Analysis of Wavefront Propagation Using the Talbot Effect

Analysis of wavefront propagation using the Talbot effect

Ping Zhou and James H. Burge* College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA *Corresponding author: jburge@optics.arizona.edu

Received 19 May 2010; revised 20 August 2010; accepted 31 August 2010; posted 3 September 2010 (Doc. ID 128711); published 24 September 2010

Talbot imaging is a well-known effect that causes sinusoidal patterns to be reimaged by diffraction with characteristic period that varies inversely with both wavelength and the square of the spatial frequency. This effect is treated using the Fresnel diffraction integral for fields with sinusoidal ripples in amplitude or phase. The periodic nature is demonstrated and explained, and a sinusoidal approximation is made for the case where the phase or amplitude ripples are small, which allows direct determination of the field for arbitrary propagation distance. Coupled with a straightforward method for calculating the effect in a diverging or converging beam, the Talbot method provides a useful approximation for a class of diffraction problems. © 2010 Optical Society of America OCIS codes: 050.1940, 070.6760.

1. Introduction periodic objects as the superposition of certain repli- – When a periodic object is illuminated by coherent cas of the initial objects using the Fresnel Kirchhoff light, an exact image of the structure will appear at formulation. Each replica of the object is shifted lat- certain distances by free-space propagation. This erally and modulated in amplitude and phase by a phenomenon is known as the Talbot effect [1]or constant coefficient, which is obtained from a series self-imaging, which is a typical Fresnel or near-field of complex numbers. Ojeda-Castaneda and Sicre [3] diffraction. In Fresnel diffraction, the spherical and Testorf and Ojeda-Castaneda [4] discussed the waves arising from point sources in the aperture are Talbot effect using the Wigner distribution function approximated as quadratic-phase surfaces. When (WDF), which provides a link between Fourier optics collimated light illuminates the periodic object, the and geometrical optics. Their approach allows calcu- self-image planes are equispaced and well defined. lating the field distribution at a fractional Talbot If the object has a period of p and the wavelength distance. Lu and Zhou use the finite-difference of the illumination is λ, then the self-images are time-domain (FDTD) method to analyze the field dis- formed in planes, which are multiples of the Talbot tribution at any plane behind the periodic object [5]. 2 When the periodic objects have either weak phase distances zT ¼ 2p =λ. The Talbot distance is valid for collimated illumination. For spherical illumina- or amplitude distribution, the calculation of the field tion, the replication of the object is amplified and distribution at any plane can be simplified. This the distance between the self-image planes varies paper reviews the calculation for this case using with propagation. the angular spectrum method [6]. With small-angle Besides those self-image planes, the diffraction approximation, it shows that the amplitude and field at any distance behind the periodic object is phase of the electric field have fairly simple expres- of interest to many researchers. Winthrop and sions: both the phase and amplitude modulation has a sinusoidal variation as a function of the propaga- Worthington [2] describe the diffraction pattern of tion distance. For a wavefront with a certain spatial frequency, the Talbot effect can be used to estimate 0003-6935/10/285351-09$15.00/0 the phase and amplitude after it propagates to any © 2010 Optical Society of America distance and can greatly simplify the calculation.

