1. Introduction

The demand for faster computers and larger data storage capacities requires smaller structures and larger fields on the respective integrated electronic circuits. This can be partially achieved by using shorter wavelengths in the lithographic process. Additionally, the numerical aperture of the stepper objectives for the demagnification of the reticle has to be increased. Therefore, it is useful to include also aspheric surfaces which deliver more free parameters for optimization of the objective [1]. The total number of lenses in the optical system can be reduced. Modern techniques allow the fabrication of aspheric surfaces with a very high surface quality. At the moment, it is even possible to fabricate aspherics with an accuracy better than can be measured. The lack of a method for absolute testing of aspheric surfaces is one of the main reasons for this discrepancy. Spherical surfaces can be absolutely tested by a three position measurement [2]. Since the wavefront coming from the null-lens for an aspheric under test has no focal point, the application of the same principle is not possible for aspherics. Now, computer generated holograms (CGHs) can be used to generate both, an aspheric and a spherical wavefront. We investigate the case of continuous phase-only holograms. In the case of superposing the respective wavefields disturbing diffraction orders appear due to omitting the amplitude information in the CGH. The other approach for dual wavefront CGHs is the alternately encoding of the two wavefields in the hologram aperture sliced into regular stripes. This also yields unwanted diffraction orders. We examine the disturbing diffraction orders with a first order approximation and discuss the obtained equations. Aspects of efficiency and intensity balancing will also be shown.

2. Quasi-absolute testing of aspherics

Absolute testing of spherical surfaces can be done by a three position measurement in an interferometric setup shown in figure 1. Two measurements are used to get the wave aberrations W₁ and W₂ for a 0° and an 180°-position of the sphere, respectively (fig. 2). A third position, called cat's eye position, is to use a mirror placed at the focal point for inverting the wavefront coming from the refractive null optics. The aberrations W₃ generated by the inverted wavefront can now be used to get rid of aberrations of the interferometer. The aberrations coming from the reference arm and of the system in the object arm are denoted by W_R and W_S, respectively. The following set of equations

W1(x,y)=WR(x,y)+WS(x,y)+P(x,y)	(1)
W2(x,y)=WR(x,y)+WS(x,y)+P(-x,-y)	(2)
W3(x,y)=WR(x,y)+½[WS(x,y)+WS(-x,-y)]	(3)

can be used to obtain the absolute errors P of the surface under test:

P(x,y)= ½ [W1(x,y)+W2(-x,-y)-W3(x,y)-W3(-x,-y)]

(4)

Fig 1: Schematic setup of an interferometer for testing aspherics using a null-lens system.

Fig 2: Scheme of the three positions for an absolute measurement of spherical surfaces.

The wavefront coming from a null optics for aspherics does not have a focal point anymore, i.e. no cat’s eye measurement can be made. Here, the ability of CGHs to generate more than one wavefront can be used to provide the missing spherical wave. The idea of one of the authors [4] allows a quasi-absolute test of rotationally symmetric aspherics. The errors measured at the cat’s eye position

W'3(x,y)= WR(x,y)+½[W'S(x,y)+W'S(-x,-y)]

(5)

contain systematic aberrations W’_S introduced by using the spherical wave. If the difference between the aspheric and the spherical wavefront is not too large, it is a good approximation that W’_S»W_S . Here, W_S are the systematic aberrations introduced by using the aspheric wave. As an alternative, an absolutely tested sphere can be used as a reference. In this case, only two measuments are necessary, W₁ and W₄(x,y)= W_R(x,y)+W_S(x,y)+Q(x,y). Since the absolute errors Q of the sphere are known, the absolute errors P of the surface under test are obtained by
P(x,y)= W₁(x,y)-W₄(x,y)+Q(x,y). Again, the approximation W’_S»W_S was used.

