2.1.

Tilted Mirror

The principle of the first method is shown in Fig. 1(a).^11,12 The cylindrical wavefront from the interferometer meets with a plane mirror. The plane mirror is placed at a cylindrical cat’s eye position, and is then tilted so the wavefront is reflected at different angles. By combining these wavefront-sheared measurements, the interferometer error contribution can be calculated.

Fig. 1

(a) Tilt mirror method (b) roof mirror method.

2.2.

Roof Mirror

This method is somewhat analogous to a method for absolute spherical surface testing.⁹ As illustrated in Fig. 1(b),¹² three relative measurements are taken. By combining these three measurements, the interferometer error can be calculated. We note that for both of these methods, the reported measurement accuracy reached is $\sim \lambda /20$.

For the tilting mirror method, it is difficult to decouple tilt, focus, and lateral motion. Unless the mirror is rotated about the line focus, light will reflect off different areas of the mirror and each area will contribute a different error to each measurement. Another disadvantage of the tilting mirror method is that it reveals only the error as a function of $\varphi$, so only errors in the $y$-direction are revealed. It is blind to any error correction in the $x$ direction.

Hence, we propose a new technique for improving the accuracy of cylindrical optic testing based on the integration of two well-known techniques.

3.1.

Analysis of Cylindrical Wavefront Error Contributions

The proposed interferometer configuration and coordinate geometry for absolute testing of a cylindrical wavefront using fiber optics are shown in Fig. 2.

Fig. 2

Configuration for the absolute testing of cylindrical wavefront using fiber optics where the fiber is oriented along the $y$-axis.

In the first pass, a collimated wavefront from the interferometer propagates to the cylindrical null optic. Then it passes through the null optic to be focused into a waist region extending along the surface of the fiber. When the fiber interacts with a collimated beam, only a small portion of the incident beam on the fiber will reflect back into the null^13–16 due to the small radius of the reflective cylindrical fiber reference. The wavefront error sources of this experiment are described by considering two cases.

3.1.1.

Case of ideal optical fiber reference

In this case, the incident wavefront picks up the error of the flat reference, the cylindrical null, and then the fiber reference as shown in Fig. 3(a). Following Geary,^13–16 the fiber filters the $x$ axis but passes the variation along the $y$ axis. Thus, the errors contribution of the wavefront reflected off the fiber can be written as

Eq. (1)

$$W(x,y)=W_{\mathrm{F}}(x,y)+W_{\mathrm{RS}}(y)+W_{\mathrm{CN}}(y),$$

where $W_{\mathrm{F}}$ (the fiber wavefront error) is twice the fiber optic surface error. $W_{\mathrm{RS}}$ is the wavefront error of the transmission flat reference, and $W_{\mathrm{CN}}$ is the error of the cylindrical null. The wavefront returns through the null and picks up its error then interacts with the flat reference surface and picks up its error. Therefore, the measured wavefront can be written as

Eq. (2)

$$W(x,y)=W_{\mathrm{F}}(x,y)+W_{\mathrm{RS}}(y)+W_{\mathrm{CN}}(y)+W_{\mathrm{CN}}(x,y)+W_{\mathrm{RS}}(x,y).$$

Fig. 3

(a) Perfect fiber surface and (b) imperfect fiber.

Equation (2) reduces to

Eq. (3)

$$W(x,y)=W_{\mathrm{F}}(x,y)+W_{\mathrm{S}}[y,(x,y)],$$

where $W_s[y,(x,y)]=W_{\mathrm{RS}}(y)+W_{\mathrm{CN}}(y)+W_{\mathrm{CN}}(x,y)+W_{\mathrm{RS}}(x,y)$ is the instrument error.

For an ideal fiber optic reference surface, the fiber adds no wavefront error, thus

$$W_F(x,y)=0.$$

Consequently, the total wavefront error sources in this case are

Eq. (4)

$$W(x,y)=W_{\mathrm{S}}[y,(x,y)].$$

3.2.

Decomposition of Possible Geometric Errors of the Fiber Reference

Creating the fiber reference begins with the production of the optical fiber and ends with mounting it on its stage, and the surface irregularities are generated throughout the whole process. During the fiber drawing process there are several factors that cause variation in the fiber cladding diameter.^17–20 To prepare the optical fiber as a reference, the plastic jacket is removed¹³ which may induce cracks, pulls, divets, or sharps microbends. The dejacketed fiber is then coated with reflective material, typically aluminum, but the coating uniformity is unknown. Finally, the fiber is gently stretched across a special mount, operating under tension to avoid sagging or bending.¹³

To discuss the form of potential geometric errors of the fiber reference, we first assume cylindrical coordinate symmetry about the fiber $(\rho ,\varphi ,y)$ as illustrated in Fig. 4. Note that, to date, one publication has discussed the accuracy of using the fiber as a reference, stating only that the fiber is $\lambda /15$ P–V or better.¹³ As cylindrical wavefront accuracy is typically quoted separately along the powered and the plano axes directions,²¹ there is a need for better data.