1 October 2010 / Vol. 49, No. 28 / APPLIED OPTICS 5351 The Talbot distance as described above is valid j2πz 1 −jπzλξ2 U ðξÞ¼e λ δðξÞþ jαe 0 ½δðξ þ ξ Þ − δðξ − ξ Þ only for collimated illumination. This paper presents z 2 0 0 an equivalent propagation distance that allows ap- j2πz 1 2 plication of the Talbot effect in converging or diver- ¼ e λ δðξÞþ jα½ ðπzλξ Þ 2 cos 0 ging light. We find that the phase and amplitude modulation as the wavefront propagates depend only 2 on the Fresnel number and the spatial frequency of − j sinðπzλξ0Þ½δðξ þ ξ0Þ − δðξ − ξ0Þ : ð4Þ the initial object. This paper focuses on the general analysis of j2πz e λ wavefront propagation using the Talbot effect and By dropping the constant phase term at any the simulation shows its validity. An example of plane, the diffraction field after propagating a distance of z can be given by taking an inverse Fourier the application in the metrology with interferom- U ðξÞ eters is provided in our other papers [7–10]. transform of z : 2 2 2πx uzðxÞ¼1 þ α½cosðπzλξ Þ − j sinðπzλξ Þ sin 2. Talbot Effect in a Collimated Beam 0 0 p The Talbot effects of sinusoidal amplitude and phase 2πx ¼ 1 þ α ðπzλξ2Þ gratings in collimated illumination are discussed in cos 0 sin p the next two sections. We follow the conventional Fourier analysis shown by Goodman [6] to derive 2πx − jα ðπzλξ2Þ : ð Þ the amplitude and phase distribution after the per- sin 0 sin p 5 iodic structure propagates to a certain distance. If this complex field is written in the form of the am- A. Sinusoidal Amplitude Grating iψ ðxÞ plitude and phase uzðxÞ¼AzðxÞe z , then the ampli- Consider a one-dimensional (1-D) sinusoidal amplitude is the square root of the imaginary and real tude field distribution with a period of p and magni- α parts. If the amplitude is described with a Taylor ser- tude of variation , which could be created by ies in α, uniform illumination upon a grating with sinusoidal transmission: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2πx 2 2 2πx AzðxÞ¼ 1 þ 2α cosðπzλξ Þ sin þ α sin 0 p p 2πx uðxÞ¼1 þ α sin : ð1Þ 2πz 2πx p ≈ 1 þ α cos z sin p T 1 2πz 2πx þ α2 2 2 þ …; ð Þ This treatment neglects edge effects, assuming a con- 2 sin z sin p 6 tinuous function over x ¼ð−∞; þ∞Þ. The angular T spectrum of this amplitude grating can be obtained by taking a Fourier transform of uðxÞ: and the phase is the arctangent of the ratio of the imaginary part to the real part and its Taylor series in α is 1 Uz¼0ðξÞ¼δðξÞþ jα½δðξ þ ξ0Þ − δðξ − ξ0Þ; ð2Þ 2 2 2πx −α sinðπzλξ0Þ sin p ψ zðxÞ¼arctan ξ j 2 2πx where is the spatial frequency, is the imaginary 1 þ α cosðπzλξ0Þ sin p δ unit, and is the delta function. This angular spectrum is nothing more than three plane-wave beams 2πz 2πx with appropriate direction and phase. The diffraction ≈−α sin sin zT p pattern after propagating a distance z can be calcu- U ðξÞ 1 4πz 2πx lated by multiplying its angular spectrum z¼0 þ α2 sin sin2 þ …; ð7Þ with the free-space transfer function HzðξÞ and then 2 zT p taking an inverse Fourier transform [6]. If the period 2 of the grating is very large compared to the wave- where zT is the Talbot distance and zT ¼ 2p =λ.Upto length λ, the free-space transfer function is this point, the only approximation we have made is to assume that the period of the grating is larger pffiffiffiffiffiffiffiffiffiffiffiffi than the wavelength. If we make another assump- j2πz 1−ðλξÞ2 j2πz −jπzλξ2 HzðξÞ¼e λ ≈ e λ e : ð3Þ tion, that this grating has a small variation in amplitude relative to the mean (α ≪ 1), then the higher order terms of the Taylor series in Eqs. (6)and(7) Applying the free-space transfer function to Uz¼0ðξÞ can be ignored. By only keeping the linear term of gives α in both AzðxÞ and ψ zðxÞ, it shows that the variation