Beside this uncertainty, which has to be investigated separately, the encoding of two wavefronts generates unwanted parasitic waves that will disturb the measurement if they cannot be filtered out. But already the measurement using one of the two wavefronts may be disturbed by the other desired wavefront, and vice versa. So, the equations given in the next sections describe the spatial frequencies of the unwanted waves in the CGH plane (fig. 1). If the spatial frequencies are larger than the cut-off frequency of the filter system the parasitic waves are filtered out. Some comments have to be made on the interferometric setup. Since one wants to measure the surface deviations, it would be best to image the surface onto the detector. But as the aspheric cannnot be imaged sharply onto the detector plane it is useful to image the CGH plane because each point of the aspheric is unequivocally connected with one point in the CGH plane due to the null-test configuration.

3. Dual wavefront CGHs

In the following, we restrict to continuous phase-only CGHs which can be nowadays fabricated with gray-tone lithography in a sufficient quality. Moreover, a wavefront U and the corresponding surface will be denoted with the same letter. The two wavefronts, A of the aspheric and the spherical B, that have to be encoded in the CGH, will be represented by their phase functions j_A and j_B, respectively. The derivation of the equations for the analysis of the parasitic waves will use the same considerations made by one of the authors for the diffraction orders of a binary phase-only hologram with one aspheric wavefront [5]. The lengthly derivation uses only small angles of the rays traced through the system, supposes similar wavefronts U, V and R, and makes some first order approximations to simplify the analytical expressions. It can be summarized as follows (fig. 3):

Fig 3: Ray-tracing scheme of the set-up used for the derivation. Here, the surface is R=U.

An incoming light ray with direction vector is traced through the CGH with the order m of the phase function j_U using the vector ray tracing equation [6] for CGHs. If the CGH is illuminated with a plane wave the vector has only a z component. Otherwise, it is assumed that has small x and y components. In the latter case, the CGH is usually illuminated by a spherical wave which is generated by a refractive component of the null-lens system. The resulting ray with direction hits the surface R and will be reflected. The surface normal at the reflection point for the desired first order will be the same as if R=U, i.e. it is used the order that hits the surface under test perpendicularly. For general a first order approximation is made using the length d of the ray from the CGH to the surface R and the lateral shifts introduced by the use of a different order or another, but similar surface under test. Finally, the obtained direction is deflected by the order m^’ of the phase function j_V when passing the CGH on the way back. Again, the approximation takes into account that U and V are similar. To get the spatial frequencies ( v_x , v_y) that result in the filter plane the refractive part of the null-lens represented by has to be subtracted from after passing the CGH the second time.

3.1 Superposed wavefronts
A phase-only hologram encoding two wavefronts A and B by superposition of the complex wavefields only takes into account the phase j of the resulting field:

(6)

The amplitude weights a_A and a_B can be used to control the intensity ratio between A and B. It can be shown [3] that the phase-only part of the resulting hologram function can be written as

(7)

The desired wavefronts A and B are generated by the diffraction orders m=1 and m=0, respectively. It should be mentioned, that the weights a₁ and a₀ are not exactly the same as the corresponding a_A and a_B, but their values can be calculated numerically [3]. Here, we only want to use the phase function F_m=mj_A+(1-m)j_B of the diffraction order m to trace rays through the object arm starting at the CGH plane. We assume that light passes the order m of F first, than hits the surface corresponding to j_A and finally passes the order m^’ of F. Using the phase function j_D= j_A- j_B, i.e. the difference between A and B, the two components of the resulting spatial frequencies, obtained by the explained derivation scheme, are:

	(8a)
	(8b)

The equations above also describe the spatial frequencies when a spherical reference is used as the reflecting surface. But in this case, A will be the sphere and B the aspheric, respectively. It is also necessary to examine the real valued amplitudes a_m or intensities I_m=a_m² of the disturbing orders. Beside the desired main orders m=0,1 the next side orders m = -1,2 have the highest amplitudes. In figure 5, the efficiencies h_m = I_m/I_total of the orders m=-2,...,3 have been plotted as a function of the input intensity weight g_A=a_A²=1-a_B². Also, the sum a₀ ²+a₁², which is the total diffraction efficiency for the two desired orders is shown. The curves were calculated numerically using the ansatz of [3]. Since the reflectivity of an aspheric made of glass is in general much lower than the reflectivity of a mirror, one can adopt the output ratio I₁/I₀ by choosing a proper input weight g_A.

