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The huge power solid-state lasers require large optical materials with high quality. The inhomogeneity must be required to be measured. Inhomogeneity measurement is often done at normal incidence by interferometer, while the size of large blanks is limited to the interferometer aperture. A five-step method to measure refractive index inhomogeneity over the interferometer aperture is proposed in this paper. The variation of the refractive index inhomogeneity of the glass blank is directly calculated using five interferograms measured at oblique incidence. The high repeatability of the results is given. The reliability of the method is further verified by comparing the same part measured at normal incidence.
© 2013 Optical Society of America
1. Introduction
The huge power solid-state lasers contains many large rectangular optical glass over 600 mm such as the laser of National Ignition Facility and the Laser of Megajoule etc [1]. The quality of the optics is specified over the full aperture [2]. The refractive index inhomogeneity of these large blanks used for these facilities has to be measured.
Inhomogeneity is a measurement of the variation in the refractive index within a material. Inhomogeneity is tested by phase measuring interferometer including a number of techniques. A well-known four-step method is developed by Twyman and Perry [3], Roberts and Langenbeck [4] and Schwider [5]. The refractive index inhomogeneity is evaluated by four interferometer’s measurements. The glass sample is polished for suppressing surface irregularities and a small wedge must be applied to the sample. A linear gradient of refractive index cannot be detected in this method. The linear gradient of refractive index of a parallel plate can be measured with Fourier Transform Phase-Shifting Interferometry (FTPSI) due to the simultaneous measurement of the surface profiles during a single wavelength scan [6]. A alternative simple method is given by Schwider using a two-step procedure with the glass sample or not where the glass plates are connected with the samples using an immersion oil liquid that has the same refractive index as sample [5]. The sample does not need to be polished for the measurement. Other improvement is described that four-step method is simplified to three steps by directly measuring the interference between the front and back surface instead of two measurements [7]. To our knowledge all the measurements are measured at normal incident angle. Therefore the size of measurements is limited to the interferometer aperture.
There are two methods to attain the inhomogeneity of full size over the instrument aperture. Stitching is well known method for enlarging the size in measuring wavefront profile. It is adapted to measure the inhomogeneity of large blanks with oil-on measurement [8,9]. Positioning inaccuracy and the algorithm is the main error source for the stitching. Another method is extrapolating which it is only suitable for some special case. The large blanks are measured on two parts, then the power and astigmatism of the inhomogeneity over the full aperture is extrapolated from the half part results [2]. The extrapolating equations are based on ideal mathematic models without considering the deviation of real materials.
In this paper, we present a method that the inhomogeneity is measured at oblique incident angle. It is suitable for polished sample with wedge. It consists of a sequence of five individual measurements. The inhomogeneity is evaluated by the five interferograms. As a result, the dimension of sample measured at horizontal direction is enlarged and inversely proportional to the cosine of the incident angle. Since there are many rectangle optics working on oblique incident angle in the laser facilities, it is significant that the inhomogeneity of these rectangle blanks can be evaluated on working angle at full aperture using the relatively small aperture interferometer. The measurement repeatability is related with the wavefront measurement accuracy of the interferometer.
2. Description of the method
The method is based on optical path difference measurements of the glass block in transmitted and reflected light by phase measuring interferometer. The interferometer has a Transmission Flat (TF) and a Reference Flat (RF). The test part of glass must be polished. It is suggested that the surfaces quality should be less than 5 wavelengths peak-to-valley. The polished glass must have a wedge about 10−3 rad so that the reflections from the different surfaces of sample can be separated on the detector plane.
Five measuring procedures are required for the algorithm as shown in Fig. 1. For the first measurement, the wavefront of the cavity consist of TF and RF is measured. The beam passes through TF and is reflected off RF. During the second measurement, the test part is inserted at oblique incident angle θ. The beam passes through TF and the test part, and is reflected off RF. The third measurement is similar to the second one, except that the test part is rotated 2θ around vertical axis. The beam passes through TF and the test part, and is reflected off RF. In the fourth measurement, the test part is kept the state of the third step, and RF is moved and adjusted to keep the light returning back exactly. Then the beam passes through TF and is reflected off the front of the test part and RF. The fifth measurement is like the fourth measurement, and the beam through TF is reflected off the back side of the test part and RF.
