Surface profile measurement of aspheric optics using
synthetic aperture interferometry
A Biswas and J Coupland
Wolfson School of Mechanical and Manufacturing Engineering, Loughborough
University, Ashby Road, Loughborough, Leicestershire, LE11 3TU, UK
A.Biswas@lboro.ac.uk
Abstract. A technique has been developed to measure aspheric optics based on synthetic
aperture interferometry. In this method, an optical probe measures the intensity of interference
between light reflected from the aspheric surface and a known reference surface (usually the
rear surface) while the probe scans the contour of the test optic. The phase difference between
the test and reference surfaces is deduced from the measured intensity and the deviation of the
aspheric surface is then calculated.
In practice the synthetic aperture technique relies on a probe design that has high numerical
aperture and is anamorphic for light efficiency. The probe consists of a separate source and
receive fibre attached to an external optical system which has the desired attributes. The high
numerical aperture of the probe facilitates the measurement of aspherics with considerable
deviation from a flat or a sphere. The anamorphic nature of the probe enhances the light
efficiency of the system considerably by increasing the photon density in the reflected light.
This method does not require any null compensator or specialized environment and has the
great potential to measure aspheric optics as they are being polished. The feasibility of this
technique has been verified by constructing an anamorphic probe and measuring the curved
surface of a plano-convex lens.
1. Introduction
In general, an optical system such as a camera or a telescope uses lenses or mirrors or a combination
of both to obtain an image of the desired object. The image quality substantially depends on the lenses
and mirrors used to build the optical system. It is well known that a single spherical surface generates
aberrations and distortions thereby degrading the image quality. These aberrations and distortions can
be eliminated by adding multiple spherical surfaces at a cost of increased weight and size. A single
aspherical surface can replace multiple spherical surfaces thereby reducing the weight of the optical
system [1], however, they are used sparingly due to the high cost involved in fabrication and testing
[2].
An optical surface is generally tested by means of interferometry. Most spherical surfaces are
tested using standard interferometers such as the Fizeau interferometer. In essence, an interferogram
is obtained between the test surface and a known reference surface [3]. Standard interferometry is not
suitable for testing aspherics because the resulting interferogram contains too many dense fringes and
is very difficult to analyze. Many methods have been discussed in the literature to overcome this
problem such as null tests [4,5], holographic interferometry [6-8] and multiple annular interferometry
[9]. To the authors knowledge, however, there is no generally applicable interferometric method that
is widely used commercially for testing aspherics. In industry, testing of aspherics is generally
accomplished using a contacting stylus probe to measure the surface profile. As a contacting
measurement it is not appropriate for soft optical glasses and is generally used for measurement of
small size rotationally symmetric aspherics [10,11].
Synthetic Aperture Interferometry (SAI) is a novel and simple method to measure aspherics [12].
SAI is a scanning interferometer where the interferogram is recreated by superposition of signals
obtained by a probe which synthesizes the measurement aperture. A schematic of the method is
shown in figure 1. In this diagram the probe consists of a bare optical fibre that scans the surface
along a defined path and the interference between the light reflected off the front and rear surface is
collected by it. The surface profile of the test surface can be deduced if the profile of the other surface
is known.
Reference Surface
(xr,yr)
y
(xf2,yf2)
(xf1,yf1)
Test Surface
(xp,yp)
x
Probe
Figure 1 Ray diagram for Synthetic Aperture Interferometer
Mono-mode fibres having a numerical aperture of 0.12 have been used in the past to serve as a
source and receiver [12]. This means that the gradient of the test surface is restricted because the
cone of light emanating from source fibre or collected by the receive fibre has to be within ± 7
defined by the fibre NA. This resulted in inability of the original method to measure surfaces which
have even a slight deviation from a flat. In this paper, we have overcome this problem by increasing
the numerical aperture of the probe extrinsically. In addition, we increase the NA only in the radial
direction using an anamorphic system, thereby increasing the light efficiency. The theory is validated
by measuring the surface deviation of the convex surface from its ideal form in a plano-convex lens.
2. Anamorphic Probe Design
Figure 2 shows the optical schematic of the anamorphic probe. The numerical aperture of the
probe has been increased to 0.53 by inserting a single aspheric element at the end of the probe which
acts as the probe objective. The objective is illuminated by a doublet placed after the bare fibre to
obtain a collimated light. This extends the surface gradient limitation to ± 32 from the original ± 7
and is suitable to test most aspherics of interest. In the tangential plane to the test optic the
illuminating beam is collimated by the doublet and brought to focus by a cylindrical, anamorphic
element. In this way a thin sheet of collimated light emerges from the objective. In our case, the light
efficiency was increased by 20 dB relative to a symmetrical system with equivalent NA. The
following section presents the theory relevant to our anamorphic probe.
Figure 2 Optical schematic of anamorphic probe in
(a) Radial Plane (b) Tangential Plane
3. Theory
With reference to figure 1 we assume that a pair of rays will retrace their path back to the probe after
reflection from the front and rear surfaces. The optical path difference between these rays can be
written as
OPD = OPLf – OPLr
(1)
where OPLf and OPLr are the corresponding path lengths for the rays reflected from the front and rear
surface.
In figure 1 we also assume that the rear planar surface acts as the reference. The front surface is
the test surface whose surface profile is to be measured and is nominally rotationally symmetric. The
probe position coordinates are represented by (xp,yp) and the ray intersection points are represented by
(xf1,yf1), (xf2,yf2) & (xr,yr) and we can rewrite equation 1 as
OPD  2[(x p  x f1 )2  (yf1  yp )2 ]1/2  2{[(x f2  x p )2  (yf2  yp )2 ]1/2  n(y r  yf2 )}
(2)
where n is the refractive index of the lens material.
The OPD can be calculated once the ray intersections are known. We have used simple line search
procedure to find the ray intersections and consequently the OPD.
The next step is to calculate the surface deviation of the test surface from the measured interference
signal. The interference signal received at the probe as a function of probe position can be written as