Fig 5: Efficiencies of a superposed wavefront CGH for the orders m=-2,...,3. The averaged efficiency h0,2 is also shown.

3.2 Alternating subapertures
Another possibility to overlay two wavefronts is to cut the hologram aperture into regular stripes and to encode both phase functions j_A and j_B alternately in these subapertures. In this encoding scheme only the stripe widths w_A and w_B in x direction can be used to control the intensity ratio between A and B (fig. 4). The resulting hologram function h_A in the subaperture grating of the wave A can be written as

Fig 4: Scheme of a slice aperture CGH with two alternately encoded phase functions.

(9)

where H_A(x,y)=exp(ij_A(x,y)) is the hologram function of A. From the Fourier transform of h_A one can see, that there will appear diffraction orders of the subaperture grating with period p=w_A+w_B, all of them containing the same information :

(10)

To apply the derivation of the spatial frequencies, we start with the hologram function for a single order m using for example j_A:

(11)

Here, the diffraction orders used to obtain the desired phase functions j_A and j_B have m=0, respectively. Again, we only want to use the phase function F_m= j(x,y)+m.s.x. But additionally, we have to choose between j_A and j_B for the two passes through the CGH. The following equations were derived for the case that light passes the order m of F_U first, than hits the surface corresponding to j_R, and finally passes the order m^' of F_V. All of them, U, V and R can be either A or B, respectively. Using the abbrevaitions j_DU= j_U - j_R and j_DV= j_V - j_R , the components of the resulting spatial frequencies appearing at the filter plane are:

	(12a)
	(12b)

To examine the intensities I_m in the case of the grating-like subapertures of the CGH, we look at equation (10). The weight a_m of order m of, for example, j_A can be written as a function of the window width w_A (fig. 4):

(13)

Figure 6 shows the efficiencies h_U,m=I_U,m/I_total for U=A,B and |m|=0,1,2; , respectively, as a function of the relative window width w_A/p. The total efficiency h_A,0+ h_B,0 has also been plotted.

Fig 6: Efficiencies of a sliced wavefront CGH for the orders m=0 for A and B , respectively. The efficiencies for higher orders |m|=1,2 do not differ for A and B.

4. Discussion of the disturbing diffraction orders

Both methods of encoding two wavefronts in the CGH enable to generate the desired wavefronts A and B, but generate additional orders that will disturb the measurements if they cannot be filtered out. The control of the intensity ratio of A and B enables to adopt to different reflectivities of the aspheric and the mirror, but also changes the ratio between all other existing diffraction orders. Both pairs of equations (8) and (12) describe the resulting spatial frequencies (v_x ,v_y) after a wave has passed a null-lens system, i.e. an optional refractive part and the order m of the CGH, was than reflected by the surface A (or B), and has finally passed the null-lens system with diffraction order m^' a second time. In the case of a sliced CGH one has additionally to choose between j_A and j_B for both passes through the CGH. The spatial filter system blocks all parts of the waves which have spatial frequencies larger than the cut-off frequency
v_Cut-off . The number of detector points N in one direction and the attached diameter D of the CGH which is imaged onto the detector determines a resonable upper border for v_Cut-off = N/2D. Lower values for the cut-off frequency of the filter system will increase the blocking of disturbing diffraction orders but also decrease the dynamic range of the interferometer. Nevertheless, this is useful if the aspheric under test is nearly ideal and as much as possible disturbing light has to be blocked.

