Laser differential confocal radius measurement method for the cylindrical surfaces

Lirong Qiu; Yang Xiao; Weiqian Zhao

doi:10.1364/OE.24.012013

1. Introduction

A cylindrical lens as a common aspheric lens can effectively reduce spherical and chromatic aberration in optical systems, simplify the construction structure, and improve the optical performance of the system. Hence, it is widely used in fields such as holographic illumination, optical information processing, and laser transmission [1–3], where a high precision measurement for the radius of a cylindrical lens is still a challenge in optical measurements.

The existing measurement methods for the radius of curvature (ROC) of a cylindrical lens are classified as contact and noncontact methods. The contact methods mainly include the chord method and fitting method. The chord method acquires the chord height and chord length through sampling points on the test cylindrical surface, and then obtains the radius of curvature through a theoretical formula. Based on the chord method, Shiue et al. designed an apparatus for measuring the curvature of spherical and cylindrical surfaces and the measurement accuracy was better than 1% for a cylindrical radius of less than 2.5 m [4]. Shukla et al. proposed a simple method for measuring the long radius of curvature of metal cylindrical surfaces, and the accuracy achieved was ± 0.3% for a metal cylindrical plate with a radius of curvature of 3228 mm [5]. Profilometer calculated the radius of the cylindrical surfaces by fitting information of the sampling points on the tested cylindrical surfaces, and the measurement accuracy is better than 0.05%. Furthermore, for the contact method, the measurement accuracy is limited by the number of sampling points and fitting precision, the measurement speed is slow, and the test cylindrical surface is easy to scratch. Therefore, interferometry has been proposed as one of the noncontact methods for measuring the radius of curvature of the cylindrical surface, which calculates the radius of the test cylindrical surface with the help of computer generated holograms (CGHs) [6–11]. However, interferometry is susceptible to environmental vibration and airflow [12–14].

In this paper, we propose a noncontact laser differential confocal radius measurement method (DCCRM) with high accuracy for a cylindrical surface. It uses a cylindrical wave generator (CWG) as the measurement lens to produce a converging cylindrical beam [15,16] and the null points of differential confocal curves to determine the cat’s eye and confocal positions, thereby achieving high accuracy measurement for the radius of the cylindrical surface by measuring the distance between the two positions. In comparison with existing measurement methods, DCCRM uses the differential subtraction of two intensity detection signals obtained from after and before focus to effectively suppress the common-mode noises caused by environmental interferences. Thus, DCCRM proposed has strong environmental anti-interference capability and high measurement accuracy.

2. Measurement principle

The DCCRM principle is shown in Fig. 1, which is based on the property that the null point of the differential confocal axial intensity curve precisely corresponds to the focus of the differential confocal measurement system [17]. The laser beam emitted from a point source S is collimated as a parallel beam through the collimating lens L_c, and then the parallel beam is converted as a cylindrical beam and converged on the test cylindrical surface by the CWG. The beam reflected by the test cylindrical surface passes through the CWG again and is converted into a parallel beam, and then the latter is focused by L_c on the two detectors D₁ and D₂ close to pinholes P₁ and P₂ placed before and behind the focus of L_c with the same offset. When the cat’s eye point of the test cylindrical surface is in the CWG focal line, the signal of the differential confocal curve I_A is null; when the vertex of the test cylindrical surface is in the CWG focal line, the signal of the differential confocal curve I_B is null. The DCCRM uses the null points of the curves I_A and I_B to identify the cat’s eye position A and confocal position B, precisely measures the distance between the two positions, and achieves a high precision measurement for the cylindrical surface radius of curvature.

Fig. 1 DCCRM principle. S is laser source, PBS is polarized beam splitter, WP is λ/4 wave plate, L_c is collimating lens, CWG is cylindrical wave generator, BS is beam splitter, P₁ and P₂ are pinholes, D₁ and D₂ are detector sand L_t is test cylindrical lens.

