizon (opposite the sun), but rather 40� (third-order) and 46� (fourth-order) around the sun. Thesebows are very weak, and difficult to see, but there area few reports of the third-order bow being observedin nature. In a laboratory, many higher-order bowscan be demonstrated, but in nature only the primaryand secondary are common, with the third-orderseen rarely, if ever.12,13,18,20,23,32 The fourth-orderbow is not visible.

Witelo and others described interference orsupernumerary bows in the 1200s.18,35 The conceptof interference must be understood to gain un-derstanding of the supernumerary bows. Interfer-ence bows do not represent interference between twolight rays but rather interference of two differentportions of the same light wave. These bows resultwhen raindrops are uniform in size and demonstratesingle internal reflection from constructive interfer-ence (two light wave crests coinciding to producea larger wave) and destructive interference (one lightwave crest and one trough negating each other) ofeach color in the spectrum, resulting in several pastel-shaded bows near the apex of the bow. Langwith in1743 is credited with the description of the in-terference bows that can be seen within the 40--42�

primary bow (Fig. 4).13,18 Other extraneous bowshave confused observers (Fig. 5). The photographdepicts four bows and a single Alexander’s phenom-ena between the four suggesting there is somethingwrong with the theory, but, in fact, there is not. Thisphotograph does indeed show two sets of bows withtwo primary bows and two secondary bows, but thereare actually two ‘‘suns.’’ The primary sun is in the skyand the secondary sun is a reflection that createsa second antisolar point around which the second setof bows is coincident.

Other distortions of the bow still puzzledobservers because the bows were different, and evendifferent within the same bow. Some bows wouldexhibit different predominant colors depending onthe position of the bow fragment. For example,bows in the late afternoon often seem brighter nearthe ground with more red or green or both (Fig. 6).This did not seem consistent with the Cartesian

Fig. 4. Supernumerary bows can be seen beneath theprimary bow and are true interference bows. Note theAlexander’s phenomenon on the outside of the bow. Thesky is much darker outside the arc. (Photograph by DavinEnigl.)

Fig. 5. Four bows can be seen in this figure with anAlexander’s phenomenon. These unexpected bows arefrom a reflected light source in the flat body of waterbehind the photographer. (Photo by Alan Laws. Permis-sion to publish by Robert Greenler. Originally publishedin Rainbows, Haloes and Glories, Cambridge Publishers)

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theory. Some additional physical difference mustaccount for the color difference.

A new messiah of the bow was needed to explainthe observations that did not fit with the Cartesiantheory. One appeared in the form of ThomasYoung. In 1803 he explained Newton’s rings usingwave theory and interference of light, applying thistheory to explain the supernumerary bows. His wavetheory explained the interference bows by troughsand peaks in the concentration of light. Herecognized that, wherever there was a concentrationof light in the form of a peak, there would beanother bow with at least some colors.13,19,21,35--37 Aslightly different angle of reflection existed witheach successive light ray emanating from a drop;hence, there would be a whole family of these raysdepending on where they struck the back of theraindrop. Some of these rays would be reflected soas to reinforce the peaks of adjacent rays and somewould cancel the peak of an adjacent beam. Thisexplained the interference or supernumerary bows

in Fig. 4. His theory matched observations betterthan the previous Cartesian theory, which was silenton the interference bows. Young discovered thatdrop size did indeed make a difference in thepredominant color of the bow, and for that matterin the color and concentration of the colors of thesupernumerary bows.36,37 Young found that thelarger the drop, the less likely interference bowswere, because the larger drops have light pathwaysthat did not interfere or reinforce other adjacentrays. The light was simply too jumbled to createinterference bows. There was a narrower angularseparation of the light rays in the larger drops, andhence less chance for interference bows.36,37

Richard Potter, a physicist who had previouslyaccepted Newton’s theory, reconsidered Young’sideas in 183529 and felt that wave theory explainedinterference bows. He utilized a caustic, a reflectedor refracted light wave front produced by a curvedsurface or object that generates a smooth but curvedwave front of light. The caustic can represent this‘‘envelope’’ of reflected or refracted light rays or theprojection of the ‘‘envelope’’ of rays on anothersurface, such as one would see with a reflection froma silver ring or the bottom of a swimming pool. Hisformulation led scientists to realize that all bows areinterference bows, including the primary bow. Healso realized that each drop was responsible for onlyone color for each eye. Nonetheless, the Young/Potter theory did not explain the white bows seen inclouds at the angle of 38� from the antisolar point(Fig. 7).

