January 1, 1990 / Vol. 15, No. 1 / OPTICS LETTERS 69

Nonlinear holography: a novel approach for creatingnear-infrared narrow-notch reflectors

Xiaohui Ning

American Optical Corporation, Southbridge, Massachusetts 01550

Received August 2, 1989; accepted October 20, 1989

It is shown that the nonlinearity in the response function of a holographic medium can lead to formation of gratingsat the sum and difference frequencies of the exposing standing-wave patterns. This nonlinear feature can beutilized to create reflection holograms at spectral regions where conventional techniques are not applicable.Specifically, it is suggested that near-infrared or infrared rejection filters that are sensor centered can be obtainedby using this approach.

Notch reflectors are useful devices for numerous ap-plications.' One way of achieving highly efficient re-flectors is to use the techniques of volume holography.Most holographic media are not sensitive to light thathas a wavelength greater than 630 nm. Holographicreflectors in the near infrared (NIR) can be fabricatedusing a combination of oblique exposure techniquesand swelling if the medium is dichromated gelatin.Exposure at an oblique angle is limiting. For exam-ple, it cannot be used to create sensor-centered holo-grams,2 which are often required for laser-protectionapplications. In this Letter we suggest a novel ap-proach for obtaining NIR holograms. This approachis based on the nonlinear response of the holographicmedia to incident radiation.

The change in the properties (index of refractionand absorption constant) of a holographic materialdue to exposure and processing can be described by itsresponse function. For a pure phase grating, the re-sponse function relates the refractive index to the totalexposure. In general, the response function is nonlin-ear because the index variation can never exceed acertain maximum value regardless of the total expo-sure. A typical response function is linear at low ex-posure and saturates exponentially as the exposureincreases. If the standing-wave pattern to which thematerial is exposed is a simple sinusoidal wave, thenonlinearity of the response function leads to forma-tions of holograms at harmonics of the standing-wavefrequency in addition to that of the linear hologram atthe fundamental frequency. This effect has been re-ported for Polariod DMP-128 material.3

If the material is exposed to multiple sinusoidalstanding waves, as in the case of multiplexing, theresultant grating structure is much more complicatedand interesting. In the following we analyze thisproblem in detail.

To create a multiplexed hologram, the material canbe exposed to multiple sinusoidal waves either simul-taneously or sequentially. For simultaneous exposurethe total effective exposure E (energy) is additive formutually incoherent beams and is given as follows:

N

E = E srIiti,i=1

(1)

where ri measures the sensitivity of the material to thelaser wavelength used for constructing grating i. Thestanding wave intensity for the ith grating is given byIs, and the exposure time is ti. If the total exposure islow, the index variation of the material is proportionalto the total exposure (for some photorefractive materi-als the index profile and the standing-wave patternmay differ by a constant phase). The response func-tion that describes the index variation An as a functionof the total effective exposure E can be expanded as

An = F(E) = F'(O)E + F"(°) E2 + ....2

(2)

If sequential exposure is used to obtain a multi-plexed hologram, the contribution of the ith exposureto the final index profile depends on all the otherexposures. The response function in general is a mul-tivariable function. In the case of two-wavelengthmultiplexing, the final index variation can be de-scribed as

An = G(E1 , E2)

= linear terms + '/2 GE,"(0, 0)E12 + l/2GE2 I'(0, 0)E2

2

+ GE1,E2"(O, O)E1E 2+ * * *,(3)

where El and E2 represent the individual effectiveexposures and the G" values represent various second-order partial derivatives.

To demonstrate the basic concept, let us assumethat the two standing-wave intensities are

I, = 1 + cos(Klz),

I2 = 1 + cos(K2z)

(4a)

(4b)

and that the sensitivities of the material and the expo-sure times are the same. If simultaneous exposure isused, the total effective exposure is proportional to the

0146-9592/90/010069-03$2.00/0 © 1990 Optical Society of America

70 OPTICS LETTERS / Vol. 15, No. 1 / January 1, 1990

0

EF'

1.0

0.8

0.6

0.4

0.2

ii.0.0 1 1 I l l l l I

300 400 500 600 700 800 900 1000 1100 1200Wavelength (nm)

Fig. 1. Reflection peak at 1138 nm due to the difference-frequency gratings.

sum of Eqs. (4a) and (4b). Substituting the totaleffective exposure into Eq. (3) and expanding the sec-ond-order terms, it can be shown that, in addition to adc bias term and the first-order terms and their har-monics, sinusoidal terms with wave numbers (K1 -K2) and (K1 + K2) are also obtained. Similarly, if thethird-order term is expanded, additional sinusoidalterms with wave numbers (2K1 - K2), (2K1 + K2), (2K2- K1 ), and (2K2 + K1 ) are obtained. It can also beshown that sinusoidal terms at the sum and differencefrequencies can also be obtained for sequential expo-sure, i.e., Eq. (3). The effect of these additional sinus-oidal components in the index profile can be analyzedusing either the coupled-wave approach4' 5 or the char-acteristic matrix formulation for modeling dielectricthin films.6 It can be shown that these terms lead tosignificant diffraction if the Bragg conditions for thesegrating structures are satisfied.