Download Full Size | PDF

When the test cylindrical lens moves near the cat’s eye position or the confocal position along the optical axis of the CWG, the measurement beam is reflected back along backtracking by the test cylindrical lens. The reflected beam passes through the CWG again and is converted into a parallel beam. Then, the parallel beam is converged by the collimating lens L_c, split into two parts by the BS, and received by two detectors D₁ and D₂ close to pinholes P₁ and P₂ with the same offset z_M from the focal plane of collimating lens L_c. the differential confocal signal can be obtained through the differential subtraction between the signals I₁(z,z_M) and I₂(z,z_M) from two detectors, and the differential signals I_A(z,z_M) and I_B(z,z_M) obtained near the cat’s eye position and the confocal position have the same expression given by

\begin{matrix} I (z, z_{M}) = \frac{1}{D^{2}} \cdot {| \int \int_{- \infty}^{+ \infty} P_{c} (ξ, η) P_{c} (- ξ, - η) P_{o} (η) P_{o} (- η) \exp (i \frac{k z}{f_{o}^{2}} η^{2}) \exp (i \frac{k z_{M}}{2 f_{c}^{2}} η^{2}) d ξ d η |}^{2} - \\ \frac{1}{D^{2}} \cdot {| \int \int_{- \infty}^{+ \infty} P_{c} (ξ, η) P_{c} (- ξ, - η) P_{o} (η) P_{o} (- η) \exp (i \frac{k z}{f_{o}^{2}} η^{2}) \exp (- i \frac{k z_{M}}{2 f_{c}^{2}} η^{2}) d ξ d η |}^{2} \end{matrix}

where D is the effective aperture of the measurement system that is equal to the minimum effective aperture of the CWG and collimating lens L_c, k = 2π/λ is the wave number of the measurement beam (λ is the wavelength of the laser source), z is the axial displacement between the test cylindrical lens and null point position, z_M is the offset of the pinholes from the focal plane of L_c, f_o and f_c are the respective focal lengths of the CWG and collimating lens L_c, and p_c(ξ,η) and p_o(η) are the pupil functions of the collimating lens L_c and CWG, respectively.

When the collimating lens and CWG have the same aperture, we have p_c(ξ,η) ≈p_c(−ξ,−η) ≈1 and p_o(η) ≈p_o(−η) ≈1. Then, after the normalization of the pupil in the polar coordinate, Eq. (1) can be simplified as

\begin{matrix} I (z, z_{M}) = I_{1} (z, z_{M}) - I_{2} (z, - z_{M}) \\ = {| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (i k (\frac{z}{f_{o}^{2}} + \frac{z_{M}}{2 f_{c}^{2}}) \frac{D^{2}}{4} ρ^{2} \cos^{2} θ) ρ d ρ d θ |}^{2} - \\ {| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (i k (\frac{z}{f_{o}^{2}} - \frac{z_{M}}{2 f_{c}^{2}}) \frac{D^{2}}{4} ρ^{2} \cos^{2} θ) ρ d ρ d θ |}^{2} \end{matrix}

where ρ is the normalized radial radius of a pupil and

ρ = \frac{2 \sqrt{ξ^{2} + η^{2}}}{D}

, and θ is the angle corresponding to variable ρ in the polar coordinate.

When D = 100 mm, f_o = 150 mm, λ = 632.8 nm and z_M = 0.22 mm, the differential confocal focusing curve is shown in Fig. 2.

Fig. 2 Differential confocal simulation curves.

Download Full Size | PDF

The cat’s eye and confocal positions can be determined precisely by the null points of differential curves I_A(z,z_M) and I_B(z,z_M), and the ROC of the test cylindrical surface is

R = Z_{A} - Z_{B}

It can be seen from Eq. (2) that the differential confocal intensity response signal is related to the effective aperture D and focal length f_o. The respective relations are shown in Figs. 3 and 4.

Fig. 3 Differential confocal simulation curves for different D.

Download Full Size | PDF

Fig. 4 Differential confocal simulation curves for different f_o.

Download Full Size | PDF

It can be seen from Figs. 3 and 4 that the null points do not change their position, whereas the detection sensitivity reduces and the width of the curve enlarges as D reduces and f_o increases.

Figure 5 shows the DCCRM simulation curves for different pinhole offset z_M. It can be seen from Fig. 5 that the identification accuracy at the null point is affected by the offset z_M. The optimized offset can be deduced by differentiating the differential confocal curve with respect to the axial displacement at z = 0. It was found that the optimized offset is z_M = 0.218 mm for f_o = 150 mm, D = 150 mm, and λ = 633 nm.