5352 APPLIED OPTICS / Vol. 49, No. 28 / 1 October 2010 of the amplitude A0 follows a cosine function with the When α ≪ 1, the sinusoidal phase described in propagation distance: Eq. (10) can be simplified to z A0 ≈ α 2π : ð Þ jα 2πx cos z 8 sin p T uz¼0ðxÞ¼e 2πx α2 2πx Also, the amplitude ripples create phase ripples and ≈ 1 þ jα sin − sin2 þ …; ð12Þ the magnitude of the phase W0 (in waves) follows a p 2 p sinusoidal function with propagation distance: z and the field distribution at the plane becomes z α πzλ 0 W ¼ −W sin 2π ¼ − sin ; ð9Þ 2 2 z 2π p2 α −jπλz 2πx α T u ðxÞ¼1 − þ jα e p2 þ z 4 · sin p 4 where α ¼ 2πW, the amplitude of the sinusoidal nz −j4πλz 4πx phase ripple in radians. At a distance z ¼ T (n is e p2 : ð Þ 2 · cos p 13 an integer), the diffraction pattern has pure ampli- ð2nþ1ÞzT tude variation; at a distance z ¼ 4 , the diffraction pattern has constant amplitude and pure phase The phase of this field can be given in terms of the variation. For a weak amplitude grating, the ampli- Taylor series expanded in α: tude and phase follow a cosine and a sine function, respectively, as the beam propagates. This simple 2πz 2πx ψ zðxÞ¼α cos sin conclusion is useful and helps estimate the ampli- zT p tude and phase after the beam propagates to any α2 8πz 4πx plane. The more general case is also interesting be- − sin cos cause the harmonics are also periodic with higher 4 zT p frequency. So propagation of a full Talbot cycle for 4πz 2πx þ 2 2 þ …: ð Þ the fundamental coincides with integral numbers sin z sin p 14 of periods for the harmonics. Talbot imaging does T not require the small amplitude approximation, α but the simplification that allows calculation of the The Taylor series of amplitude in is field at arbitrary propagation distances does. α2 2πz 2πx B. Sinusoidal Phase Grating AzðxÞ¼1 − þ α sin sin 4 zT p A 1-D pure sinusoidal phase grating with a period of 2 p W α 8πz 4πx and magnitude of in waves illuminated by a col- þ cos cos limated beam can be described as an infinite sum of 4 zT p Bessel functions [6]: 2πz 2πx þ 2 cos2 sin2 þ …: ð15Þ zT p jα sin 2πx Xþ∞ p j2πqx uz¼0ðxÞ¼e ¼ JqðαÞ · e p ; ð10Þ The term quadratic in α corresponds to the second q¼−∞ harmonics, and they have much smaller magnitude because α ≪ 1. By keeping only the term linear in α, where α ¼ 2πW in radians. By following the proce- the phase reduces to dure described for the amplitude grating [Eqs. (2)– (5)], the diffraction field, after propagating a distance z 2πx z ψ zðxÞ¼α cos 2π sin ; ð16Þ of , can be calculated as zT p 2 q and the amplitude becomes Xþ∞ −jπzλ p j2πxq uzðxÞ¼ JqðαÞ · e · e p: ð11Þ q¼−∞ z 2πx AzðxÞ¼1 þ α sin 2π sin : ð17Þ zT p The field distribution at any plane z is an infinite sum of Bessel functions. If we make the same as- Again, the harmonics are periodic with integral num- sumption, that the phase ripple is much less than ber of cycles within one Talbot cycle, so the Talbot 1 rad (α ≪ 1), then we can approximate the field dis- imaging phenomenon remains valid for large phase tribution at any plane z by looking only at the lower ripples. As a sinusoidal phase pattern propagates, it orders Bessel functions. If α ≥ 1, then the higher will cycle through a reverse contrast amplitude pat- order Bessel functions may be larger than the lower tern, a conjugate phase pattern, a pure amplitude order ones and cannot be neglected. pattern, and then back to the original phase pattern.

1 October 2010 / Vol. 49, No. 28 / APPLIED OPTICS 5353 Compared to the original phase ripple, the magnitude of the phase ripple is approximately modulated by a cosine function. If the magnitude of the original phase ripple is W in waves, then the attenuation of the phase ripple as it propagates a distance z can be described as z πzλ W0 ¼ W 2π ¼ W : ð Þ cos cos 2 18 zT p

For many applications, this propagation distance is small compared to the Talbot distance. In this case, π2λ2z2 W0 ≈ W 1 − : ð19Þ 2p2

