1. Introduction

The fabrication of high-precision optical surfaces is qualitatively connected to an accurate quality control by means of an interferometer. They are well established for testing plane and spherical surfaces. However, in the last few decades aspherical surfaces are more and more used in applications such as night vision devices, aerospace optics, miniaturized cameras and astronomical telescopes. The application of lenses and mirrors with aspherical surfaces allow the design of optical systems with a decreased number of elements, which results in reduced aberrations and a broader field view.

Aspherical surfaces need to be tested with null-correctors that convert plane or spherical wavefronts into aspherical ones. The accuracy of the interferometric test with a computer-generated hologram (CGH) [1] is defined by: CGH diffractive structure patterning error, the alignment error of the test setup and transmitted wavefront distortion (TWD). During the last years, the quality of CGH patterning was seriously improved by using specifically adapted CGH-writing systems [2] and by development of CGH certification and writing process control [3]. Using auxiliary holograms that can be fabricated on the same substrate decreases the time of the adjustment process and improves the measurement results due to a more precise alignment [4].

At present the most significant errors are introduced by the substrate imperfectness. The flatness and the homogeneity of the substrate cannot be improved below a certain limit. It requires substrates with a TWD in the range of λ/20, which increases the price significantly. Thus, the elimination of the TWD increases the precision and makes the testing much cheaper due to the possibility to use substrates with lower quality. This is very important since the size of a CGH can reach up to 300 mm or more.

In this paper we present a new type of combined CGH that eliminates TWD from interferometric measurements. The idea is to include the substrate in the area of the common path of the reference and test beams. In this case, substrate errors will be compensated and will not affect the measurement results. Our CGH was realized by combining two diffractive structures: a phase microstructure, that generates the test wavefront, and an amplitude microstructure, that generates the reference wavefront.

2. Interferometer with combined CGH

Interferometrical testing is well developed for flat and spherical surfaces. That is based on a comparison of the wavefront reflected from the surface under test and the wavefront reflected from a Fizeau reference surface. A Fizeau interferometer has the advantage that in the area of the common path of the test and reference waves (Fizeau cavity), vibrations, air turbulences and lens inhomogeneities are compensated and do not contribute to the measurement results. In the case of asphere testing with a CGH, the substrate is placed outside the Fizeau cavity and introduces errors to the measurement results.

 

Fig.1. Layout of a CGH test of an asphere. BS is a beam splitter, TS is a transmission sphere or any focusing objective.

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The idea of our method is to expand the Fizeau cavity up to the CGH surface to eliminate the TWD from the measurements. Our goal is to generate the reference wavefront at the same surface where the diffractive structure for generating the test wavefront is placed. So the Fizeau cavity needs to be extended up to the external face of the CGH (Fig. 1). In this case all disturbing optical effects will be compensated. This type of CGH is known as Diffractive Fizeau Null Lens (DFNL) [5].

The proposed DFNL presented in previous works was realized as a combination of an off-axis diffractive null lens combined with an amplitude linear grating. DFNLs of that type can operate only with plane incident wave. They need to encode differences between the plane and aspherical wavefront. So in case of a steep aspherical surface under test they are not applicable, since a very small period of diffractive structure is required.

In this paper we present a new DFNL type that can operate with convergent spherical wavefronts. This type of CGH is a hybrid reflective and transmissive diffractive element. They are needed to transform a spherical wavefront into an aspherical one and hence can be made with much larger periods of microstructures (Fig. 2).

 

Fig. 2. Phase diffractive structures for the generation of the test wavefront (a), amplitude diffractive structures for the generation of the reference wavefront (b) in reflection, combination of two structures (c), fabricated diffractive amplitude-phase structure (d).

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The diffractive structure of the combined CGH was realized as superposition of two diffractive structures that generate test (Fig. 2(a)) and reference wavefronts (Fig. 2(b)). The diffractive structure for the test wavefront shows low spatial frequencies, because the optical power of the surface under test can be compensated with a transmission sphere (TS). On the other hand, the structures of the reference wavefront has high spatial frequencies, as it operates in retro-reflection and needs twice the optical power as the initial TS.

The combined phase-amplitude structure is shown in Fig. 2(c). The phase diffractive structure is fabricated by dry etching. In order not to be sensitive to etching inhomogenieties, we place the amplitude structure only on top of each phase structure, as shown in Fig. 2(d).