Fig. 5 Differential confocal simulation curves for different pinhole offset z_M.

Download Full Size | PDF

In practical DCCRM system, the pinhole size is not considered as an ideal point. When the normalized radius of pinhole size is v_P, the DCCRM axial intensity signal detected from the detectors is

\begin{matrix} I (ν_{p}, z, z_{M}) = \int_{0}^{ν_{p}} [I_{1} (ν, z, z_{M}) - I_{2} (ν, z, - z_{M})] \cdot ν \cdot d ν \\ = \int_{0}^{ν_{p}} [{| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (i k (\frac{z}{f_{o}^{2}} + \frac{z_{M}}{2 f_{c}^{2}}) \frac{D^{2}}{4} ρ^{2} \cos^{2} θ) J_{0} (ρ ν) ρ d ρ d θ |}^{2} - \\ {| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (i k (\frac{z}{f_{o}^{2}} - \frac{z_{M}}{2 f_{c}^{2}}) \frac{D^{2}}{4} ρ^{2} \cos^{2} θ) J_{0} (ρ ν) ρ d ρ d θ |}^{2}] \cdot ν \cdot d ν \end{matrix}

Using Eq. (4), DCCRM focusing curves for different v_P are shown in Fig. 6(a).

Fig. 6 Differential confocal simulation curves for different v_P. (a) Intensity curves (b) focusing sensitivity.

Download Full Size | PDF

It can be seen from Fig. 6 that the DCCRM focusing sensitivity detected reduces as pinhole size increases. But the intensity detected increases as pinhole size increases. In order to ensure enough the focusing sensitivity and intensity detected, the optimal pinhole size selected is v_P = 2. Then, the focusing sensitivity reduces 8% than that of ideal point pinhole, so the effect of pinhole size on measurements can be negligible.

3.Uncertainty analysis

3.1 Effect of distance measurement error

An XL-80 laser interferometer produced by the Renishaw Company is used as the distance measurement instrument (DMI) in the DCCRM system. The measurement accuracy is ± 0.5 ppm (k = 2) and the standard uncertainty introduced by the DMI is calculated by the Eq. (5).

u_{1} = \frac{0.5 \times 10^{- 6} \times R}{2} = 2.5 \times 10^{- 7} R_{D M I}

3.2 Effect of difference between two pinholes offsets

As shown in Fig. 7, when the offsets of two pinholes are different in the laser DCCRM system, the difference between the two offsets has an influence on the differential confocal intensity response curve and the respective null points o_A and o_B of the differential confocal response signals deviate from the cat’s position A and confocal position B. Assuming the offsets of two detectors are + z_M and −z_M−z_δM, the deviations of the differential confocal intensity response curves caused by these two offsets are Δz_A and Δz_B, respectively.

Fig. 7 Light path schematics with different offsets.

Download Full Size | PDF

Then, the differential confocal response signals I_A´(z,z_M) and I_B´(z,z_M) are given by

\begin{matrix} I_{A}^{'} (z, z_{M}) = {| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (\frac{i 2 π}{λ} \cdot (\frac{z}{f_{o}^{2}} + \frac{z_{M}}{2 f_{c}^{2}}) \cdot \frac{D^{2}}{4} \cdot ρ^{2} \cos^{2} θ) \cdot ρ d ρ d θ |}^{2} - \\ {| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (\frac{i 2 π}{λ} \cdot (\frac{z}{f_{o}^{2}} - \frac{z_{M} + z_{σ_{M}}}{2 f_{c}^{2}}) \cdot \frac{D^{2}}{4} \cdot ρ^{2} \cos^{2} θ) \cdot ρ d ρ d θ |}^{2} \end{matrix}

\begin{matrix} I_{B}^{'} (z, z_{M}) = {| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (\frac{i 2 π}{λ} \cdot (\frac{z}{f_{o}^{2}} + \frac{z_{M}}{2 f_{c}^{2}}) \cdot \frac{D^{2}}{4} \cdot ρ^{2} \cos^{2} θ) \cdot ρ d ρ d θ |}^{2} - \\ {| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (\frac{i 2 π}{λ} \cdot (\frac{z}{f_{o}^{2}} - \frac{z_{M} + z_{σ_{M}}}{2 f_{c}^{2}}) \cdot \frac{D^{2}}{4} \cdot ρ^{2} \cos^{2} θ) \cdot ρ d ρ d θ |}^{2} \end{matrix}

It can be seen from Eqs. (6) and (7) that the deviations ΔZ_A and ΔZ_B caused by the offsets of two pinholes are the same in sign and magnitude, and hence their effects on the radius measurements in the DCCRM system can be eliminated.