The magnitude of the phase becomes smaller than (if not equal to) that of the original phase. We call this the phase smoothing effect. The pure phase ripples Fig. 1. (Color online) Talbot effect illustrated as three-beam interference. The angular spectrum of a laser beam transmitted also create amplitude ripples accordingly, which through a periodic object consists of three plane waves, which com- have a sinusoidal distribution, as shown in Eq. (17). bine together to form the periodic replication of the original per- In conclusion, for a wavefront with periodic, weak iodic structure along the z axis. amplitude or phase variations, the Talbot effect be- haves as expected, with the periodic cycling between amplitude and phase. While the periodic behavior is the self-imaging effect can be simulated as a sum general, we show that this behavior is approximated of three plane waves. Consider a phase ripple with as sinusoidal for cases with amplitude ripples much amplitude of 0:05λ; the phase and amplitude distri- less than the mean, or phase ripples much less than butions for propagating the Talbot distance two 1 rad. In general, higher harmonics are generated, times are simulated using the method of the three- which are about a factor of α less than the fundamen- beam interference, as shown in Fig. 2. The replica- tal where α is the ratio of the amplitude variation to tion of the original wavefront can be obtained by sim- the mean, or it is the amplitude of phase ripples in ply adding the complex amplitude of the three units of radians. plane waves. Note that the Talbot analysis is for one single frequency, but the method is general, because any dis- 4. Talbot Effect in a Spherical Converging or tribution can be decomposed into a set of sinusoidal Diverging Beam functions. The propagation using the Talbot analysis In some applications, the periodic structure propa- allows us to take advantage of the linear behavior of gates in a spherical beam [12]. The field distribution diffraction. Since the Fourier components do not in- of a periodic structure propagating in a spherical teract, we decompose the distribution into Fourier beam can be obtained using the Fresnel–Kirchhoff components, then propagate each component with integral or other approaches [13,14,12]. Those ap- a simple transfer function. The resulting distribution proaches require extensive mathematical treatment. can be reconstructed by integrating over the Four- In this paper, we present an equivalent propagation ier space. method for the weak periodic structure that allows calculating the phase and amplitude distribution 3. Physical Insight of the Talbot Effect with the simple expressions derived in Section 2. When the periodic object has weak phase or ampli- The Talbot distance zT is defined for a collimated tude distribution, the Talbot effect can be understood light. For a spherical beam, it is convenient to con- in terms of three-beam interference [11]. The form of vert it into an equivalent collimated beam and then Eq. (2) is that of three plane waves (the three lowest use the Talbot effect to predict how wavefronts harmonics of the object), as shown in Fig. 1.One change due to propagation. The diffraction pattern plane wave propagates along the z axis, and the other for a spherical beam is the same as that observed two are off-axis and symmetric about the z axis. The for a collimated beam, except that the diffraction pat- resulting field is the addition of these three plane tern occurs at the effective propagation distance Le, waves. This three-beam interference causes the am- and it is scaled in the transverse dimension. plitude and phase to vary periodically in space. At As shown in Fig. 3, a converging wavefront start- the Talbot distance, the pattern is the same as the ing with radius of curvature R1, diameter 2a1, and original one. The pattern is shifted sideways at the ripples with period p1, propagates to a position where half-Talbot distance. it has radius of curvature R2. To convert this propa- The simplicity of three-beam interference helps gation to equivalent propagation in a collimated understand the physics of the Talbot effect, and space, an ideal lens with an arbitrary focal length

5354 APPLIED OPTICS / Vol. 49, No. 28 / 1 October 2010 Fig. 2. Simulation of the phase and amplitude distribution using the method of three-beam interference. The original phase ripple has an amplitude of 0:05λ. f is used to convert the light into collimated light. R1ðR1 − R2Þ Le ¼ ; ð23Þ Then we use geometric imaging relationships to cal- R2 culate equivalent dimensions. Assuming an ideal imaging system, the amplitude and phase variations and Eq. (22) becomes are correctly imaged through the lens. In collimated space, the equivalent propagation πλ L W0 ¼ W · e : ð Þ distance is · cos 2 24 p1 1 1 The period of the phase ripple p varies as it propa- ΔZ0 ¼ Z0 − Z0 ¼ f 2 − ; ð Þ 2 1 20 gates in a spherical beam. To avoid the scaling issue, R2 R1 the ripple period can be normalized by 2a, the and the phase ripple has a period of diameter of the aperture. Thus, the normalized frequency f normalized ¼ 2a1=p1 ½cycles=diameter, remains unchanged as the wavefront propagates in 0 f p ¼ p1: ð21Þ any type of illumination. By replacing the period R1 p1 with the normalized frequency f normalized, Eq. (24) becomes The attenuation of the phase ripple due to propagation, via Eq. (18), in the equivalent collimated space πλL f 2 W0 ¼ W e normalized : ð Þ becomes cos 2 25 4a1 πλ ΔZ0 πλR ðR − R Þ W0 ¼ W · ¼ W 1 · 1 2 : By using the Fresnel number Nf, · cos 02 · cos 2 p R2 · p1 a2 ð22Þ 1 Nf ¼ ; ð26Þ λLe The focal length f in Eq. (22) falls off, and this makes sense because the lens is arbitrarily selected to con- Eq. (25) can be further reduced to vert a spherical beam to a collimated one. Comparing R1ðR1−R2Þ Eq. (22) with Eq. (18) shows that is equiva- 2 R2 πf lent to z. Therefore, the effective propagation dis- W0 ¼ W normalized : ð Þ cos 4N 27 tance Le can be defined as f

