1. Introduction

Owing to their excellent processing properties, spherical optical elements are widely used as essential elements in optical systems such as laser fusion optical systems, ultraviolet lithography lenses, interferometer objectives, and microscopy objectives [1,2]. The imaging characteristics of a spherical lens can be determined by its radius of curvature (ROC) r, refractivity n, central thickness t, and surface figure w. High-accuracy measurement of the parameters of a spherical lens is important in order to ensure desired performances of the spherical lens and the entire optical system. However, there are still two problems in the existing measurement techniques for the parameters of the spherical lens [3–9]:

Therefore, to improve the measurement accuracy of the existing measuring methods and multi-parameter comprehensive measurement capability, we proposed a new laser differential confocal interference multi-parameter measurement (DCIMPM) method with the integration of the laser differential confocal focusing technique and laser interference technique [10]. The proposed DCIMPM method, which integrates the optical arrangements of laser differential confocal focusing and laser sphere interference, uses the laser differential confocal parameter measuring technique to measure the values of r, n, and t of a spherical lens, and then uses the laser sphere interference technique to measure its value of w. This process, for the first time, achieves high-accuracy comprehensive measurement of these parameters of a spherical lens on a single apparatus. The DCIMPM method provides a new means for comprehensive measurement of multiple parameters of a spherical lens.

2. Measurement Principle

2.1 DCIMPM principle

The DCIMPM principle is shown in Fig. 1. The laser beam emitted from a point source (P) passes through a polarized beam splitter (PBS) and a quarter-wave plate (λ/4). It is then collimated as a parallel beam by a collimating lens (L_c) and enters a standard lens (L_S). The reference beam reflected back by the reference surface of the L_S passes through the L_C, λ/4, PBS, beam splitter 1 (BS1), and reflector R_M into a phase-shifting interferometry system. The measurement beam transmitted by the reference surface of the L_S is converged as a spherical beam illuminating the test lens by L_S. When the test lens lies at the cat’s eye A or confocal position B, as shown in Fig. 1, the measurement beam illuminating the test lens is reflected back along the path of the test lens and passes through L_S, L_C, and λ/4 again. After the PBS reflects the beam, BS1 then splits the beam into the phase-shifting interferometry system and differential confocal measurement system (DCMS).

 

Fig. 1 DCIMPM principle. The PBS is the polarized beam splitter,P is a point source, L_c is the collimating lens, L_S is the standard lens, R_M is the reflector, L_W is the test spherical lens, BS1 and BS2 are the beam splitters, P₁ and P₂ are pinholes,D₁ and D₂ are detectors,CCD is the detector,M is the offset of the pinhole from the focal plane of L_C, and L_i is the imaging lens.

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The last surface of the standard lens L_S, coated with a reflective film, is used as the reference surface, and the center of curvature of the reference surface and the converging point O of the measurement beam are in agreement. Then, the reference beam reflected by the reference surface and the measurement beam reflected by the test surface return along their path into a phase-shifting interferometry system and interfere with each other, and the interference pattern formed is used to measure the figure of the test surface.

The beam reflected by BS1 enters into the laser differential confocal measurement system and is split into two parts by BS2. The beam reflected by BS2 passes through the pinhole P₁ before the focus with an offset of M and is received by detector D₁ close to P₁, where the detected axial intensity response is I₁(u,u_M). The beam transmitted by BS2 passes through the pinhole P₂ behind the focus with an offset of -M and is received by detector D₂ close to P₂, where the detected axial intensity response is I₂(u,-u_M). When the test lens lies near the cat’s eye position A or confocal position B, the response signal I_D(u,u_M) of the laser differential confocal measurement system is obtained through subtraction of the intensity signals detected simultaneously from detectors D₁ and D₂, and can be expressed by Fresnel integration [11] as

When the figure error is w(ρ, θ) < λ/4, the effect of the figure error on the response signal I_D(u,u_M) of the laser differential confocal measurement system is neglected, and the signal I_D(u,u_M) can be simplified as:

Differentiating Eq. (2) with respect to u, the sensitivity S(u_M) of the normalized differential confocal response curve is obtained as shown in Eq. (3).