The application of our combination method is limited by the minimal period of the amplitude structure of about d_a = 1 μm, which is defined by the fabrication process. The minimal period defines the maximal possible angle at that light diffracts back from an amplitude microstructure, sinα = mλ/2d_a, where m is the diffractive order and λ = 633 nm is the interferometer wavelength. Hence, an incident spherical wave has a maximal possible numerical aperture (NA) of NA = 0.31 for m = 1 and NA = 0.95 for m = 3. The period of the phase microstructure d_p is limited by the requirement that the phase microstructure needs to contain 3-4 amplitude zones to avoid crosstalk interference between test and reference beams. That defines the maximal possible asphere under test departure slope, which can be estimated as λ/d_p ~ 90 μm/mm.

Intensities of the reference and test beam can be attenuated by changing the duty cycles of appropriate diffractive gratings. Of course, that is possible only if the periods are not too small. The reference beam is produced by diffraction in first order on the amplitude grating with the reflection index R_chr. The intensity of the reference beam is defined by the diffractive amplitude of the linear grating: I_R=I ₀ R_chr(D_P)² sin(D_A)², where I ₀ is incident beam intensity, D_A and D_P are duty cycles of amplitude and phase microstructures, correspondingly. The duty cycle is defined as the ratio between the linewidth and the period.

 

Fig. 3. The dotted lines are the normalized intensities of the test beam due to the surface refractive index Rsurf=20% (a), Rsurf=10% (b), Rsurf=5% (c). The solid line is the normalized intensity of the reference beam.

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The test beam diffracts at the phase microstructures in first order, reflects from the surface under test with the index of reflection R_surf and passes the CGH in the same way and in the same manner: I_T=I ₀(η_P)² R_surf[(1 +D_A)/2]², where η_P is the diffractive efficiency of the diffractive phase structure. Fig. 3 shows the dependence of the reference beam intensity on D_A. The intensity of the reference beam is normalized to the incident beam intensity. The duty cycle of the microstructure for the object wave D_P was D_P = 0.5. It can be seen that an optimal contrast of the interference fringes can be obtained with D_A = 0.5 for R_surf = 20% (Fig. 3(a)), and D_A = 0.34 for R_surf = 10% (Fig. 3(c)), and D_A = 0.23 for R_surf = 5% (Fig. 3(c)).

3. Hologram fabrication and experimental results

The proposed method of an interferometric test with a combined CGH was investigated experimentally. The aim of the experiment was to demonstrate, that it is possible to nearly eliminate the TWD of a CGH substrate.

The optical layout of the experimental setup is shown in Fig. 1. Measurements were made with a Fizeau type interferometer Intellium Z100, using a transmission sphere objective with f-number f/5 using a standard spherical mirror with f/0.68 from a Zygo interferometer toolkit as surface under test, having a surface figure error of 0.05λ PV and 0.005λ rms.

The combined CGH was written on a fused silica substrate covered with a thin chromium film. The substrate diameter was 60mm with 8.06mm thickness and a small wedge between the plane surfaces of ~1 arc sec. Before CGH fabrication, the substrate was interferometrically tested. The flatness of both substrate sides was 0.05λ PV and 0.01λ rms. The transmitted wavefront distortion (TWD) in the area of the CGH pattern was 0.1 1λ PV and 0.02λ rms.

The CGH pattern as it is shown in Fig. 2 consists of two axially-symmetrical structures: a binary phase structure and a reflective chrome structure that was placed on the top of each phase zone. The diffractive phase structure for the generation of the test wavefront was calculated using a common raytracing method [6]. The reflective microstructure that was forming the reference wavefront was also designed using this method. The CGH pattern had a diameter of 50mm, the distance from CGH to the focusing point was 360mm, the distance between TS and CGH was L₂ = 70mm, and the distance from the focusing point to the reference mirror was L₃ = 25mm.

The combined CGH pattern was fabricated using a circular laser writing system (CLWS) [3,4]. The complex diffractive structure of the combined CGH was fabricated with a thermo-chemical technology by direct laser writing on the substrate into a thin chromium film. The process of the hologram fabrication consisted of five consequent steps.