Fig. 3. Propagation in a converging space is converted to equivalent propagation in a collimated space.

1 October 2010 / Vol. 49, No. 28 / APPLIED OPTICS 5355 Physically, the Fresnel number represents the geometric difference between the distance from the edge of the aperture to the observation point, and the distance from the center of the aperture to the observation point, divided by λ=2. Equation (27) shows that the phase smoothing effect of the periodic structure depends only on the Fresnel number, which is a function of effective propagation distance, aperture size, and wavelength. The larger the Fresnel number, the less the smoothing effect is. For any given geometry, the Fresnel number can be calculated, allowing the change of the wavefront magnitude due to propagation to be estimated at any given frequency. A transfer function describing the magnitude of the phase smoothing can be defined as W0 πf 2 TF ¼ ¼ cos normalized : ð28Þ W 4Nf

Another useful form of the transfer function is to R describe it in terms of F-number, which is 1 in this Fig. 4. Transfer function due to a defocused image plane for 2a1 case. Rewrite Eq. (21) with F-number and normal- interferometric measurement of a 50 mm optic with 500 mm 632:8 ized spatial frequency, and we have defocus error. The wavelength is nm. The defocus causes loss of information (phase smoothing) for a frequency above 40 cycles=diameter and causes severe problems for higher fre- πλ ΔZ0 πλR2 1 1 TF ¼ · ¼ 1 − quencies, which show phase reversal. cos 02 cos 2 p p R2 R1 1 1 1 ¼ πλF2f 2 − : ð Þ creates a new wavefront. The new wavefront propa- cos n normalized 29 R2 R1 gates into the geometric shadow of the aperture and looks like diffraction “ripples” around the edge of the Note that the phase attenuation varies with spatial aperture [13,15]. It will also be seen when an aper- frequency. In general, any wavefront aberration can ture is not in focus in an imaging system. The edge be decomposed to a collection of sinusoidal ripples diffraction pattern can be studied by dividing the with variable spatial frequencies. For each frequency aperture into Fresnel zones and calculating the Fres- component, the phase attenuation can be estimated nel number. The severity of the edge diffraction de- by the transfer function. pends on the Fresnel number, which is a function As mentioned in Subsection 2.B, the amplitude of the wavelength and propagation distance. A larger variation follows a sine function as a periodic struc- Fresnel number (closer to the periodic structure) ture propagates in a collimated beam. In spherical means less edge diffraction. The edge diffraction illumination, the irradiance varies proportional to becomes more significant as the Fresnel number the inverse beam size as the beam propagates. The becomes small (further away from the periodic amplitude variation will follow a sine function if it structure). ~1 is scaled by the beam size (R2). The diffraction near the edge of the aperture can be The transfer function shows how wavefront irregu- modeled as Fresnel knife-edge diffraction. The real larities with high spatial frequency are filtered by and imaginary parts of the electric field distribution propagation. This is of great importance to interfero- can be found by evaluating the Fresnel integrals [13]. metry, where the high-frequency phase information Figure 5 shows the amplitude and phase fluctuations is needed. As an example, Fig. 4 shows a phase trans- due to edge diffraction. Both amplitude and phase fer function due to a defocused image plane for an have rapid oscillations as the distance from the edge interferometric measurement of a 50 mm optic with becomes large. The central 80% of the aperture will a defocus error of 500 mm. It is clear that this defocus be virtually free of edge effects when the distance causes loss of information (phase smoothing) for a from the edge corresponding to 10% of the aperture 1=2 frequency above 40 cycles=diameter (period of radius is 5ðλLe=2Þ (about six ripples from the 1:25 mm) and causes severe problems for higher fre- edge), or quencies, which can show phase reversal at about rffiffiffiffiffiffiffiffi 90 cycles=diameter (period of 0:56 mm). 2 λLe a 5 < 0:1a; Nf ¼ > 1250: ð30Þ 5. Edge Diffraction 2 λLe The analysis above neglects all edge effects. Edge diffraction occurs when the wavefront strikes the edge The Fresnel number of the geometry has to be of an aperture, which acts as a secondary source and greater than 1250 in order to have six or more ripples

