White-light-modified Talbot array illuminator with a ... - OSA Publishing

White-light-modified Talbot array illuminator with a variable density of light spots Enrique Tajahuerce, Elvira Bonet, Pedro Andre´s, Carlos J. Zapata-Rodrı´guez, and Vicent Climent

A flexible array illuminator, comprising only two conventional optical elements, with a variable density of bright white-light spots is presented. The key to our method is to obtain with a single diffractive lens an achromatic version of different fractional Talbot images, produced by free-space propagation, of the amplitude distribution at the back focal plane of a periodic refractive microlens array under a broadband point-source illumination. Some experimental results of our optical procedure are also shown. © 1998 Optical Society of America OCIS codes: 050.1970, 070.6760, 070.2580, 070.2590, 230.3990.

1. Introduction

Many applications in optics make use of array illuminators ~AI’s!, i.e., optical systems that are able to transform a plane wave front into a two-dimensional array of bright spots with equal amounts of light. There are different approaches for implementing a monochromatic AI, as reviewed in Ref. 1. Among them, in this paper we focus our interest on Talbot AI’s, first introduced by Lohmann2 and based on the fractional Talbot effect provided by certain specially designed phase gratings.3 Recently, several Talbot AI’s have been reported that make use of binary phase gratings,4 multilevel phase gratings,5 or binary phase gratings in cascade.6 A general approach for designing Talbot AI’s has been presented by Hamam.7 On the other hand, the use of microlens arrays as phase gratings in a Talbot AI configuration, called by Lohmann a modified Talbot AI,8 seems also to be very productive.9 The idea is to combine the high compression and splitting ratio provided by the lenslet array and the flexibility and improved uniformity provided by the fractional Talbot effect. Advantages

When this study was performed, E. Tajahuerce, E. Bonet, and V. Climent were with the Department de Cie`ncies Experimentals, Universitat Jaume I, 12080 Castello´, Spain. P. Andre´s and C. J. ´ ptica, UniverZapata-Rodrı´guez were with the Departamento de O sidad de Valencia, 46100 Burjassot, Spain. Received 7 October 1997; revised manuscript received 25 February 1998. 0003-6935y98y04366-08$15.00y0 © 1998 Optical Society of America 4366

APPLIED OPTICS y Vol. 37, No. 20 y 10 July 1998

and limitations of this procedure are discussed by Besold and Lindlein.10 A modified Talbot AI is based on fractional Talbot images—also called Fresnel images—produced by the set of focal points of a periodic refractive lenslet array under monochromatic illumination. These Fresnel images are replicas of the amplitude distribution at the back focal plane of the lenslet array but with a reduced period compared with the original one, as depicted in Fig. 1. Although each individual focal pattern is the Fourier transform of the pupil function of a single microlens, the corresponding focal intensity distribution looks like a set of sharp intensity peaks. Under parallel illumination with wave number s ~s 5 1yl, where l is the wavelength of light!, the Fresnel images are obtained, by free-space propagation at distances R from the back focal plane of the lenslet, as given by9

S D

R 5 2d2s Q 1

N , M

(1)

where d is the period of the lenslet array, Q is an integer, and M and N ~N , M! are natural numbers with no common factor. The reduction factor r is equal to M when M is odd, and r 5 My2 when M is even.11 The reduction factor is unity at distances R such that N 5 0. In this case the self-images of the focal distribution are achieved. Thus the multiplicity m, i.e., the ratio of the number of bright spots per unit of area of the Fresnel image to that of the original intensity distribution at the focal plane, is m 5 r2. It is important to note that different Fresnel images have, in general, different multiplicities.

Fig. 1. Fresnel image generated by the focal amplitude distribution of a periodic refractive microlens array under parallel monochromatic illumination.

In spite of their potential interest, Talbot AI’s that are based on diffraction are restricted in principle to monochromatic illumination because of the wavelength dependence of light propagation in free space. Since the distance R in Eq. ~1! is wavelength dependent, note that, under parallel white-light illumination, each Fresnel image generated by the fractional Talbot effect is located in a different plane for each wavelength of the incident light, producing strong chromatic blurring at the output of the AI ~see Fig. 2!. In another context, an optical architecture constituted by a single diffractive lens has been reported to achieve an achromatic Fresnel diffraction pattern of a transparency under white-light illumination.12