It should be mentioned that the spatial frequencies calculated with equations (8) and (12) are defined in the plane of the null-lens which is assumed for these first order calculations to be a thin element. Therefore, the diameter D of the CGH is taken to calculate the cut-off frequency. Of course, it has been also assumed that the CGH is imaged onto the detector in such a way that it fills the whole detector size. Another property of equations (8) and (12) has to be stated. That is, their derivation was only made for the case of a nearly perpendicular incidence onto a surface under test. The derivation will change if the cat’s-eye position is considered. Since the wavefront will be inverted, the phase function j has to be taken into account at the coordinates (x,y) and (-x,-y). Moreover, the distance d will be much larger for a cat’s-eye measurement. The equations for this case will not be shown in this proceeding.

We have to determine a lot of paramters for a certain case to discuss, for example the intensity ratio, whether the used surface R is A or B, or the degree of similarity between A and B. Since this could become a little unclear, we examine only some basic properties of ( v_x, v_y). For small numerical apertures of the aspheric the dependance of d on the coordinates (x,y) can be neglected. This dependance should be kept in mind although it is omitted in the following to simplify the equations.

4.1 Properties of ( v_x , v_y ) for superposed wavefronts
Let (m_in,m_out) denote the combination that m_in is the order used during the first pass through the CGH and m_out is the order used during the second pass, respectively. For a rotationally symmetric null-lens all first partial derivatives vanish on the axis. This generates the wellknown disturbing interferences on the axis for an on-axis testing setup with a CGH although no binary CGH has been used.

In the case that either only m=1 or only m^’=1 the second terms of equations (8) vanish and only the first terms can provide the separation of the diffraction order combinations:

(14)

where either m=1 or only m^’=1. This implies the use of a carrier frequency for the considered phase functions. Supposing a carrier in x direction we can decompose j_D and obtain:

(15)

where j_C= j_D-c.x, i.e. j_C is the remaining difference when the carrier has been subtracted. Now, the carrier frequency c can be adjusted in order to supress interference from this type of combinations (m,m’). In the case that m+m’ =2 the equations (8) will yield

	(16a)
	(16b)

So, the distance d from the CGH to the aspheric may not be too small as separation has to be achieved. It is also required that the second partial derivatives of equations (16) are not zero, i.e. j_C has to have a defocus term. These results are similar to the case of a binary CGH with one wavefront [5]. The desired combination (1,1) will have the most disturbing interference from the combination (0,2). To see the influence of changing the intensity balance between A and B the averaged efficiency is calculated, and is compared to h₁ (fig. 5). Supposing now that A is the mirror, a input weight g_A<0.5 would be choosen to adjust for the different reflectivities of A and B. But as can be seen from figure 5 this would increase the ratio h_0,2/h₁ and the disturbances would be worse. Nevertheless, the much larger d in the cat’seye case gives a hint that the separation will be quite good although equations (16) cannot be used. Indeed, the equations for ( v_x, v_y) in the cat’s eye case will be different from (8), but have similar dependencies and will provide the desired separation.

4.2 Properties of ( v_x , v_y ) for alternating subapertures
In the case of a sliced CGH aperture, (U,m,R,V,m’) denotes the combination that the order m of j_U is used during the first pass through the CGH, than surface R reflects the wavefront, and the order m’ of j_V is passed on the way back. Here, the desired diffraction orders are m,m’=0.

From equation (12a) we know that for m+m’¹0 the seperation in x direction is automatically achieved. The period of the subaperture grating p=p/s has only to be choosen small enough to avoid overlaping of adjacent combinations like (U,m,R,V,m’) and (U,m±1,R,V,m'). But if we consider m = -m' we have to look more into detail. All together four different types of combinations can be found, namely (A,m,A,A,-m), (A,m,A,B,-m), (B,m,A,A,-m), and (B,m,A,B,- m). The role of A and B may also exchange. To simplify equations (12), the phase function j_D= j_A- j_B will be used again. In the following, the more general cases of m ¹ -m^' will be included again. In the cases (A,m,A,B,m') and (B,m,A,A,m') the first terms of equations (12) always yield

(17)

And in the case of (B,m,A,B,m') the term with j_D will even be doubled. This again implies the use of a carrier frequency in order to separate the disturbing orders.