Fig. 4

Cylindrical coordinate system of the fiber.

We begin by defining four possible forms of geometric error on the fiber surface. The first three are fiber diameter variations $S_y(y)$, longitudinal error $S_{\varphi }(\varphi )$, and random bumps $S_{\varphi ,y}(\varphi ,y)$, Fig. 5. The tensioned fiber is effectively straight, so the fourth geometric error, bending, is assumed to be negligible and is ignored, though it could be accommodated with $S_{\varphi ,y}(\varphi ,y)$.

Fig. 5

The fiber geometric error forms (a) diameter variation, (b) random bumps, and (c) longitudinal error.

3.3.

Measurement Algorithm Development

The proposed absolute interferometric testing is illustrated in Fig. 6. The fiber is precisely located at the cylindrical null focal line so the number of fringes is minimized in the test aperture. An interferogram is taken and the wavefront data stored. We start with the basic assumption that the measured wavefront $W$ is the sum of system error and fiber surface error

Fig. 6

The proposed method setup for absolute cylindrical testing.

According to Sec. 3.2, we can write $W_{\mathrm{F}}$ as the sum of three wavefront error sources:

Therefore, the measured wavefront in Eq. (5) can be written as

Perfect cylindrical optics has three degrees-of-freedom that do not change the reflected wavefront. They are translation along the cylindrical axis, rotations about the $y$-axis, and flipping the fiber $y\to (-y)$. Based on this principle, we propose three steps to quantify and remove the fiber geometric error contribution from the actual measured wavefront.

4.1.

Fiber Error Modeling

The fiber cladding surface shape is unknown; no published papers discuss the diameter variation of the cladding, especially at the spatial periods over which we are interested. Additionally, the effect of de-jacketing and metalizing the optical fiber cladding is unknown. We, therefore, chose a simple model of fiber surface shape errors which matches the physically appropriate error decomposition described above. This allows us to both exercise and check our algorithms and conceptually prove the proposed method. The surface errors were described by the sum of sinusoidal errors in $y$ and $\varphi$ each with random amplitudes and phase, and random Gaussian bumps or divits whose position, amplitude, and widths were also randomly generated:

We used parameters that match the pending experimental values. The total fiber length is $Y_{\mathrm{F}}=100\mathrm{mm}$ and the range of rotation ${\mathrm{\Phi }}_{\mathrm{F}}$ is $[\pi /4,-\pi /4]$. The long dimension of the cylindrical null in Fig. 6 is $Y_p=40\mathrm{mm}$ and it generates a cylindrical wavefront with $\mathrm{NA}=0.1$. Thus, the size of each measurement patch on the fiber is $Y_p=40\mathrm{mm}$ and ${\mathrm{\Phi }}_p=0.1\mathrm{rad}$ as shown in Fig. 7.

Fig. 7

The measurement sampling method (a) fiber optic and (b) unrolled fiber surface.

4.2.

Result and Discussion

A series of 50 simulated fiber references surfaces, each with different spatial surface deformation, were randomly generated in order to evaluate the effect of applying this “random fiber” approach. Based on the only published data on the fiber reference quality and the fiber manufacturing tolerance of $\pm 0.001\text{-}\mathrm{mm}$ diameter, we have only accepted randomly generated surfaces which have no more than $\sim 1\mu \mathrm{m}$ of error. Regardless of the magnitude of the error, what matters is the reduction resulting from the multiple measurements.

From this simulation, we have found that this method reduces the contributed P-V and RMS error by a factor, on average, $15.9\times$ and $15.9\times$, respectively. The best case yielded a reduction in the PV of $29.5\times$ and an RMS of $31.7\times$, where the worst case yielded a reduction in PV of $6.5\times$ and an RMS of $4.8\times$.

One example of simulated fiber surface errors with a $1\text{-}\mu \mathrm{m}$ peak-to-valley distance and 0.18waves RMS along the entire fiber length is shown in Fig. 8. Simulation shows that the averaging of 231 patches leaves a residual fiber error of $\sim 0.0393\text{-}\mu \mathrm{m}$ peak-to-valley, and an RMS error of $0.0087\mu \mathrm{m}$. This is a $16.5\times$ reduction in P-V and a $17.8\times$ reduction in RMS compared to a single patch. Thus, this method appears feasible in eliminating the error contribution of the fiber reference to the measurement.

Fig. 8

Simulated result (a) fiber with a $1\text{-}\mu \mathrm{m}$ peak-to-valley and (b) averaged random fiber result.