This configuration is able to superpose, in a firstorder approximation, the monochromatic versions of a selected Fresnel diffraction pattern in a single plane and with the same size for all the wavelengths of the incident light. In this paper the above-described achromatic Fresnel transform setup is used for partially compensating the chromatic dispersion associated with the Fresnel images of the amplitude distribution at the back focal plane of the lens array. In this way our optical proposal is simply formed by two conventional elements: a refractive lenslet array and a diffractive lens. The separation between both optical components permits the selection of the multiplicity of the

Fig. 2. Chromatic dispersion of the Fresnel image in Fig. 1 under white-light illumination. 10 July 1998 y Vol. 37, No. 20 y APPLIED OPTICS

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Fig. 3. Single-zone-plate achromatic Fresnel transformer: optical arrangement.

replica of the array of focal points to be achromatized. For the first time, to our knowledge, we obtain a very simple AI with a variable density of bright whitelight spots at the output plane. In addition, our configuration preserves the high compression ratio provided by the lenslet array and the flexibility and homogeneity provided by the fractional Talbot effect. The residual chromatic aberration of our white-lightmodified Talbot AI is low even when the temporal spectrum of the point source spreads over the entire visible spectrum. In Section 2 we review the configuration and the basic properties of the achromatic diffractive Fresnel transformer. In Section 3 we describe how to apply these ideas to designing our flexible white-light AI based on a single refractive microlens array. Some experimental results of the proposed optical setup are given in Section 4. In what follows we restrict ourselves to the paraxial scalar theory of diffraction. When an input transparency is illuminated by a parallel light beam having a bandwidth of Ds 5 s2 2 s1, each Fresnel diffraction pattern appears axially chromatic dispersed. The Fresnel diffraction pattern located at a distance R0 from the diffracting screen for a reference wave number s0 is found at a distance R for a wave number s, such that s . s0

(2)

As shown in Ref. 12, a single on-axis blazed diffractive lens ~DL! is able to recombine in a first-order approximation the monochromatic versions of the above Fresnel diffraction pattern in a single picture and preserve the same size for all of them, providing an achromatic representation of the diffraction pattern. To this end the diffractive lens, whose focal 4368


z5

21

Îa

Z0,

(3)

where the dimensionless parameter a is defined as the ratio a 5 Z0yuR0u.

(4)

The geometry under consideration is depicted in Fig. 3. The achromatic Fresnel diffraction pattern is then obtained at a distance D09 from the DL such that D09 5

2. Variable Achromatic Fresnel Diffraction Pattern

R~s! 5 R0

distance for the reference wave number s0 is Z0, must be inserted at the source plane, something that implies converging spherical wave illumination, and the transparency must be located at a distance z from the DL, as given by

1

2 2 Îa

Z0.

(5)

The magnification M9 of this achromatic pattern with respect to the case of parallel illumination is M9 5

Îa D0 9 52 . z 2 2 Îa

(6)

It can be shown that, to obtain a real achromatic Fresnel diffraction pattern, i.e., D09 . 0, it is necessary that the diffractive lens be a convergent one and a must satisfy the inequality 0 , a , 4. We wish to emphasize that this optical configuration permits the achromatization of different Fresnel diffraction patterns in a sequential way with a single diffractive lens. After the diffractive lens with a focal length Z0 for s0 is chosen, the index R0, which characterizes the diffraction pattern to be acromatized, determines the value of the dimensionless parameter a @see Eq. ~4!#. And a fixes, by virtue of Eq. ~3!, the separation z between the diffracting screen

Fig. 4. White-light-modified Talbot array generator.

and the DL. In this way the Fresnel diffraction pattern to be achromatized can be selected in a simple way by a change in the position of the input along the optical axis. 3. White-Light-Modified Talbot Array Illuminator

The achromatic Fresnel transformer described in Section 2 is used for achromatizing different Fresnel images generated by the amplitude distribution at the back focal plane of a periodic microlens array, as shown below. First, the lenslet array is supposed to be perfectly nondispersive and aberration free. In this way, the location of the back focal plane and the spot spacing are maintained for every l. However, the focal amplitude distribution is not completely wavelength independent. The size of each light spot, albeit small, is not the same for all the wavelengths. In fact, it is related to the scale associated with the optical Fourier transform of the pupil of an individual microlens, and, consequently, it is proportional to lfyd. We discuss the influence of this last effect below. With these assumptions we recognize that the chromatic dispersion associated with the above Fresnel images is the same as that of a Fresnel diffraction pattern. In mathematical terms, the distance R in Eq. ~1! can be rewritten as in Eq. ~2! if the parameter R0 is selected such that