It remained for George Biddle Airy in 1836 topropose a ‘‘complete theory’’ of the bow thatincluded the ‘‘Airy Integral’’ that allowed others tocalculate the location of the light for the primary

Fig. 7. The 38� bow or white bow can be seen in the leftportion of this figure as a partial large white ring. A glorycan be seen near the center of the photograph.

Fig. 6. The more robust colors of the bow nearer theground can be seen because drops tend to coalesce toapproximately 1--2 mm as they fall. This is the optimal sizefor primary bow colors, and especially favor red andgreen.

718 Surv Ophthalmol 54 (6) November--December 2009 SCHWAB ET AL

bows as well as the interference bows for each of thehigher order bows.1 Airy also considered that lightpassing through the raindrop produced a causticand, applying wave theory, he understood that therewould be interference bows. He also understooddiffraction and realized that light scattering wouldoccur within the previously sacrosanct area ofAlexander’s phenomenon, the area devoid of colorsbetween the primary and secondary bows, due totheir deviation angles (see Fig. 1). His integralproved this theory as well as that of the interferencebow.

The theory was still not truly ‘‘complete’’ becausethe 38� bow of Fig. 7 (white bow) could not beexplained as the basic nature of light was still notunderstood. Lord Rayleigh, who spent little time onthe bow, developed the theory of light scattering(and explained why the sky is blue) and extendedAiry’s theory to account for this effect from verysmall, almost molecule-sized drops.27 The ‘‘com-plete’’ theory was being altered and was comingcloser to an approximation of truth. Certainly withthese alterations in the ‘‘complete’’ theory of Airy,the white bow could be explained. Mascart in 1892explained that, as the drops became very small (on

the order of 30 microns), the white bow would becreated and, with even smaller droplets, the colorswould reverse.

Poincare and Watson devised the theory ofcomplex angular momentum of photons that wouldlater lead Feynman and others to propose thequantum electrodynamic theory of light.16

The Eye and Perception

A study of the eye and its visual mechanisms isimportant to gain an appreciation of the discoveriesconcerting perception. Aristophanes thought therewere three basic colors----purple, green, and yellow.Aristotle thought the three principal colors werered, green, and blue, perhaps the first to recognizethis concept.16,17,38,39

Thomas Young, the same who first used wavetheory to explain the interference bows, was alsoinstrumental in advancing the understanding ofcolor vision. In a rather brief description ina Bakerian Lecture before the Royal Society ofLondon in 1803, he enunciated the trichromacytheory of vision.36,37 He correctly surmised thatthere were three basic photoreceptors in the eyeable to see red, green, and blue. However, hethought that the rays of energy were otherwisecolorless and that color was only seen at the retinallevel and interpreted by the brain. Helmholtz andJames Maxwell expanded on Young’s ideas, and thisformulation is now called the Young-Helmholtz orYoung-Helmholtz-Maxwell theory of color vision.6

Maxwell contributed the confirmation of threeprimary photoreceptors in the eye related to thethree primary colors and recognized that colorblindness resulted from a deficiency of thesereceptors. He explained how primary colors werethe building blocks to generating additional colors,which ultimately led to the world’s first colorphotograph. Clearly, by this time in history, scientists

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had adequately described the optics of the bow andwere closing in on the next horizon—perception. AsYoung and others realized, the energy waves we callcolor could only create a colorful bow in the eye ofa beholder capable of deciphering those waves.

Once light waves strike the retina, the individualcones interpret the color depending on which of thethree types is stimulated. Three separate wave-lengths stimulate the three different photoreceptorsthat are responsible for all color sensations. Theoverlying ganglion cell travels along the visualpathways to the visual cortex within the occipitallobe and synapses in the color sensitive V1 cortexblobs. The signal eventually passes to the V4 areaoutside the visual cortex. The brain probably hasa center, or separate cortical area, that deciphersand interprets color utilizing the frequency of pulsesinitiated by each ganglion cell.16,26,38,39 It is here inthe visual cortex and additional cortical regions thatthe remaining mysteries of perception still reside,such as color constancy.15 Study of these mecha-nisms remains important to the complete under-standing of the bow.

