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Abstract
A shearing interferometer finds a derivative of the wavefront along the direction of the shear parameter. While the general belief was held that the sensitivity of the RSI increases with the shear angle, we find that just the opposite is true. We examine the range of validity of the derivative function for the rotationally shearing interferometer (RSI). We find that all terms of the aberration polynomial are multiplied by the sine of the half-shear angle multiplied by the order of polynomial, l. This factor may be expanded in series for small angles, or for angles close to π and π/2. We propose steps to evaluate the errors in the determination of the magnitude of individual aberrations. We apply findings to the planet-detection problem.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Shearing interferometry
Many techniques have been proposed and successfully implemented for the detection of planets outside our solar system [1]. They tend to be based on the existing instruments, modified for the challenging requirements of extra-solar system exploration. We developed the theoretical foundation that the RSI may detect a faint exo-planet next to a bright star. The novel technique compares two signals captured within a short time interval, based on the rotationally shearing interferometry (RSI) [2]. It incorporates a temporal element under control of the experimentalist that allows for causal response when two point-sources are within the instrument field of view [3].
The traditional Mach-Zehnder (MZ) interferometer, that requires a reference surface for its functioning, is transformed into an RSI when a rotatable Dove prism is inserted in one of its arms, as shown in Fig. 1. Thus, a wavefront in one interferometer arm is compared, that is subtracted and interfered, with the duplicated and rotated copy of itself.
Fig. 1. A schematic diagram of the rotationally shearing interferometer (RSI) implemented in a Mach–Zehnder configuration with two Dove prisms. A Dove prism DP that is rotated by an angle about the optical axis introduces the wavefront shear in the RSI. M denotes a mirror, BS, a beam splitter, and OP the observation plane. An RSI rotates a wavefront through an arbitrary shear angle.
Using this technique, the presence of an invisible, off-axis object next to a bright object has been confirmed by detecting straight fringes [4]. In the RSI a change in the angle of orientation of the Dove prism generates a change in the detected fringe pattern. This feature allows for the confirmation of the presence of a faint object next to a bright one. Straight fringes arise due to interference of radiation, emitted by two objects. Thus, straight fringes also confirm the presence of a faint point source. Simulations indicate an increase in the fringe density, and a decrease in the fringe inclination angle when the shear angle is increased [5].
In our experiment though, the fringe density decreases while the fringe inclination angle increases upon increasing the shear angle [4], in disagreement with the simulations. The objective of this research was to explain the inconsistency between the simulation and experiment. The theory of the RSI is formulated as a derivative-forming instrument, starting from the first principles of shearing interferometry, and examining three special cases. The theory is valid for all shear angles. Furthermore, it is applicable to the wavefront measurements, for testing the optical aberrations or for assessing the RSI to detect an extra-solar system.
The paper is organized as follows. In the third section we present RSI both as a wavefront subtractor and as a derivative-forming instrument. We present conclusions in Section 4. In the next section, we develop a general theory, dealing with the derivative and shearing features of the RSI. We apply it to several low-order Seidel aberrations. We compare it to the theory of a traditional MZ for two-beam interference.
2. Theory of the rotationally shearing interferometer
We study the effects of the RSI on an arbitrary wavefront, with an arbitrary shear angle. It applies to the wavefront from a planetary system or aberrations of an optical system. A general wavefront W may be expanded in a polynomial form [6].
Here anl and bnl are coefficients, measured in fraction of wavelength, or µm [m] in general. We are using ρ and φ coordinates. Letters n and l are dummy indices.
2.1 Two-beam interference
As a reference, we include the change in wavefronts in the MZ interferometer.
We include only those aberrations that the RSI also detects, that is, tilt, astigmatism, and coma for two orientations, corresponding to the a- and b-coefficients. Terms in the first raw are tilt with respect to the x-axis, 0-degree stigmatism, and coma flare in the x-direction, also seen or detected by the RSI. In the second raw, there is tilt with respect to y-axis, 45-degree astigmatism, and coma − flare in the y-direction.
We model the incremental change that corresponds to two experimentally obtained measurements separated by an angle Δφ. To simplify the equations and to maintain symmetry, we divide the change in angle of orientation in two equal parts, Δφ/2. We rotate the wavefront in two arms by +Δφ/2, and −Δφ/2. We subtract the wavefronts and find the incidence pattern.
Here n and l are both even or both odd. Thus, the RSI is insensitive to azimuthally symmetric aberrations. The cosine terms for l = 0 cancel out, while sine terms of zero (0) angle are zero (0).
Next, we consider a single aberration term, ΔWnl(ρ,j) under the double summation.
2.2 RSI aberration polynomial for a given set of subscripts
Using the trigonometric identities for the sums and differences of angles, the cosine terms cancel out in the first line in Eq. (4). Similarly, the sin-cos terms cancel in the second line. Finally, we factor out [-2 sin(lΔφ/2)]. The wavefront difference for a single (n,l) term then becomes,
When single terms in the wavefront expression in Eq. (5) are substituted into Eq. (4) for several low-order aberrations, we obtain the wavefront difference, detected by the RSI.