The variation of the refractive index Δn(x,y) is related to the transmission path. The mean refractive index is n0, thus the refractive index n(x,y) is written as:
The wave aberrations measured in the different step of the five-step procedure are defined as C, T1, T2, A and B, respectively. The Interferogram Scale Factor of 0.5 is used during five-step measurement. Their relationships are deduced as following:
where Z1 and Z2 are the surface deviations of the auxiliary mirror TF and RF, Z2f is the horizontally flipped data set of Z2, Za and Zb are the front and the back surface deviations of test part, respectively. The direction is defined in Fig. 1. Δn1 and Δn2 are the variations of the refractive index corresponded to the different transmissions of the second and the third measurement; d is the mean depth of the test part; θ is incident angle and θ′ is refractive angle in glass; Za′ is the data set of Za with a horizontal shift of x0, and x0 is the displacement on X direction between the light input and output of the front surface of the test part in the fifth measurement. x0 is expressed as Eq. (7).Coordinate of interferogram is corresponded to certain points of the front and back surfaces. The relationship is shown in Fig. 2. Light come from the interferometer returns back in every measurement. In the case of measuring T2, A and B, the interferometer light passing through the reference point “O” returns back, and it is naturally projected on the same pixel of CCD. A (ξ0, η0) is supposed to be related to point “O”. T2 (ξ0, η0) is relate to points “O” and “Q”. B (ξ0, η0) is response to points “O”, “Q” and “P”. T1 is measured at the inverse angle. Interferogram T1 is so shifted horizontally that T1 (ξ0, η0) relates to points “P” and “Q”. It is realized by shifting the right border of T1 horizontally to the corresponded edge of B.
The surface deviation of point O, P and Q is supposed as Za, Za′ and Zb respectively. T1, T2 and B relate to same Zb. So there is only one variable Zb shown as Eqs. (3), (4) and (6). Then inhomogeneity at direction OQ can be solved by eliminating influence of direction PQ.
The expression for oblique incidence is more complicated. Fortunately the variation of the refractive index Δn2 can be solved by eliminating the variables Z1, Z2, Z2f, Za, Za′, Zb and Δn1. It is written as following:
The result Δn2 is the deviation from mean refractive index n0 when the light passes through the path of the third step. If the incident angle θ is the working angle the result is expressed as the condition in situ.
The variation of the refractive index is separated from the surface deformations. This method is similar as the technique used in normal incidence configuration. This five-step measurement is suitable for oblique incidence, so the measurable size is enlarged over the aperture of the interferometer horizontally.
3. Experiments and discussion
A ZYGO MST 24” interferometer is used in our experiments. The surface deviations of the auxiliary mirror TF and RF (Z1 and Z2) are all better than 53nm (PV). The test sample is a fused silica glass with the dimensions of 428mm × 428mm × 54.7 mm. The mean refractive index n0 is 1.5.
The five-step measurement is implemented using the method of Fig. 1. The wavefront result of C, T1, T2, A and B is recorded by the interferometer, respectively. It is easy to determine whether the surface under test corresponds to the front surface or the back one. When one touch the back surface with a finger, the back surface fringes will change slightly. The incident angle θ is chosen to be 28 degree. The value is guaranteed by a mask edited on the MASK interface of the application. The horizontal length of the mask lm is calculated as Eq. (9).
where l is the horizontal length of the test part, d is the depth of the test part, θ is the incident angle. In the second and the third measurement, the test part is so placed that the horizontal shadow of the test part in the light is equal to lm. Then the incident angle equals to θ. The accuracy of θ can be evaluated by length measurements of the empty cavity.The refractive angle in glass is calculated by Snell’s law.
The result Δn2 is calculated from Eq. (8) by the commercial software of Zygo corp. It is also conveniently obtained using other technical computing software.
Results of C, T1, T2, A and B in five-step procedure are shown in Fig. 3, respectively. Inhomogeneity map Δn2 calculated from Eq. (8) is shown in Fig. 4, and the horizontal size of 34.7 cm is decided by the size X in Fig. 3 (e). Therefore, the final Horizontal dimension of 39.3 cm is retrieved by the size divided by cos28°.