5356 APPLIED OPTICS / Vol. 49, No. 28 / 1 October 2010 Fig. 5. (Color online) (a) Amplitude and (b) phase variation from a knife edge. at 10% of the aperture radius from the edge. In ad- rectly represent the errors in the test surfaces. Edge dition, the period under consideration should be long diffraction from the limiting aperture usually has the enough that the Fresnel number for propagation is most dominant effect in a measurement, like the larger than the number of ripples across the aper- transmission sphere/flat. ture. Otherwise, the edge diffraction effect will strongly interact with the Talbot effect of the periodic 6. Simulation structures. Figure 6 shows the period of the edge diffraction To verify the equations derived above for calculating ripples, the amplitude envelop, and the phase envel- the phase and amplitude behind a weak periodic op, which drop quickly with respect to the distance structure, a simulation, based on wave propagation, was performed using the physical model of the ASAP, from the edge of the diffraction pattern. It is interest- an optical modeling software [16]. ing to note that the three curves have exactly the A sinusoidal phase ripple with a magnitude of same shape and they can be obtained one from the 0:05λ (m ¼ 0:31 rad) and the amplitude of unity other by only scaling and translating. propagates one Talbot distance in the collimated Edge diffraction is often seen in interferometry. beam. The phase and amplitude distribution are There are many apertures inside an interferometer plotted in Fig. 7. The analytical solution is obtained that cannot be in focus because interferometers often using the Eqs. (17)and(18). They agree with the image the surface under test on the detector to cor- ASAP simulation.

Fig. 6. (a) Period of the edge diffraction ripples, (b) the amplitude envelop, and (c) the phase envelop due to edge diffraction.

1 October 2010 / Vol. 49, No. 28 / APPLIED OPTICS 5357 Fig. 7. (Color online) (a) Phase and (b) amplitude distribution at x ¼ 0 as a weak phase ripple propagates one Talbot distance. The results from the Fourier analysis were validated with numerical simulation using ASAP.

Numerical simulation using ASAP was also used to R2 when the effective propagation distance Le equals verify the effective propagation distance for a conver- some fraction of the Talbot distance. An ASAP simu- ging spherical beam. If a spherical wavefront (λ ¼ lation is also performed to verify the calculation, and 0:5 μm) starting with radius of curvature 20 mm both the amplitude and phase variations are shown and diameter 2 mm passes through a sinusoidal in Fig. 8. The data from the ASAP simulation are zT zT zT 3zT shown at the appropriate propagation distance. phase grating with period 0:1 mm, the 8 , 4 , 2 , 4 , and zT occur at 4.0, 2.7, 3.3, 2, and 1:3 mm from Note that ASAP calculates the complex field in a one another. This can be calculated by solving the plane instead of a sphere, so the light near the edge of the pupil propagates a longer distance than the on- axis one and there is a cosine effect. For this F=10 system, the cosine effect causes a propagation distance error of about 0.1%. In Fig. 8, the amplitude variation is shown on a plane where ASAP calculates the field distribution. The phase variation is shown on a curved surface. The ripples are scaled to show the phase ripples clearly. The pure phase variation occurs right after the grating, at both the half and one Talbot distances. The pure amplitude variation happens at a quarter zT and three-quarters of Talbot distance. At 8 , there are both amplitude and phase variations. The calculated equivalent propagation distance for converging wavefront with a phase grating matches the simulation using ASAP.