Using Eqs. (2) and (3), the sensitivity S(u_M) and differential confocal curve I_D(u,u_M) are shown in Fig. 2. It can be seen from Fig. 2 that the sensitivity S(u_M) is maximum and S_max = 0.54 when u_M = 5.21, i.e., the differential confocal curve I_D(u,u_M = 5.21) has the best sensitivity of S_max = 0.54 at zero position; therefore, the corresponding offset of the pinhole is M = 10.42λf_c'²/(πD²), which is inversely proportional to the square of the effective aperture of Lc.

 

Fig. 2 Sensitivity of Differential confocal response curves for different u_M. (a). Sensitivity S(u_M) (b). Differential confocal curve I_D(u,u_M)

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Substituting $u=\frac{\pi D^2}{2\lambda f_o{\text{'}}^2}z$ and u_M = 5.21 into Eq. (2), the differential confocal response curve is

2.2 Measurement principle for the parameters of a lens

2.2.1 Figure measurement

As shown in Fig. 1, when the DCIMPM method is used to measure the surface figure of the test lens, the test lens lies at the confocal position, and L_S is adjusted until the reference beam reflected by the reference surface and the parallel beam collimated by L_C have the same optical axis. This causes the reference beam and measurement beam reflected back along their path by the reference surface and test surface, respectively. The reflected beams again pass through L_S, L_C, and λ/4, are reflected by the PBS, and are reflected by the reflector R_M after passing through BS1 into a phase-shifting interferometry system. In the phase-shifting interferometry system, the two beams interfere and the resulting interference pattern is recorded by the CCD.

In order to improve the measurement accuracy of the surface figure, a five-step phase shifting measurement is adopted for the reference wave front of L_S. A piezoelectric ceramic phase shifter drives L_S to obtain the phases shifted by 0, π/2, π, 3π/2, and 2π, the intensity distributions I(x,y,t_n) of the interference pattern recorded by the CCD are I₁, I₂, I₃, I₄, and I₅, respectively. Then, the phase φ(x,y) of the light beam reflected from the test surface is:

The phase Φ(x,y) is obtained by unwrapping the phase φ(x,y), and the surface figure w(x,y) of the test surface obtained from the phase Φ(x,y) is:

2.2.2 Parameter measurements

As shown in Fig. 3(a), when the DCIMPM method is used to measure r of a lens, the DCMS uses the null points Q_A and Q_B of the differential confocal response curves I_A(z) and I_B(z), respectively, to identify the cat’s eye and confocal positions of the test lens. Then, it uses a distance measurement instrument with high accuracy to measure the cat’s eye and confocal positions z_A and z_B, and the test r is obtained by calculating the difference between the positions z_A and z_B, as shown in Eq. (7).

 

Fig. 3 Measurement of the ROC, thickness, and refractivity of a spherical lens. (a) ROC measurement, (b) thickness and refractivity measurement.

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As shown in Fig. 3(b), when the DCIMPM method is used to measure n and t of a lens, L_S is moved to cause the measurement beam to converge on the vertices of the front and back surfaces of the test lens and on the reflector L_R with or without the test lens L_W in the measurement light path, respectively. The DCIMPM method uses the null Q₀, Q₁, Q₃, and Q₂ of the differential confocal response curves I₀(z), I₁(z), I₃(z), and I₂(z) to determine the positions z₀, z₁, z₃, and z₂ of L_S. Then, the corresponding optical distances between these positions are calculated as d₁, d₂, and d₃ using the positions z₀, z₁, z₃, and z₂, and high-precision measurements of t and n for the test lens L_W are achieved by ray tracing.

The detailed ray tracing measurement process is shown in Fig. 4.

 

Fig. 4 Ray tracing used for refractivity and central thickness measurement.

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As shown in Fig. 4, an optional ray in the pupil plane of the L_S is used for ray tracing, where the angle between this ray and the optical axis is θ₁. Equations of thickness t and refractivity n related to the optical distance d_i (i = 1, 2, 3) are deduced by ray tracing, the equations for which are shown in Eq. (8).