3.3 Effect of axial misalignment

As shown in Fig. 8, the DMI axis t and optical axis m of the system in the plane perpendicular to the generatrix of the test cylindrical lens should be coincident in DCCRM system, whereas a deviation angle between the two axes always exists in practice. When the angle between the axes m and t is α, the deviation of the radius ΔR_A caused by the angle α can be calculated using the below equation.

Fig. 8 Angles between DCCRM axes.

Download Full Size | PDF

Δ R_{A} = R_{M} (1 - \cos α)

In the adjustment of a practical system, a laser beam emitted from DMI illuminates on an observation marker, and the position deviation of laser beam between the ends of air bearing slider can be controlled within 1.5 mm by the careful adjustment. The effective length of the used air bearing slider is 1.3 m, then the angle α can be easily controlled to be less than 5′ by a careful adjustment of the system. Assuming the error caused by the angle α obeys a uniform distribution, the standard uncertainty caused by the axial misalignment is given by

u_{2} = \frac{Δ R_{A}}{\sqrt{3}} = \frac{R_{M} (1 - \cos α)}{\sqrt{3}}

3.4 Effect of figure error

In practice, the test cylindrical lens has a figure error due to the limitation of processing and testing ability, and the figure error has an effect on the radius measurement of the cylindrical lens. The figure error w(ρ,θ) can be expressed as the orthogonal Zernike circle polynomials [18] as

w (ρ, θ) = \sum_{j} a_{j} Z_{j} (ρ, θ)

Where a_j is the expansion coefficient and Z_j(ρ,θ) is the orthogonal Zernike circle polynomial.

When the figure error exists, the differential confocal intensity response curves at the cat’s eye position A and confocal position B will deviate. The respective different signals I_A(z,z_M) and I_B(z,z_M) at the cat’s eye position and confocal position are obtained by substituting Eq. (10) into Eq. (2).

\begin{array}{l} I_{A} (z, z_{M}) \\ = {| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (i k (\frac{z}{f_{o}^{2}} + \frac{z}{2 f_{c}^{2}}) \frac{D^{2}}{4} ρ^{2} \cos^{2} θ) \exp (- i k [w (ρ, θ) + w (ρ, π + θ)]) ρ d ρ d θ |}^{2} - \\ {| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (i k (\frac{z}{f_{o}^{2}} - \frac{z}{2 f_{c}^{2}}) \frac{D^{2}}{4} ρ^{2} \cos^{2} θ) \exp (- i k [w (ρ, θ) + w (ρ, π + θ)]) ρ d ρ d θ |}^{2} \end{array}

\begin{matrix} I_{B} (z, z_{M}) = {| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (i k (\frac{z}{f_{o}^{2}} + \frac{z}{2 f_{c}^{2}}) \frac{D^{2}}{4} ρ^{2} \cos^{2} θ) \exp (- 2 i k w (ρ, θ)) ρ d ρ d θ |}^{2} - \\ {| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (i k (\frac{z}{f_{o}^{2}} - \frac{z}{2 f_{c}^{2}}) \frac{D^{2}}{4} ρ^{2} \cos^{2} θ) \exp (- 2 i k w (ρ, θ)) ρ d ρ d θ |}^{2} \end{matrix}

The DCCRM simulation curves with the figure error w(ρ,θ) are shown in Fig. 9. It can be seen from Fig. 9 that among these orthogonal Zernike circle polynomials, the defocus Z₄ = $\sqrt{3} (2 ρ^{2} - 1)$ shown in Fig. 9(a) and primary astigmatism at zero degree Z₆ = $\sqrt{6} ρ^{2} \cos 2 θ$ shown in Fig. 9(b) have the most influence on the deviation of the differential confocal intensity response curve.