The human eye/brain combination can distin-guish 8,000 colors at a single luminance, and a totalof 8 million shades and tints.9,14,21--23,26,30 The visualprocess somehow allows us to perceive a lemon asa lemon in almost any light level, the phenomenonof color constancy, even though the photometricmeasurement of the wavelength given off by thelemon in bright sun is vastly different from thewavelength of light by the same lemon in twilight.Part of this color constancy is the result of thebrain’s ability to compare and contrast adjacentcolors and to make those assumptions that tell usthis object is a lemon. This also helps us perceive thesame colors in a bow in many different situa-tions.9,14,16,17,21--23,26,30

Fig. 8. A and B: The author is shown using a simplepolarizing camera filter to prove the polarization of thebow.

The Family of Bows

The various members of the family of bows areseen in several different manifestations. The pri-mary bows seen after a summer thunderstorm areperhaps the best known. Other bows include thesecond-order 10--11� above the primary bow, thethird- and fourth-order bows around the sun, andhigher order bows found only in the labora-tory.18,20,28 Supernumerary interference bows areusually found just below the apex of the primarybow, as mentioned earlier.18,28 This 38� bow or fogbow (sometimes called the circle of Ulloa because ofhis eloquent description in 1748) is part of the bowfamily.20 Dew bows, seen on wet grass with the sun atyour back, are another member of this family. These

beautiful and subtle bows radiate away from you asa true conic section.

All of these bows are polarized (Figs. 8A and 8B),individual (i.e., you and your neighbor do not seethe same bow), uniocular (i.e., your right eye doesnot see the same bow as your left eye), andstereoscopic (i.e., the bows can be seen in three-dimensional quality), especially if some of the dropsthat produce the colors are closer to you thanothers.

The perceived size of a bow depends on ourperspective and the actual location of the dropletsreflecting and refracting the light. The bow can bea matter of feet or a mile or more away, but theangle that the bow subtends is the same regardlessof its position. That is, you cannot capture a bowwith a 50-mm photographic lens whether the bow iscreated in your backyard with a sprinkler, or 2 milesaway by a colossal thunderstorm. The bow subtendsthe same arc, 41� to 42�, regardless of whether it isclose or far away, and to get it all requires a wideangle lens. Even though the perception of its sizedepends upon comparison to familiar objects nearthe bow, the angle the bow subtends remains thesame. Color saturation also determines how close we

720 Surv Ophthalmol 54 (6) November--December 2009 SCHWAB ET AL

perceive the bow to be. The more saturated thecolors, the closer the bow appears, and the morelikely we are to see it in ‘‘stereo.’’

The Bow’s Future

The study of bows is a continuum from Aristotle’s‘‘mirrors in the clouds’’ to our current attempts tounderstand cortical perception. Bows have enteredour scientific understanding as tools. The sizes ofbows can be used to predict accurately the size ofdrops in clouds and thus the amount of water theycontain.

While the 17th (Descartes, Newton and others) andthe 19th (Young, Airy, Maxwell, Helmholtz, Hertz andothers) were important centuries for the understand-ing of light, refraction, and bow theory, we still argueabout the basic nature of light. Although we un-derstand the angular momentum of photons, wecannot both measure this force and determine itslocation. We seem to be on the threshold of un-derstanding visual perception, but still do not knowhow the eye/brain achieves color constancy. Perhapsthe 21st century will foster another major break-through in understanding of the bow.

References

1. Airy GA. On the intensity of light in the neighborhood ofa caustic. Camb Phil Trans. 1838;6:379--403

2. Andriesse CD. Huygens: The Man Behind the Principle[Translated by Sally Miedema]. Cambridge, CambridgeUniversity Press, 2005

3. Aristotle. Meteorologica. In Ross WD (ed): The Works ofAristotlevol, 3. Cambridge, MA, Harvard University Press, 1952

4. Boyer CB. The Rainbow: From Myth to Mathematics.Princeton, NJ, Princeton Universtiy Press, 1987

5. Boyer CB. Robert Grosseteste on the rainbow. Osiris. 1954;11:247--58

6. Campbell L, Garnett W. The Life of James Clerk Maxwell.London, Macmillan & Co, 1882

7. Descartes R. Discourse on Method, Optics, Geometry, andMeteorology [Translated by Paul Olscamp]. New York,Bobbs-Merrill Co, Inc, 1965