The practical implications of these nonlinear effectsare as follows: (i) By a suitable choice of K1 and K2,the rejection wavelength corresponding to a grating at(K1 - K2 ) can be in the NIR or IR spectral range.Thus, this technique can be used for creating NIR orIR holographic reflectors. (ii) By a suitable choice ofK, and K2, the sum and difference terms, (K1 + K2),(2K 1 - K2 ), etc., can be in a spectral range where theholographic material is not sensitive or where a laserhaving that wavelength is not available.

To study this concept further a hypothetical re-sponse function was chosen that is based on the as-sumption that the refractive index saturates exponen-tially at high exposure for all photorefractive materi-als. This function is

an(E) = nmax[1 - exp(--yE)],

that is due to the difference-frequency grating as aresult of nonlinear mixing. The diffraction efficiencyof this grating depends on the second-order nonlinearcoefficient as well as the wavelength, the thickness ofthe film, and the average refractive index of the mate-rial. For NIR or IR holograms, thicker films may berequired to obtain sufficient peak diffraction efficien-cies because of longer wavelengths. To optimize thenonlinear effects, further understanding of the nonlin-earities of the currently available holographic materi-als is needed. Experimental approaches should alsobe taken to determine the optimum processing andexposure techniques for making the nonlinear holog-raphy practical.

These results were experimentally verified. Thesimultaneous exposure method was used to create amultiplexed hologram in Dupont HRP-352-26 photo-polymer film. The line at 514 nm from an argon-ionlaser was split into two beams. One beam struck theflat film at normal incidence, while the other beam wasdirected to strike the film at an angle of incidence ofapproximately 54° in order to create a larger gratingperiod.

The measured transmission spectrum of the multi-plexed hologram is shown in Fig. 2. The two primary

1.0

0.8

0

.ECI

00

0.6

0.4

0.2

0.04-400 450 500 550 600

Wavelength (nm)

Fig. 2. Experimental result indicating alinear grating at 436 nm.

1.0

0.8

C0CI,

C0,(5)

where nmax is the maximum attainable index changeand y is the rate of saturation. Two standing waves ofthe form of Eqs. (4a) and (4b) were assumed. Thetotal index n is the sum of a constant dc index and theindex variation. The characteristic matrix method6

was used to calculate the transmission spectrum. Theresult is shown in Fig. 1. In this figure, in addition tothe two peaks at 368 and 545 nm, which are due to thelinear gratings, a third peak can be seen at 1133 nm

0.6

0.4

0.2

0.0 4-400

Fig. 3. Calculatedsponse function.

450 500 550 600Wavelength (nm)

650 700

third-order non-

650 700

spectrum based on the hypothetical re-

January 1, 1990 / Vol. 15, No. 1 / OPTICS LETTERS 71

peaks at 505 and 600 nmn are due to the linear part ofthe response function. The blue shift in the wave-length is attributed to the photopolymer shrinkageduring and after exposure and processing. It can beshown that the smaller third peak at 436 nm is due tothe third-order coupling term 2K1 - K 2, where K1corresponds to the 505-nm line and K 2 corresponds tothe 600-nm line. The rest of the second-order andthird-order terms cannot be observed experimentallyfor the following reasons. Most of them are in the UVrange of the spectrum, where the photopolymer andsubstrate absorb strongly. The two lines that are atlonger wavelengths (a third order at 739 nm and asecond order at 3189 nm) are too weak to be observed,partly owing to the small nonlinear coefficient andpartly owing to the fact that for the 3189-nm line thereare only a few grating periods in a 25-Am-thick film.The theoretical spectrum was also calculated using theresponse function given by Eq. (5). The result isshown in Fig. 3. The total exposure was assumed to bethe sum of Eqs. (4a) and (4b). The other parametersused for the calculations are average index of 1.5, dcbackground index of 1.48, nmax = 0.08, and y = 1.8.

Theory and the experiment agree well when the back-ground absorption of the material and the Fresnelreflections from the surfaces of the film are ignored.

Even though the current research is concerned withtraditional volume holography, the concept of a non-linear response function also applies to other hologra-phy-related areas, such as real-time (dynamic) holog-raphy, photorefractive phenomena, and holographicdata storage and processing. This will be the topic offuture publications.

References

1. The literature is extensive; see, e.g., H. J. Caufield, Hand-book of Optical Holography (Academic, New York,1979); R. Lytel and G. F. Lipscomb, Appl. Opt. 25, 3889(1986).

2. J. R. Magarinos and D. J. Coleman, Appl. Opt. 26, 2575(1987).

3. R. T. Ingwall and M. Troll, Opt. Eng. 28, 586 (1989).4. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).5. S. K. Case, J. Opt. Soc. Am. 65, 724 (1975).6. M. G. Moharamn and T. K. Gaylord, J. Opt. Soc. Am. 72,

187 (1982).