S D

R0 5 2d2 Q 1

N s0. M

(7)

From the above identification we conclude that the process of achieving an achromatic copy of a Fresnel image is equivalent to the achromatization of a Fresnel diffraction pattern with an index R0 as given by Eq. ~7!. Therefore, for implementing the whitelight-modified Talbot AI, the lenslet array is illuminated following the prescriptions in Section 2 by a converging broadband beam, and the DL is placed at

the source plane, as shown in Fig. 4. The achromatic condition for the Fresnel image that appears at a distance R0 given by Eq. ~7! fixes the distance h between the DL and the lenslet array, which is given approximately by h 5 2f 1 z 5 2f 2

1

Îa

Z0,

(8)

where the last equality is derived by the insertion of the form for z given by Eq. ~3!. If we are dealing with microlenses with a short focal length f compared with z, the distance h can be rewritten as h ' 2Z0y=a. In this way the achromatic Fresnel image appears at a distance D09 from the DL given by Eq. ~5!. Besides, if we take into account the magnification M9 of the achromatic pattern given by Eq. ~6!, which still holds, the separation d9 between two consecutive achromatic light spots results in d9 5 uM9u

Îa d d 5 , r 2 2 Îa r

(9)

where r is the reduction factor associated with the achromatized Fresnel image. This reduction factor results in an AI with a spliting ratio s, the number of beams generated out of one incoming beam,1 given by s 5 r2n,

(10)

where n is the total number of microlenses in the array. Note that this value is independent of the curvature of the spherical beam illumination. In our case, when nonparallel illumination is employed, it also seems convenient to use the multiplicity parameter m, which is the ratio of the number of white-light point sources per unit of area of the achromatized Fresnel image to that of the original focal 10 July 1998 y Vol. 37, No. 20 y APPLIED OPTICS

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Table 1. Multiplicity m at the Output Plane of the White-Light-Modified Talbot Array Generator Shown in Fig. 4 for Different Fresnel Images Characterized by the Value of Q 1 NyM

Q

NyM

m

0 0 0 0 0 1 0 1 0 2 0 2 0 0 1 0 0 0 1 0 1 0 0 1 0

1y2 1y3 3y10 2y7 3y8 0 2y5 1y2 3y4 0 2y3 1y2 3y7 5y8 1y4 5y6 3y5 4y9 3y4 7y10 1y6 7y8 4y7 1y3 4y5

0.17 0.22 0.23 0.23 0.81 1.00 1.75 2.10 2.14 3.34 3.61 4.68 4.69 5.40 6.11 6.14 7.54 9.00 10.83 11.33 12.12 12.13 12.84 15.43 15.56

CA~s! 5

intensity distribution. The multiplicity is then given by m 5 d2yd92. Therefore from Eq. ~9! we have m5

d2 ~2 2 Îa!2 2 r. 5 d92 a

(11)

To match the separation d9 of light spots at the output plane of the AI with the spacing d1 of a periodic micro-optical component array we want to illuminate, it is necessary that the value of m coincide with the quantity ~dyd1!2. It is very important to recognize that the multiplicity at the output plane of the achromatic AI achieved in this manner can be modified by the selection of different Fresnel images. Note that the selection of a different Fresnel image results in a different value for Q 1 NyM in Eq. ~1!, and, consequently, also for R0 in Eq. ~7!. In this way, on the one hand, the reduction factor r is different; on the other hand, as was pointed out in Section 2, the variation in R0 modifies the value of a, fixed by Eq. ~4!, and then a different distance z @see Eq. ~3!# is needed to achromatize the new Fresnel image. The variation of r and a in Eq. ~11! leads to a different multiplicity at the output plane of the AI. So a white-light AI with variable multiplicity can be achieved simply by a shifting of the lens array along the optical axis of the system. To make clear the ability of our proposal to vary the density of white-light spots at the output plane, we show the value of the multiplicity in terms of the selected Fresnel image characterized by the parameter Q 1 NyM in Table 1. In this table we assume 4370


that the focal length of the DL is Z0 5 2d2s0, and thus combining Eqs. ~4! and ~7! yields a 5 ~Q 1 NyM!21. In Table 1 we consider only Fresnel images with M less than or equal to 10 and Q less than 3. Table 1 shows, ordered by increasing multiplicity, the lower values of m. The irradiance distribution at the output plane of the white-light-modified Talbot AI suffers from residual chromatic aberrations. The resulting chromatic aberrations, both longitudinal and transversal, expressed as the difference in location and magnification, respectively, between the monochromatic version of the array of spots for s with respect to that obtained for s0 are given, in percentage, by the same equation12: 100 ss0 1 1 ~2 2 Îa! ~s 2 s0!2