Microimages of the fabricated CGH surface structure are shown in Fig. 4 and are corresponding to the structure depicted in Fig. 2(d). The images were made with the optical profiler Zygo NewView 6000. One can see that the axially-symmetrical zones of the first CGH are superimposed on the top of the phase zones of the first CGH with a profile depth of ~400nm. Each diffractive zone of the first CGH contains 3-4 chromium zones of the second CGH that are generating the reference wavefront by diffraction in retro-reflection. The measured diffractive efficiency of the first hologram was about 30% and the diffractive efficiency (in reflection) of the second amplitude hologram was about 1%.

The wavefront error is defined by the accuracy of the diffractive structure fabrication operating in diffraction order m and can be calculated as [7] ∆W(x,y) =-mλs(x,y)/S(x,y), where ε(x,y) is the position error of the diffractive zones in the direction perpendicular to the diffractive zones, S(x,y) is the period of the diffractive zones. In our experiment, the diffractive zones were fabricated with an accuracy of ε <50 nm. Thus, for the diffractive structure of the first hologram with a minimal period S_min=15μm and operating in the first diffractive order m=1, the error of the test wavefront is theoretically not exceeding 0.003λ PV. For the diffractive structure of the second hologram, S_min=1.3 μm, m=1 and the wavefront error is theoretically not exceeding 0.03λ PV. The experimental investigation of the interferometer was made in three steps:

 

Fig.4. 3D images (a, c) and plots (b, d) of the central part and periphery of the fabricated combined CGH. Zygo NewView 6000 microscope, objectives 20x (a,b) and 50x (c,d) Mirau.

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It is well-known that the measurement accuracy for a Fizeau interferometer is defined by the accuracy of the master surface [8]. Similarly, in our proposed interferometer measurement, the accuracy is defined by the accuracy of the reference wavefront, formed by the reflective part of the combined CGH. In our experiment a focusing objective transmission sphere (TS) was used (Fig. 1). The accuracy of this reference wavefront was defined by the analysis of the interference between the reflected reference wavefront and the Fizeau reference surface of the transmission sphere. A typical interferogram is shown in Fig. 5(a). The measured wavefront error was 0.16λ PV and 0.022λ rms which corresponds to the substrate TWD. Relative to this value the additional error contribution due to CGH pattern fabrication errors can be neglected. Using a diffractive retroreflective spherical wavefront on the CGH leads to an additional advantage: It serves automatically as an adjustment hologram for aligning the combined CGH relative to the incident spherical wavefront of the interferometer. Thus, there is no need for an additional alignment ring hologram around the main hologram, so that the full aperture can be used for generating the test wavefront.

In the second step the spherical reference mirror was set in the predefined distance from the combined hologram surface, and the TS was slightly tilted in order to remove its backreflection into the interferometer. Thus, the TS was used only as a focusing objective. An interferogram of a reference sphere using this configuration is shown in Fig. 5(b). In the central area of the interference pattern, a “hot spot” is visible, which is typical for axially-symmetrical holograms, having large (~ 1mm) diffractive zones in the CGH center, leading to a superposition of higher orders in the center. If this area is excluded from the analysis, the measured surface error was 0.08λ PV and 0.01λ rms. It can be seen, that this value is below the reference wavefront error, produced by the combined CGH. This can be explained by the effect of the substrate TWD compensation, because here the common path area has been included into the substrate.

 

Fig. 5. Ref. wavefront generated by combined CGH, measured with TS (a), Ref. sphere measured with combined CGH (b), Ref. wavefront generated by combined CGH and artificial TWD, measured with TS (c), Ref. sphere measured with combined CGH and artificial TWD (d). White arrows show the border of the fused silica plate, which covers the right part of the field.

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In order to further prove this effect, we artificially increased the TWD by introducing a thin fused silica wafer with 1mm thickness and low optical quality, covering approximately half of the aperture as shown in Fig 1. The resulting interferograms are shown in Fig. 5(c, d).

When using the reference wavefront of the Fizeau surface of the TS objective, the introduced wavefront distortion where clearly visible (Fig. 5(c)). The measured wavefront error in the distorted area was 0.58λ PV and 0.13λ rms. Using the combined CGH with an extended Fizeau cavity up to the hologram surface, these distortions could be compensated and the measured wavefront error was reduced to 0.12λ PV and 0.029λ rms (Fig.5d).

Conclusions

It is shown that the presented combined CGH is a new DFNL type, which allows the compensation of the substrate TWD when testing steep aspheres. The adaptation of the combined CGH microstructure duty cycle facilitates an optimum contrast of interferograms, depending on the specific reflectivity of the surface under test.