The expressions t(r₁, d₁, n, ρ, NA) and n(r₁, r₂, d₁, d₂, d₃, n, ρ, NA) of ray tracing for the central thickness t and refractivity n can be obtained using Eq. (8). The expressions for t and n of the test lens are obtained by integration of all rays in the pupil plane, as shown in Eq. (9).

Substituting the parameters (r₁,r₂) and d_i (i = 1, 2, 3), which are obtained using the measurement method shown in Fig. 3, and NA into Eq. (9), t and n are then calculated using the iterative calculation.

3. DCIMPM system development

3.1 Main structure of the DCIMPM system

The DCIMPM system shown in Fig. 5 is established based on the DCIMPM principle shown in Fig. 1, which consists of a laser differential confocal interference system, four-dimensional precision adjusting stage, precision air-bearing linear driving system, laser distance measurement, environment compensation system, computer measurement and control system. Here, DMI is a distance measurement instrument.

 

Fig. 5 Diagram of the DCIMPM system

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The laser differential confocal interference system is mainly used to identify the positions of the surfaces of the test lens and measure the figure of the test lens; the four-dimensional precision adjusting stage is used to precisely adjust the alignment of the optical axes between the test lens and DCIMPM system; the precision air-bearing linear driving system drives the working table along the optical axis to achieve precise scanning and positioning of the test surfaces; the laser distance measurement and environment compensation system is used to measure and record the movement distance and positions of the working table, and the whole measurement system can achieve automatic measurement and data processing and assessing by the computer measurement and control system.

Based on Fig. 5, the DCIMPM system was designed as shown in Fig. 6.

 

Fig. 6 Designed DCIMPM system

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Based on the designed DCIMPM system shown in Fig. 6, the DCIMPM system shown in Fig. 7 was developed, where the laser differential confocal interference system, four-dimensional precision adjusting stage, and precision air-bearing linear driving system were developed.

 

Fig. 7 Developed DCIMPM system

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In the DCIMPM system, the laser source has a wavelength λ = 632.8 nm, the collimating lens L_C used is an achromatic lens with a focal length of 1000 mm and an aperture of 100 mm, the standard lens L_S used is a standard sphere (Zygo Corporation, USA) with a focal length of 151.74 mm and an aperture of 100 mm. The laser distance measurement and environment compensation system used is an XL-80 laser interferometer system (Renishaw Corporation, UK), and the distance measurement accuracy is ± 0.5 ppm.

3.2 Measurement and control system

The laser differential confocal interference multi-parameter comprehensive measurement system comprises a laser differential confocal multi-parameter measurement system (DCMPMS) and a phase-shifting interferometry measurement system for the figure surface.

3.2.1 Measurement and control system of DCMPMS

Figure 8 shows a measurement and control diagram of the laser DCMPMS, where the measurement and control module is used for acquiring the positions of the DMI and intensity information of laser differential confocal system, and achieving the adjustment and position measurement of the test lens by the motor driving controller. The focus scanning and aiming module is used to control synchronous data acquisition from the DMI and two detectors, and the positioning signal of the test lens by the differential subtraction of the signals from the two point detectors. The data processing module is used to process the differential confocal signal and information obtained from the DMI, and then to obtain the measured parameters of the test lens.

 

Fig. 8 Measurement and control diagram of the DCIMPM system

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Figure 9 shows the data processing interface of the differential confocal parameter measurement system, which comprises the parameter setting for data processing, the selection of curve filtering approach, the converging spot monitoring near the focus, and the displays of differential confocal curve and measurement results.

 

Fig. 9 Data processing interface of differential confocal parameter measurement system

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3.2.2 Phase shifting interferometry measurement system for the figure of spherical surface

Figure 10 shows a diagram of the phase shifting interferometry measurement software; the measurement and control module consists of the image acquisition module and movement control module, where the image acquisition module is used to collect and display the phase-shifting interference image in real time, and the movement control module is used for controlling the movement of the air-bearing linear guide to adjust the positions of the test surface and optical axis. The interference image acquisition module achieves the five-step phase shift of the reference surface using a piezoelectric ceramic driving module. The interference image-processing module acquires the measurements by phase extraction, phase unwrapping, and Zernike fitting polynomial to five fringe patterns captured through the five-step phase shift.