,

(12)

which depends on only s0 and a. To minimize the chromatic aberration, it is convenient to choose the reference wave number s0 in such a way that s0 5 =s1s2 and to select a as small as possible. If we assume that the spectral content of the incident light is the whole visible region, it is enough to select a value of a such that a , 1 to obtain a maximum chromatic error of less than 7%. In this case we identify the values for s1 and s2 with those corresponding to the Fraunhofer lines C and F, respectively. That is, s1 5 1.52 mm21 and s2 5 2.06 mm21; then s0 5 1.77 mm21. It is important to recognize that our achromatic processor reduces drastically the chromatic dispersion inherent to the fractional Talbot effect ~of the order of some millimeters! but does not alter the different sizes of the colored spots at the focal plane ~of the order of some tenths of a millimeter!. Nevertheless, this fact does not limit our setup. If we assume circular microlenses with a radius d and we focus our attention on the smallest central lobe of its Fourier transform, which corresponds to s2 5 2.06 mm21 for white light, the encircled energy in this area corresponding to the most unfavorable ~s1 5 1.52 mm21! is 95% of that corresponding to s2. In mathematical terms we have

*

0.61

@ J1~2pAx!yAx#2A22pxdx

0

*

0.61

5 0.95,

(13)

@ J1~2px!yx# 2pxdx 2

0

where, of course, A 5 s1ys2 and J1 denotes the Bessel function of the first kind and the order one. The conclusive experimental results shown in Section 4 confirm the validity of this mathematical argument. 4. Experimental Results

To verify the performance of our proposed white-light AI, we constructed the optical configuration shown in

Fig. 5. Irradiance distribution at the one-fourth fractional Talbot image under white-light illumination: ~a! Gray-level picture of the irradiance at the output plane of the conventional AI. ~b! Irradiance profile along a horizontal line in ~a! for each RGB component of the incident light. ~c! Achromatic pattern in gray levels provided by our optical configuration in Fig. 4. ~d! RGB irradiance profiles along a horizontal line in ~c!.

Fig. 4 by using a refractive lenslet array, manufactured by melting photoresist, with a period of d 5 0.399 mm and f 5 0.4 mm. The DL, made by multimask-level photoresist technology, has a principal focal length of Z0 5 91.1 mm for the reference wave number s0 5 1.77 mm21. A conventional modified Talbot AI was also constructed by substitution of the DL in Fig. 4 by a nondispersive refractive objective with the same focal length, in comparison. In both optical configurations we use a high-pressure xenon lamp as an equienergetic white-light source. In Fig. 5 we show the result achieved at the output plane of both the conventional and the achromatic AI for a Fresnel image of the amplitude distribution characterized by Q 5 0, N 5 1, and M 5 4. For this selection, h 5 2111 mm, a 5 0.67, and m 5 8.3. The result obtained when we select a second Fresnel image with Q 5 0, N 5 1, and M 5 3 is shown in Fig. 6. In this case, h 5 2131 mm, a 5 0.48, and m 5 32.0. Figures 5~a! and 6~a! correspond to a gray-level picture of the irradiance distribution at the output of the conventional AI, whereas Figs. 5~c! and 6~c! show the

achromatic result provided by our optical configuration. These images were recorded by placement of a color CCD camera at the output plane of the AI. Figs 5~b! and 6~b! and 5~d! and 6~d! show the irradiance profiles @for each red– green– blue ~RGB! component of the incident light# along a horizontal line obtained by integration of the corresponding twodimensional irradiance distribution with a vertical slit. Two findings are clear from Figs. 5 and 6: Our setup provides a set of real, bright peaks of white light, and at the same time the densities of the light spots are variable. The efficiency at the output of the achromatic modified Talbot AI was calculated and compared with that of a conventional array generator. To this end the irradiance of each spot was measured by integration of the two-dimensional irradiance distribution in a small circular region ~with a diameter one fourth of the spot spacing! around the intensity peak. In this way we define the efficiency as h 5 IsyIt, where Is is the sum of the spot irradiances corresponding to the 5 3 5 light points contained in the central region and 10 July 1998 y Vol. 37, No. 20 y APPLIED OPTICS

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Fig. 6. Same as Fig. 5 but for the one-third fractional Talbot image.