Fig. 9 DCCRM simulation curves with figure error w(ρ,θ). (a) Defocus. (b) Primary astigmatism at 0°.

Download Full Size | PDF

The deviation of radius ΔR caused by the figure error is equal to the difference between the measurement errors ΔR_A at the cat’s eye position and ΔR_B at the confocal position.

Δ R = Δ R_{A} - Δ R_{B}

The deviation of radius ΔR can be precisely compensated by the figure data obtained from ZYGO GPI phase shift interferometer with a measurement accuracy of up to λ/8. Hence, the compensated measurement uncertainty u₃ caused by the figure error is

u_{3} = \frac{λ}{8 k} = 0.04 μ m

3.5 Effect of offset of laser source

In practices, when laser source S and the focus of collimating lens L_C do not coincide, the laser beam emitted from L_C is misaligned, and the identification of cat’s eye and confocal positions will deviate in the measurements. As shown in Fig. 10, let the offset between laser source S′ and the focus S of collimating lens L_C be ΔS,the respective deviations of cat’s eye and confocal positions are Δz_A and Δz_B_.

Fig. 10 Light path schematics with offset of laser source.

Download Full Size | PDF

Then, the differential confocal intensity signals at positions o_A and o_B can be obtained by Eqs. (15) and (16).

\begin{matrix} I_{A}^{'} (Δ z_{A}, z_{M}) = {| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (\frac{i 2 π}{λ} \cdot (\frac{Δ z_{A}}{f_{o}^{2}} + \frac{Δ S}{2 f_{c}^{2}} + \frac{z_{M}}{2 f_{c}^{2}}) \cdot \frac{D^{2}}{4} \cdot ρ^{2} \cos^{2} θ) \cdot ρ d ρ d θ |}^{2} - \\ {| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (\frac{i 2 π}{λ} \cdot (\frac{Δ z_{A}}{f_{o}^{2}} + \frac{Δ S}{2 f_{c}^{2}} - \frac{z_{M}}{2 f_{c}^{2}}) \cdot \frac{D^{2}}{4} \cdot ρ^{2} \cos^{2} θ) \cdot ρ d ρ d θ |}^{2} \\ = 0 \end{matrix}

\begin{matrix} I_{B}^{'} (Δ z_{B}, z_{M}) = {| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (\frac{i 2 π}{λ} \cdot (\frac{Δ z_{B}}{f_{o}^{2}} + \frac{Δ S}{2 f_{c}^{2}} + \frac{z_{M}}{2 f_{c}^{2}}) \cdot \frac{D^{2}}{4} \cdot ρ^{2} \cos^{2} θ) \cdot ρ d ρ d θ |}^{2} - \\ {| \frac{1}{π} \cdot \int_{0}^{2 π} \int_{0}^{1} \exp (\frac{i 2 π}{λ} \cdot (\frac{Δ z_{B}}{f_{o}^{2}} + \frac{Δ S}{2 f_{c}^{2}} - \frac{z_{M}}{2 f_{c}^{2}}) \cdot \frac{D^{2}}{4} \cdot ρ^{2} \cos^{2} θ) \cdot ρ d ρ d θ |}^{2} \\ = 0 \end{matrix}

Thus,

Δ z_{A} = Δ z_{B} = - \frac{Δ S}{2} \cdot \frac{f_{o}^{2}}{f_{c}^{2}}

The deviations of cat’s eye and confocal positions coincide exactly, thus the effect of offset of laser source on radius measurement can be negligible.

3.6 Combined standard uncertainty

Besides the aforementioned errors, several random errors such as the positioning errors at the cat’s eye and confocal positions may also be observed in the DCCRM. Radom errors may be obtained through repeated measurements, and the measurement uncertainty caused by random errors is denoted as u₄. Thus, the combined standard uncertainty of the DCCRM is

u = \sqrt{u_{1}^{2} + u_{2}^{2} + u_{3}^{2} + u_{4}^{2}}

4. Experiments

The DCCRM main structure shown in Fig. 11 is based on Fig. 1. The experimental setup shown in Fig. 12 is established based on Fig. 11 to verify the DCCRM. A He–Ne laser with a wavelength of 632.8 nm is used as the light source. The CWG used is a CGH Cylinder Null H95F3C (produced by Diffraction International) with focal length of 281 mm. An achromatic lens (produced by Linos Photonics GmbH & Co.) with a focal length of 1000 mm and a diameter of 100 mm is used as the collimating lens Lc. The pinhole size used is 8 μm. An XL-80 laser interferometer (produced by Renishaw Company) is used as the DMI. And the effective aperture of the system is D = 95mm. The environmental conditions of the experiments are: relative humidity = (50 ± 0.4) %, temperature = (21 ± 0.2) °C and pressure = (99600 ± 70) Pa.