 

Fig. 10 Diagram of phase shifting interferometry measurement software

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Figure 11 shows the data processing interface of the phase-shifting interferometry measurement software, which comprises the parameter setting of fringe pattern processing, the interference image acquisition of the five-step phase shift, the phase extraction, phase unwrapping, Zernike fitting polynomial, 3D display, and display of the PV and RMS values of the fringe patterns.

 

Fig. 11 Data processing interface of phase-shifting interferometry measurement software

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4. Error analyses

4.1 Measurement uncertainty of radius

In the DCIMPM system, the main errors influencing the radius measurement include the measurement error Δ_d of the DMI, the error Δ_γ caused by the angle γ between optical axes of the DCIMPM system and DMI, the position adjustment offset Δ_a of the test spherical surface, the error Δ_w caused by the figure w of the test lens and the error Δ_z caused by the focusing error and the detailed uncertainties caused by these errors are as follows.

1) Uncertainty caused by distance measurement error

The standard uncertainty caused by distance measurement error of DMI used in the DCIMPM system satisfies

2) Uncertainty caused by misalignment between DMI and DCIMPM axes

In practice, the angle γ between the optical axes of the DMI and DCIMPM system always exists; this will cause measurement error of the ROC. The adjustment accuracy of angle γ can be controlled within 4′ by CCD, and the measurement uncertainty of the ROC caused by angle γ is:

3) Uncertainty caused by adjustment offset of test lens position

In the ROC measurement, the measurement uncertainty caused by the offset of test lens position satisfies

4) Uncertainty caused by of figure error

The measurement uncertainty of the ROC caused by figure error can be compensated for by measuring the actual figure. Considering the error obeys the uniform distribution, the uncertainty after the compensation is

5) Uncertainty caused by focusing error

The signal-noise ratio of the detection system is SNR = 150:1. When the largest slope of the differential confocal focusing curve is S_max = 0.54, the relation between the focusing error Δz and the relative aperture D/f_o' of lens L_S satisfies

6) Uncertainty caused by the difference between offsets of two pinholes

In practice, two offset of two pinholes are impossible to be equal in a laser differential confocal interference measurement system. As shown in the Fig. 12, assuming the offset of pinhole P₁ is M + δ and the offset of pinhole P₂ is –M, the corresponding normalized offset are u_M + u_δ and –u_M respectively, and the resulting position offset of zero point are ΔZ_A and ΔZ_B, respectively.

 

Fig. 12 DCIMPM with different pinhole offset.

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Then, the differential confocal signal described by Eq. (2) can be written as I_DA(u, u_M + u_δ) and I_DB(u, -u_M) below.

Therefore, it can be known from the above equations that the position offset ΔZ_A and ΔZ_B of zero points resulted from the different offset of two pinholes are the same in value and direction. Consequently, the effect of difference between offsets of two pinholes on radius measurement can cancel each other out.

7) Synthetic standard uncertainty

Assuming the effects of the aforementioned errors on the ROC measurements are independent, the total measurement standard uncertainty of the DCIMPM system is obtained using Eq. (17).

4.2 Measurement uncertainty of the lens thickness

In the DCIMPM system, the main errors influencing the thickness measurement include radius measurement error Δ_r1 of the front surface of the test lens L_w, the position error Δ_d1 of the movement distance d₁ of the test lens L_w, the error Δ_NA of the calibrated numerical aperture of the DCIMPM system, the error Δ_n of the refractive index n, and the detailed uncertainties caused by these errors are as follows.

1) Uncertainty caused by radius measurement error Δ_r1

The standard uncertainty caused by radius measurement error Δ_r1 of the front surface of the test lens L_w can be obtained by Eq. (17).