7. Conclusion Fig. 8. (a) Amplitude and (b) phase distributions at some frac- This paper uses conventional Fourier analysis to de- tional Talbot distance in a converging spherical beam. The calcu- monstrate the Talbot effect for a sinusoidal ripple in lated equivalent propagation distance for converging the amplitude or phase. A sinusoidal approximation is wavefront with a phase grating matches the simulation using made for the case where the phase or amplitude rip- ASAP. The data from the ASAP simulation are shown at appropriate ples are small, which allows direct determination of propagation distances. It is clear that the results have a character- the field for an arbitrary propagation distance. istic period defined using the equivalent propagation distance. Coupled with a straightforward method for calculat- Also, the presence of higher harmonics is seen in the nonsinusoidal ing the effect in a diverging or converging beam, the shape at zT =8. The amplitude variation is scaled by the inverse area in this figure, and the phase variation is also scaled to show Talbot method provides a useful approximation for a its variation. Only ten cycles at the middle of the aperture are class of diffraction problems. The derivation in this shown in both pictures. The numbers in the figures have units paper is very useful for solving or explaining the of millimeters. phase smoothing and edge diffraction effects in

5358 APPLIED OPTICS / Vol. 49, No. 28 / 1 October 2010 coherent imaging systems, especially with small 4. M. Testorf and J. Ojeda-Castaneda, “Fractional Talbot effect: variations in amplitude or phase. analysis in phase space,” J. Opt. Soc. Am. A 13, 119–125 (1996). The authors acknowledge help from several collea- 5. Q. Lu and C. Zhou, “Rigorous electromagnetic analysis of Tal- gues at the College of Optical Sciences, University of bot effect with the finite-difference time-domain method,” Arizona: Chunyu Zhao for many discussions on this Proc. SPIE 5638, 108–116 (2005). topic in general, Matt Dubin for his help with the 6. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. ASAP simulation, and Tom Milster for his insight (Roberts, 2005). 7. C. Zhao and J. H. Burge, “Imaging aberrations from null cor- on three-beam interference. The authors also would ” 6723 like to thank Ulf Griesmann from the National Insti- rectors, Proc. SPIE , 67230L (2007). 8. P. Zhou, J. H. Burge, and C. Zhao, “Imaging issues for inter- tute of Standards and Technology (NIST) for reading ferometric measurement of aspheric surfaces using CGH null the manuscript and providing thoughtful comments. correctors,” Proc. SPIE 7790 (2010). The authors are grateful for partial support from 9. J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for inter- National Institute of Standards and Technology ferometry with CGH null correctors,” Proc. SPIE 7739, (NIST), U.S. Department of Commerce, under Amer- 77390T (2010). ican Recovery and Reinvestment Act award 10. P. Zhou and J. H. Burge, “Limits for interferometer calibration 60NANB10D010. using the random ball test,” Proc. SPIE 7426, 74260U (2009). 11. T. Milster, “505 Classnotes,” University of Arizona, http://www .optics.arizona.edu/Milster/. References 12. H. O. Carmesin and D. Goldbeck, “Depth map by convergent 3D-Talbot-interferometry,” Optik (Jena) 108, 3 (1998). “ ” 1. W. H. F. Talbot, Facts relating to optical sciences. No. IV, 13. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge 9 – Philos. Mag. , 401 407 (1836). U. Press, 1999). “ 2. J. T. Winthrop and C. R. Worthington, Theory of Fresnel 14. R. Jozwicki, “The Talbot effect as a sequence of quadratic ” images. I. Plane periodic objects in monochromatic light, J. phase corrections of the object Fourier transform,” Opt. Acta 55 – Opt. Soc. Am. , 373 381 (1965). 30,73–84 (1983). “ 3. J. Ojeda-Castaneda and E. E. Sicre, Quasi ray-optical ap- 15. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. proach to longitudinal periodicities of free and bounded wave- Am. 52, 116–130 (1962). ” 32 – fields, Opt. Acta ,17 26 (1985). 16. http://www.breault.com/.

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