Fig. 11 DCCRM main structure schematic.

Download Full Size | PDF

Fig. 12 Experimental setup.

Download Full Size | PDF

Figure 13 shows single measurement result for the plane-concave cylindrical lens. The respective position coordinates of the cat’s eye and confocal positions for the test cylindrical lens corresponding to the null points Z_A and Z_B of the differential confocal curves I_A(z) and I_B(z) are o_A = 0.023781mm and o_B = −25.834429 mm obtained by the conic fitting on measurement data near zero, respectively. Thus, the ROC of the test plane-concave cylindrical lens was R = −25.858210mm.

Fig. 13 Single measurement result of the test cylindrical lens.

Download Full Size | PDF

The measurement uncertainty obtained by 10 measurement results shown in Fig. 14 is u₄ = 0.003166 mm.

Fig. 14 10 measurement results of the test cylindrical lens.

Download Full Size | PDF

The figure of the test cylindrical lens is measured using the ZYGO GPI interferometer. Moreover, the main orthogonal Zernike circle polynomial coefficients are shown in Table 1.

Table 1. Figure measurements of the test cylindrical lens

View Table

Substituting the above measurements into Eqs. (11) and (12), the measurement errors at the cat’s eye position and confocal position are ΔR_A = 0.0087 mm and ΔR_B = 0.0061 mm, respectively. Then, the measurement deviation of radius ΔR caused by the figure error is

Δ R = Δ R_{A} - Δ R_{B} = 0.0087 - 0.0061 = 0.0026 m m

Hence, the corrected radius is

R = - (25.85821 - 0.0026) = - 25.85561 m m

the combined standard uncertainty obtained using Eq. (18) is

u = \sqrt{u_{1}^{2} + u_{2}^{2} + u_{3}^{2} + u_{4}^{2}} = 0.003167 m m

and the expanded uncertainty is

U = \frac{2 u}{R} \times 100 % = \frac{2 \times 0.003166 m m}{25.85561 m m} \times 100 % = 0.024 %

5. Conclusion

A new laser differential confocal cylindrical radius measurement method is proposed based on the differential confocal focusing technology. The proposed DCCRM uses the null points of the differential confocal axial intensity curves to precisely identify the cat’s eye position and confocal position of the test cylindrical surface, thereby realizing a high precision ROC measurement of the cylindrical surface. Compared with other methods, the proposed DCCRM has high measurement precision and strong anti-interference capability. The preliminary experiments show that the DCCRM has a relative expanded uncertainty better than 0.03%. Thus, the DCCRM is an effective method for the high precision radius measurement of a cylindrical lens, and provides a new technical approach for the ROC measurement of cylindrical surfaces.

Acknowledgments

The authors gratefully acknowledge support from the National Natural Science Foundation of China (No. 61327010 and 51405020).

References and links

1. M. Schellhorn and H. Bülow, “Improvement of the beam quality of a gasdynamically cooled CO laser with an unstable resonator by use of a cylindrical mirror,” Opt. Lett. 20(12), 1380–1382 (1995). [CrossRef] [PubMed]

2. O. Momtahan, C. Hsieh, and A. Adibi, “Multifunctional and compact spectrometers based on cylindrical beam volume holograms,” Opt. Lett. 32(22), 3269–3271 (2007). [CrossRef] [PubMed]

3. P. Z. Fan, Z. Q. Zhang, J. Z. Zhou, R. S. Jin, Z. Z. Xu, and X. Guo, “Stigmatic grazing-incidence flat-field grating spectrograph,” Appl. Opt. 31(31), 6720–6723 (1992). [CrossRef] [PubMed]