2) Uncertainty caused by the position error Δ_d1

The standard uncertainty caused by the position error Δ_d1 of the movement distance d₁ of the test lens L_w can be obtained by Eq. (10).

3) Uncertainty caused by error Δ_n

The refractive index n of the test lens L_w can be obtained by the minimum deviation method with accuracy of 3 × 10⁻⁶, so the standard uncertainty caused by the error Δ_n of the refractive index n is

4) Uncertainty caused by error Δ_NA

The NA of the DCIMPM system is calibrated by an optical flat using Eq. (21).

The standard uncertainty caused by calibration error of the NA of the DCIMPM system is calculated by Eq. (22).

5) Synthetic standard uncertainty

Differentiating Eq. (9) on r, d_, NA and n, the synthetic uncertainty obtained by the above measurement errors is:

4.3 Measurement uncertainty of refractive index

In the DCIMPM system, the main errors influencing the refractive measurement include the radius measurement errors Δ_r1 and Δ_r2 of the two surfaces of the test lens L_w, the position errors Δ_di (i = 1, 2, 3) of the movement distances of lens L_w, the error Δ_NA of the calibrated numerical aperture of the DCIMPM system, and the detailed uncertainties caused by these errors are as follows.

1) Uncertainty caused by radius measurement error Δ_ri

The standard uncertainty caused by the radius measurement errors Δ_ri (i = 1,2) of the two surfaces of the test lens L_w can be obtained by Eq. (17).

2) Uncertainty caused by the position error Δ_di

The standard uncertainty caused by the position error Δ_di (i = 1,2,3) of the movement distance d_i (i = 1,2,3) of the test lens L_w can be obtained by Eq. (10).

3) Uncertainty caused by error Δ_NA

The standard uncertainty caused by calibration error of the NA of the DCIMPM system can be calculated by Eq. (23).

4) Synthetic standard uncertainty

Differentiating Eq. (9) on r, d and NA, the synthetic uncertainty obtained by the above measurement errors is:

4.4. Measurement error of surface figure

In the DCIMPM system, the main errors influencing the surface figure measurement include the geometry error M_θ and the vibration error M_v of phase shifting, as well as the detection error of the detector M_D and measurement error M_Φ of surface figure. In the DCIMPM system, the Fizeau common light path structure is used; therefore, the effects of the aberrations of the optical measurement system on measurements can offset each other, and the errors affecting surface figure measurement mainly comprise the figure error of the standard lens and environmental disturbance. When the standard lens used is an aplanatic lens with λ/20, it causes an uncertainty M_Φ of λ/20.

5. Experiments

5.1 Test sample preparation

As shown in Fig. 13, a triple prism and a plano–concave lens are used to verify the DCIMPM and are made of glass with a peak to valley of 10⁻⁶ in the refractivity fluctuation. The triple prism has a length of 50 mm and a height of 30 mm, and the plano–concave lens has a nominal ROC of 100 mm and a diameter of 30 mm. The plano–concave lens shown in Fig. 13(b) is also used for the figure measurement experiment. The experimental conditions are 22°C ± 0.2°C for temperature, 48% ± 0.4% for relative humidity and 100400Pa ± 50 Pa for pressure.

 

Fig. 13 Test samples: (a) triple prism, (b) plano-concave lens

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Owing to the facts that the measurement uncertainty of the minimum deviation method (MDM) is 3 × 10⁻⁶ and the refractivity fluctuation of the glass has a peak to valley of 10⁻⁶, the refractivity of the triple prism obtained by MDM can be considered to be a standard value of the refractivity of the plano-concave lens with a 10⁻⁵ accuracy level; the true value of the refractive index of the plano-concave lens measured by the Chinese National Institute of Metrology (CNIM) is n_sd = 1.513601 ± 0.000003. The central thickness of the lens is measured by high-precision contact measurement.

5.2 Experiments

Figure 14 shows the measurement curves of the ROC for the test plano-concave lens obtained using the developed DCIMPM system shown in Fig. 7, where the axial coordinate corresponding to the null point Q_A gotten by linear fitting of the differential confocal response curve near the cat’s eye position is z_A = –0.008915 mm, the axial coordinate corresponding to the null point Q_B gotten by linear fitting of the differential confocal response curve near the confocal position is z_B = –100.103872 mm, and the measured ROC of the test plano-concave lens is r₁ = |z_B – z_A| ≈100.094957 mm.