Inhomogeneity of the block is defined as peak-to-valley value (PV). Six inhomogeneity measurements are used to calculate repeatability. The six results are list in Table 1. The mean of the PV is 1.43 × 10−5, and the repeatability (1σ) of the PV is 1.4 × 10−7. Inhomogeneity RMS is 1.43 × 10−6, and the repeatability of RMS is 1 × 10−8. The measured length of the test part is the size X divided by cos28°. Horizontal dimension of 39.3cm is obtained by measuring beam size of 34.7cm.
Table 1. Results of Inhomogeneity Measured at 28 Degree Incidence
When the sample is test under oblique incidence, there is loss in size. The percent of measureable size to full horizontal length is expressed as t as following:
Where d is the depth of the part, l is the horizontal length of the test part, θ is the incident angle, n0 is the refractive index. Better than 91 percent can be obtained in our measurement. The ratio of thickness to size d/l is less, the measurable percent is larger. If the part is used in laser facility at the same angle as being measured, the loss section can be omitted.
4. Verification of the inhomogeneity result
The inhomogeneity result of oblique incidence is compared with the result of normal incidence obtained by four-step measurements [[5]]. This normal-incidence result can be corrected for angle of incidence [[10]]. It is divided by cosθ′, where θ′ is refractive angle shown as Eq. (10). Four results of wavefronts are C′, T′, A′ and B′ as shown in Fig. 5, respectively. The refractive profile Δn′ is calculated using Eq. (12) and shown in Fig. 6.
The size is selected to be 39.3 cm corresponding to the size 34.7 cm of 28 degree incidence. The inhomogeneity measured at normal incidence is 1.41 × 10−5 (PV). The inhomogeneity RMS is 1.7 × 10−6. The repeatability of normal incidence is evaluated by the accuracy of the wavefront measurement [[11]]. The meaning of 5 nm wavefront accuracy is 1 × 10−7 if the sample is 50 mm thick. It is the same order as the oblique measurement repeatability.
The measured zones are plotted in Fig. 7. The light passes through the part at different directions. The normal incidence specification is corrected to be 1.49 × 10−5 (PV) and 1.88 × 10−6 (RMS) for ease of comparison. The inhomogeneity difference between the results measured at different incidences, e.g. Figure 4 and corrected Fig. 6, is 7 × 10−7 (PV) and 4.4 × 10−7 (RMS). When the aperture is selected to be 350mm × 400mm (normal inciedence) shown as Fig. 8, e.g. 309mm × 400mm shown as Fig. 9 (oblique-incidence), the inhomogeneity difference is decreased. The wavefront map is inhomogeneity map multiplied by depth d. The result of Fig. 8 has been corrected by dividing cosθ′. The cross profiles at the same reference point are shown on the right side of the Figs. 8 and 9. The up curve is the wavefront profile at x-line and the down curve is the wavefront profile at y-line. The x-line wavefront PV is displayed at below. The two x-lines of Fig. 8 and Fig. 9 are similar. The difference of PV values is 1.33nm between 220.59nm and 219.26nm which is corresponded to 2.4 × 10−7 of inhomogeneity PV.
Fig. 8 Wavefront map measured at normal incidence. The wavefront map is homogeneity map multiplied by depth d and divided by cosθ′. The aperture is 350mm × 400mm excluding edge influence. The right curves are wavefront profiles at sampled lines. The up curve is the wavefront profile at x-line and the down curve is the wavefront profile at y-line. The PV and rms values below the curves are corresponded to x-line.
Fig. 9 Wavefront map measured at oblique incidence. The wavefront map is homogeneity map multiplied by depth d. The aperture is 309mm × 400mm excluding edge influence. The right curves are wavefront profiles at sampled lines. The up curve is the wavefront profile at x-line and the down curve is the wavefront profile at y-line. The PV and rms values below the curves are corresponded to x-line.
5. Conclusion
The measurement of the inhomogeneity at oblique incidence is presented and explained. Five-step wave aberrations measurements including cavity C, transmittance T1, T2, reflectance A and B are used for deducing the inhomogeneity profile of the sample at oblique incidence. The repeatability of the results is 1.4 × 10−7. The reliability is verified by comparing with the result measured at normal incidence.
Acknowledgment
The author gratefully acknowledges the support of the Chinese Academy of Science and the discussions with Dr. Shijie Liu and Prof. Deyan Xu.
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