It is the total irradiance measured by integration of the irradiance in the entire region. From the data obtained with the color CCD camera, the efficiency corresponding, for instance, to the Fresnel image shown in Fig. 5 is shown for each RGB component of

the incident light of Fig. 7. This measurement gives an estimation of the overall efficiency of the optical setup. In connection with this result, we point out that the diffraction efficiency of the DL used in our experiment is limited to approximately 70%. Thus

Fig. 7. Efficiency ~in percent! measured for each RGB component of the incident light at the central region of the output plane provided by both the conventional and the achromatic AI.

Fig. 8. Uniformity error ~in percent! of the white-light spots provided by the achromatic AI. Measurements are made on the Fresnel image in Fig. 5.

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the efficiency of our achromatic system performs near the theoretical limit. Note also that the efficiency of any DL depends on the wavelength and decreases as we move away from the design wavelength l0. This fact has been neglected in this study, but its effect can be appreciated in the slight reduction of the efficiency for the chromatic components R and B in Fig. 7. The uniformity error eu of the white-light point sources provided by the achromatic AI was also measured in the same central square region of the output plane. It is defined as eu 5 ~Imax 2 Imin!y~Imax 1 Imin!, where Imax and Imin are the maximum and minimum irradiances, respectively, of the light peaks in the region under consideration. The result corresponding to the same Fresnel image ~Fig. 5! is shown for each RGB component of the light in Fig. 8. 5. Conclusions

We have presented a versatile achromatic modified Talbot AI based on a refractive microlens array that has noteworthy features. The achromatic capability of our optical configuration allows us to generate a two-dimensional periodic distribution of white-light point sources with low residual chromatic aberration, and thus it permits the utilization of the whole spectral content of the light source. Moreover, the axial separation between the refractive microlens array and the compensating DL allows us to select the multiplicity of the light points at the output, preserving the achromatic correction. The combination of the lenslet array with the fractional Talbot effect in our achromatic AI offers, as in the monochromatic case, several additional advantages, such as high efficiency and uniformity of the spots. The experimental verifications prove the good performance of our proposal. This research was supported by the Direccioń General de Investigacioń Cientı´fica y Tećnica ~grant

PB93-0354-C02-02!, Ministerio de Educacioń y Ciencia, Spain. The permanent address for E. Bonet is the Departamento de Fı´sica Aplicada, Universidad Politećnica de Valencia, 46071 Valencia, Spain. References 1. N. Streibl, “Multiple beamsplitters,” in Optical Computer Hardware, J. Jahns and S. H. Lee, eds. ~Academic, San Diego, Calif., 1993!, pp. 227–248. 2. A. W. Lohmann and J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337– 4340 ~1990!. 3. J. R. Leger and G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes,” Opt. Lett. 15, 288 –290 ~1990!. 4. V. Arrizon and J. Ojeda-Castan˜eda, “Talbot array illuminators with binary phase gratings,” Opt. Lett. 18, 1–3 ~1993!. 5. V. Arrizon and J. Ojeda-Castan˜eda, “Multilevel phase gratings for array illuminators,” Appl. Opt. 33, 5925–5931 ~1994!. 6. H. Hamam and J. L. de Bougrenet de la Tocnaye, “Multilayer array illuminators with binary phase plates at fractional Talbot distances,” Appl. Opt. 35, 1820 –1826 ~1996!. 7. H. Hamam, “Talbot array illuminators: a general approach,” Appl. Opt. 36, 2319 –2327 ~1997!. 8. A. W. Lohmann, “Array illuminators and complexity theory,” Opt. Commun. 89, 167–172 ~1992!. 9. E. Bonet, P. Andre´s, J. C. Barreiro, and A. Pons, “Self-imaging properties of a periodic microlens array: versatile array illuminator realization,” Opt. Commun. 106, 39 – 44 ~1994!. 10. B. Besold and N. Lindlein, “Fractional Talbot effect for periodic microlens arrays,” Opt. Eng. 36, 1099 –1105 ~1997!. 11. J. T. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–381 ~1965!. 12. J. Lancis, E. Tajahuerce, P. Andre´s, V. Climent, and E. Tepichıń, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Commun. 136, 297–305 ~1997!.

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