 

Fig. 14 Measurement curves of the ROC for the test plano-concave lens

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Figure 15 shows ten measurement results of the ROC for the test plano-concave lens obtained using the developed DCIMPM system shown in Fig. 7. The 10 repeated measurements of the ROC resulted in an average measurement of r_avg ≈100.094736 mm with a standard deviation of 0.247 μm and a relative error of δ = U/r × 100% = 2.5 ppm.

 

Fig. 15 Ten measurement results of ROC for the test plano-concave lens using the DCIMPM system

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Using the developed system shown in Fig. 7, Fig. 16 shows the focusing curves at each position obtained according to the measurement processes for the thickness and refractivity of the test lens shown in Fig. 4. The null points Q₀, Q₁, Q₂, and Q₃ correspond to the L_S positions P₀, P₁, P₂, and P₃, respectively, when the measurement beam is converged on the vertices of the front and back surfaces of the test lens for measuring t and in the reflector L_R with or without the test lens in the measurement light path for measuring n. The coordinates of the measured positions P₀, P₁, P₂, and P₃ of lens L_S are z₀ = 0.006259 mm, z₁ = 3.359773 mm, z₂ = 105.218142 mm, and z₃ = 218.819452 mm, and the corresponding optical distances between these positions are d₁ = z₁ - z₀ = 3.353514 mm, d₂ = z₂ - z₀ = 105.211883 mm, and d₃ = z₃ - z₀ = 218.813193 mm. Substituting d₁ = 3.353514 mm, d₂ = 105.211883 mm, d₃ = 218.813193 mm, r₁ = −100.094736 mm, r₂ is infinite, and NA = 0.0767 obtained from the calibration into Eq. (9), the n and t of the test lens L_W obtained by the ray tracing calculation are n = 1.513764 and t = 5.079634 mm. The 10 repeated measurements of the test plano-concave lens resulted in average measurements of refractivity of n_avg = 1.513764 with a difference of refractivity between the standard value and the measurement value of Δn = n_sd − n_avg = 0.000163 and a relative error of δ_n = (Δn/n_avg) × 100% = 1.08 × 10⁻⁴, as well as an average central thickness t_avg = 5.079720 mm with a measurement standard deviation of σ_t = 0.648 μm and a relative error of δ_t = (σ_t/t_avg) × 100% = 1.27 × 10⁻⁴.

 

Fig. 16 Measurement curves of central thickness and refractivity for the test plano-concave lens

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Figure 17 shows the surface figure of the test plano-concave lens measured by the developed DCIMPM system shown in Fig. 7 and the Zygo interferometer. The figure average measured using the DCIMPM system is 0.154λ in Fig. 17(a) and that measured using the Zygo interferometer is 0.157λ in Fig. 17(b); hence, the measurements are in agreement. The standard lens of the Zygo interferometer has a peak to valley measurement of λ/20; therefore, the DCIMPM setup also has a peak to valley measurement accuracy of λ/20.

 

Fig. 17 Figures measured by (a) the developed DCIMPM system and (b) the Zygo interferometer

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6. Conclusion

This paper proposes a new laser differential confocal interference multi-parameter measurement method for spherical lens that combines the laser differential confocal measurement technique with the laser Fizeau interference technique. It uses the differential confocal measurement technique to measure the ROC, refractivity, and central thickness of a spherical lens, and then uses the laser Fizeau interference technique to measure the surface figure of the spherical lens, achieving high-accuracy comprehensive measurements of the parameters for the spherical lens on a single instrument. Based on this proposed measurement method, the first laser differential confocal interference multi-parameter comprehensive measurement system was successfully developed, thereby achieving, for the first time, high-accuracy comprehensive measurements of the parameters of a spherical lens on a single instrument, which provides an effective approach to the comprehensive measurement of spherical lens parameters.