If a carrier is applied in x direction, the combinations like (A,0,A,A,0) and (A,0,A,B,±1) may overlap in the filter plane. Using j_C= j_D-c.x the component lv_x for (A,0,A,B,m’) forms to

(18)

So, it is clear that the best separation is achieved for c_x = s/2. The alternative of using a carrier c_y in y direction has some advantage. In all combinations except (A,m,A,A,m’) the carrier c_y appears in the component lv_y, i.e. the carrier c_y has only to be choosen large enough to provide the separation. In this case, s can be choosen more independently. Unfortunately, the spatial frequencies will vanish for all combinations (A,m,A,A,m’) if m = -m’. In practice it may still be possible to separate the disturbing orders due to higher order effects, but the separation cannot be described anymore by the equations (12) and the approximations made to derive them. The equations only give a hint that separation in x direction may be achieved if d is not too small. The degree of separation has to be examined by a higher order approximation or by exact raytracing via software tools.

Looking at figure 6 one can see that the orders |m|=1 still have quite high efficiencies. If one increases, for example, the efficiency h_A,0 for A, corresponding to an aspheric, by adjusting the relative window width w_A, the ratio h_A,±1/h_A,0 decreases very much. This provides better suppression of combinations (A,m,A,A,-m) with m¹0. At the same time, the problem becomes worse for the spherical wavefront B. The ratio h_B,±1/h_B,0 tends to 1, i.e. large disturbances could appear for the reference measurement. In the case of using a spherical reference surface, the intensity adjustment does not provide any improvements. But in the case of using a mirror in the cat’s eye position, the much larger distance d to the mirror again gives a hint that separation may be achieved in x direction. The equations for ( lv_x, lv_y) in the cat’s eye case will be different from (12) and will provide the desired separation.

5. Conclusion

The idea of a quasi-absolute test for rotationally symmetric aspherics requires an additional spherical wavefront in the null-lens of the interferometer. This enables a cat’s-eye measurement, i.e. a mirror placed in the focus of the spherical wave, similar to the 3-positions absolute test for spherical surfaces. Two methods of designing dual wavefront CGHs have been investigated. The approximative expressions for the spatial frequencies appearing in the CGH plane were derived for the case of nearly perpendicular incidence onto the surface under test. Aspects of adjusting the intensity balance of the two desired diffraction orders were examined. The influence of the balancing on the parasitic waves has also been shown. Some basic properties of the disturbances for different combinations of diffraction orders were given. They allow faster estimations whether filtering of disturbances can be achieved for a particular practical application. A principle problem is that the approximation of quasiabsolute test, i.e. similar wavefronts, competes with the necessary difference for the separation of disturbations. The errors of the quasi-absolute test introduced by using only similar wavefronts still have to be examined. In spite of these problems the approach of using dual wavefront CGHs for testing aspherics is promising. Up to now, we think that the superposition of the wavefronts has some advantages over slicing the hologram aperture because the second method includes a case, where separation cannot be achieved by the described first order terms. Only higher order effects, which are in general smaller, may provide the separation.

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5. N. Lindlein, "Analysis of the disturbing diffraction orders of computer generated holograms used for testing optical aspherics," accepted by Appl. Opt., scheduled for June 1 issue 2001.
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This Paper was presented at Fringe 2001 "The 4th International Workshop on Automatic Processing of Fringe Patterns" held in Bremen, Germany, 17-19 September 2001. Proceedings edited by Wolfgang Osten, BIAS, Germany. Please contact Wolfgang Osten for full set of proceedings at wolfgang@uni-bremen.de.

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