 2OPD( x p )  
I( x p )  2I r 1  cos 





(3)
where  is the wavelength and Ir is the reflected intensity from the front and rear surface which are
considered to be approximately equal. The actual optical path difference can be obtained by inverting
equation 3 so that
OPDinv ( x p ) 



cos 1 I( x p ) / 2I r  1
2
(4)
where it is noted that 2 phase jumps are removed in the usual way such that the inverse cosine in
equation 4 can be interpreted as a phase unwrapping operator [13]. The surface form error is deduced
by first calculating the optical path difference corresponding to the ideal surface profile, OPD for. Then
we formulate a cost function defined by


C   OPDfor (x p )  OPDinv (x p ) dx p
2
(5)
The surface deviation is then calculated by solving the multi-variable optimization problem so as to
achieve the minimum cost function. The actual experimental process is described in the next section.
4. Experimental Results
The validity of the theory was assessed by constructing a probe based on the anamorphic design.
This probe was used to test the curved surface of a plano-convex lens. The block diagram of the
experimental set-up is shown in figure 3. The reflected interference signal from the front and rear
surface was collected by the probe and passed to a photomultiplier as it scanned the test optic. The
linear movement of the probe was monitored using a Michelson’s interferometer as shown in figure 3.
The intensity data obtained was then converted to unwrapped phase data. An algorithm based on
phase-locked-loop [14] was written to evaluate the phase data. The surface deviation of the test
surface from the ideal surface is then calculated numerically using equations 4 and 5 for three different
set of data and is shown in figure 4. It can be seen that the maximum surface form error is within ± 1
for a lens diameter of 28mm.
Reference
surface
Probe
Photomultiplier
Low pass filter
Fibres
Test surface
He-Ne Laser
ADC
Corner cube
reflector
Computer
Beam splitter
Photodiode
Glass plate
He-Ne Laser
Polarizer
Figure 3 Block diagram of the experimental set-up
-3
1
x 10
Surface Deviation (mm)
0.5
0
-0.5
data1
data2
data3
-1
-1.5
0
2
4
6
8
Lens Radius (mm)
10
12
14
Figure 4 Surface Deviation
5. Summary
In this paper, the basic synthetic aperture interferometer was modified to enable it to measure steep
aspherics by increasing the numerical aperture of the probe. The light efficiency of the system was
increased by using an anamorphic design. An experiment was carried out to assess the feasibility of
the technique. This method is very simple and can be used for measuring aspherics on a CNC
polishing machine without using any null compensator or specialized environment.
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ADC
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