8. Dijksterhuis FJ. Lenses And Waves: Christiaan Huygens AndThe Mathematical Science Of Optics in the SeventeenCentury. Dordrecht, Kluwer Academic Publishers, 2004

9. Duke-Elder S. System of Ophthalmology. The Physiology ofthe Eye and of Vision, vol. 4. London, Henry Kimpton, 1968

10. Eliade M (ed): The Encyclopedia of Religion, vol. 12. NewYork, MacMillan, 1989

11. Fauvel J, Flood R, Shortland M, Wilson R, et al (eds): LetNewton Be: A New Perspective on his Life and Works.Oxford, Oxford University Press, 1989

12. Fraser AB. Inhomogeneities in the color and intensity of therainbows. J Atmospheric Sci. 1972;29:211

13. Greenler R. Rainbows, Halos and Glories. New York,Cambridge University Press, 1991

14. Hubel DH. Eye, Brain, and Vision. New York, ScientificAmerican, Inc., 1988

15. Jacobs GH. The distribution and nature of color vision amongthe mammals. Biol Rev Camb Philos Soc. 1993;68:413--71

16. Kenrick W, Pickard GW. Summary of progress in the study ofradio wave propagation phenomena. Proc IRE. 1930;18:649--68

17. Lamb R, Bourriau J. Colour: Art & Science. New York,Cambridge University Press, 1995

18. Lee RL. The Rainbow Bridge. University Park, PA, PennState Press, 2001

19. Longair MS. Theoretical Concepts in Physics. New York,Cambridge University Press, 1992

20. Lynch DK, Schwartz P. Rainbows and fogbows. Appl Optics.1991;30:3415--20

21. Lynch DK, Livingston S. Color and Light in Nature. NewYork, Cambridge University Press, 1995

22. Maxwell JC. On colour vision. Proc R Instn GB. 1872;6:260--7123. Minnaert MGJ. Light and Color in the Outdoors. New York,

Springer Verlag, 199324. Newton I. Opticks: Or, A Treatise of the Reflexions,

Refractions, Inflections and Colours of Light. London,Smith and Walford, 1704

25. Newton I. Opticks: Or, A Treatise of the Reflexions,Refractions, Inflections and Colours of Light. Based onthe Fourth edition, 1730. New York, Dover Publishing, 1952

26. Nicholls JB, Martin AR, Wallace BG. From Neuron to Brain.Sunderland, MA, Sinaure Associates, Inc., 1992

27. Nussenzveig HM. The theory of the rainbow. Sci Am. 1977;236:116

28. Palmer F. Unusual bows. Am J Phys. 1945;13:203--429. Potter R. Mathematical considerations of the problem of the

bow, showing it to belong to physical optics. Camb PhilTrans. 1838;6:141--52

30. Ryan SJ. Basic Science and Inherited Retinal Disease, vol. 1.Retina. St Louis, MO, CV Mosby Co., 1989

31. Sabra AI. Theories of Light from Descartes to Newton.London, Oldbourne Book Co., 1967

32. Sayili AM. Al Qarafi and his explanation of the rainbow. Isis.1940;32:16--26

33. Sayili AM. The Aristotelian explanation of the rainbow. Isis.1939;30:65--83

34. The Holy Bible, containing the Old and New Testaments,King James Version. New York, American Bible Society, 1999

35. Tricker RAR. Introduction to Meteorological Optics.London, Mills and Boon, 1970

36. Young T. The Bakerian Lecture: On the theory of lights andcolours. Phil Trans R Soc Lond. 1802;92: pp. 12--48

37. Young T. An Account of some Cases of the Production ofColours, not hitherto described. Phil Trans R Soc Lond.1802;92: pp. 387--97

38. Zeki S. A Vision of the Brain. London, Blackwell ScientificPublication, 1993

39. Zeki S. Colour coding in the cerebral cortex: the reaction ofcells in monkey visual cortex to wavelengths and colours.Neuroscience. 1983;9:741--56

The authors reported no proprietary or commercial interest inany product mentioned or concept discussed in this article. Thisarticle was funded in part by an unrestricted grant from Researchto Prevent Blindness, Inc.

Reprint address: Ivan R. Schwab, MD, Department of Ophthal-mology, 4860 Y Street, Suite 2400, Sacramento, CA 95817. e-mail:irschwab@ucdavis.edu.