4. S. G. Shiue, T. S. Liao, S. T. Chang, C. F. Kao, and Y. F. Chen, “Apparatus for measuring the curvature of spherical and cylindrical surfaces,” Rev. Sci. Instrum. 67(4), 1688 (1996). [CrossRef]

5. R. P. Shukla, D. V. Udupa, and A. K. Sinha, “Simple method for measuring long radius of curvature of metal cylindrical surface,” J. Opt. 30(2), 73–83 (2001). [CrossRef]

6. L. A. Selberg, “Radius measurement by interfermoetry,” Opt. Eng. 31(9), 1961–1966 (1992). [CrossRef]

7. S. Brinkmann, R. Schreiner, T. Dresel, and J. Schwider, “Interferometric testing of plane and cylindrical workpieces with computer-generated holograms,” Opt. Eng. 37(9), 2506–2511 (1998). [CrossRef]

8. N. Lindlein, J. Lamprecht, K. Mantel, and J. Schwider, “Interferometrical measurement of cylindrical lens with the help of computer generated holograms,” Proc. SPIE 440, 127–134 (2001). [CrossRef]

9. R. Schreiner, T. Herrmann, J. Roder, S. Muller-Pfeiffer, and O. Falkenstorfer, “Computer generated holograms for optical shop testing of aspheres,” Proc. SPIE 5856, 503–508 (2005). [CrossRef]

10. S. Reichelt, C. Pruss, and H. J. Tiziani, “New design techniques and calibration methods for CGH-null testing of aspheric surfaces,” Proc. SPIE 4778, 158–168 (2002). [CrossRef]

11. H. Liu, “Measurement accuracy verification of aspheric surface test with computer-generated hologram,” Chin. Opt. Lett. 10(7), 0171201 (2012).

12. T. L. Schmitz, A. D. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius of curvature,” Proc. SPIE 4451, 432–447 (2001). [CrossRef]

13. T. L. Schmitz, C. J. Evans, A. Davies, and W. T. Estler, “Displacement uncertainty in interferometric radius measurements,” CIRP. Annals-manufacturing technology. 51, 451–454 (2002). [CrossRef]

14. C. B. Lin, S. H. Yan, Z. G. Du, G. C. Wang, and C. H. Wei, “Symmetrical short-period and high signal-to-noise ratio heterodyne grating interferometer,” Chin. Opt. Lett. 13(10), 100501 (2015). [CrossRef]

15. J. Peng, D. Ge, Y. Yu, K. Wang, and M. Chen, “Method of misalignment aberrations removal in null test of cylindrical surface,” Appl. Opt. 52(30), 7311–7323 (2013). [CrossRef] [PubMed]

16. J. Peng, H. Xu, Y. Yu, and M. Chen, “Stitching interferometry for cylindrical optics with large angular aperture,” Meas. Sci. Technol. 26(2), 025204 (2015). [CrossRef]

17. W. Zhao, J. Tan, and L. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express 12(21), 5013–5021 (2004). [CrossRef] [PubMed]

18. D. Malacara, Optical Shop Testing (Wiley Interscience Publication, 2007).

Polynomial	Coefficient (λ)	Representation	Aberration name
Z₄	0.039	$2 ρ^{2} - 1$	Defocus
Z₅	0.128	$ρ^{2} \sin (2 θ)$	Primary astigmatism at 45°
Z₆	0.369	$ρ^{2} \cos (2 θ)$	Primary astigmatism at 0°
Z₇	0.064	$(3 ρ^{3} - 2 ρ) \sin θ$	Primary y coma
Z₈	−0.010	$(3 ρ^{3} - 2 ρ) \cos θ$	Primary x coma
Z₁₁	0.001	$6 ρ^{4} - 6 ρ^{2} + 1$	Primary spherical

Laser differential confocal radius measurement method for the cylindrical surfaces

Abstract

1. Introduction

2. Measurement principle

3.Uncertainty analysis

3.1 Effect of distance measurement error

3.2 Effect of difference between two pinholes offsets

3.3 Effect of axial misalignment

3.4 Effect of figure error

3.5 Effect of offset of laser source

3.6 Combined standard uncertainty

4. Experiments

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (14)

Tables (1)

Equations (22)

Optics Express