Terms in the first raw are tilt with respect to the x-axis, 0-degree stigmatism, and coma flare in the x-direction, as detected by the RSI. In the second raw, there is tilt with respect to the y-axis, 45-degree astigmatism, and coma flare in the y-direction. These terms correspond to the non-rotationally symmetric aberrations, the only low-order aberration seen by the RSI.
Upon comparison of Eqs. (2) and (6), we observe that the cosφ −terms (the terms with a-coefficients) in the MZ-difference in the wavefront expression became sinφ-terms in the RSI. Similarly, the sinφ-terms (the terms with b-coefficients) in the MZ difference in the wavefront expression became cosφ-terms in the RSI. For this reason, researchers say that the RSI transforms each aberration term in a MZ wavefront configuration into a complementary aberration: horizontal tilt becomes a vertical tilt, and vice versa; zero-degree astigmatism becomes a 45-degree astigmatism, and vice versa; and coma flare in horizontal direction transforms into coma flare in y-direction, and vice versa.
We list the primary aberration polynomials up to the third order spherical aberration in Table 1. We also include the change in the aberration provoked by the change in the shear angle that in all cases includes the shear factor.
Table 1. Aberration terms, their polynomials, derivatives, differentials, and infinitesimals.a
Furthermore, in the third column we list the derivative with respect to the angle φ that has no shear factor, indicating that the RSI functions as a derivative only for a very small amount of shear, which in the limit approaches to zero. In the fourth column we include the change in wavefront function for incremental change in angle of orientation. Lastly, we indicate the infinitesimal change in wavefront detected when the shear angle is changed infinitesimally.
2.3 Shear factor
Each term in Eq. (5) or Eq. (6) features a factor whose argument includes a summation index l. We call it the shear factor, although some researchers have referred to it as sensitivity [7]. Sensitivity in measurement theory is related to the change in the response caused by the change in the input variable and must be independent of the measured quantity.
Figure 2 shows the shear factor as a function of the shear angle for tilt, astigmatism, and coma. Zero shear angle results in zero shear factor. The shear factor of 2 is obtained for the shear angle of π for tilt and coma, and for shear angle of π/2 for astigmatism. For none of these values, may it be assumed that the shearing function of the RSI could be described as a derivative. This further negates the interpretation of the shear factor as sensitivity. We prefer to call it shear factor because it contains the information about the magnitude of the shear.
Fig. 2. The shear factor as a function of the shear angle for tilt, astigmatism, and coma. Angle may also be measured from the maximum of the shear factor curves (above).
When the shear angle increases from 0, the shear factor increases as a sine. We consider three special cases: one for very small shear angle, δφ, one near π/2, and the last one near π. For the special case 1, we expand the sine function, keeping only the first term. The shear factor for small angles is δφ for tilt and coma, and 2δφ for astigmatism. For the special cases 2 and 3, we expand the sine of difference of angles. For the maxima of the shear factor, we introduce a change of variables, Δφ= π−Δϕ for l = 1 (tilt, coma), and Δφ= π/2−Δϕ for l = 2 (astigmatism), indicated in Table 2. The new variables are also indicated in the top abscissa of Fig. 2.
When the shear angle is very close to the maximum of the shear factor for the respective aberration, the shear factor may be described as a quadratic function of the shear angle that is measured from the angle where the maximum is attained, upon cosine expansion in series.
Table 2. Special cases for the shear factor values
3. RSI both as a wavefront subtractor and as a derivative-forming instrument
We insert special cases 1, 2, and 3, into Eq. (6) and list the individual aberration terms in Tables 3 and 4. Table 3 lists the aberration terms, their derivatives, increments and differentials the shear angle close to zero.
Table 3. Aberration terms, their derivatives, increments and differentials for zero shear angle.a
Examining Table 3, we conclude that an exact derivative is found only for very small shear angles that approach to zero.
Table 4 presents the aberration terms, their derivatives, increments and differentials for shear angle at maximum shear factor: Δφ at π/2 for astigmatism, and Δφ at π for tilt and coma.
From Table 4, we learn that for a shear angle that is near either π/2 (astigmatism) or π (tilt and coma), the shear factor may be expressed as a series expansion of cosine function. In these configurations the interferograms may not be integrated to reconstruct the wavefront.
4. Conclusions
A general expression has been derived for the change in the wavefront generated by any shear angle. Its general applicability significantly expands the suitability of the RSI for the measurement of rotationally asymmetric aberrations for optical testing. A variable factor has been identified that depends not only on the shear angle but also on the order of the polynomial l. This factor, called a shear factor, multiplies each wavefront term.
The RSI may be considered a derivative-taking instrument only for very small changes between the wavefront orientation angles. In those cases, a wavefront may be reconstructed from its measured derivative, with the error increasing with the magnitude of the shear angle. The change in wavefronts must be interpreted as a simple difference when the shear angle Δφ is no longer infinitesimal, or specifically when it approaches π/2 or even π.
Additionally, for the largest values of the shear factor (2), the zero value of the shear angle may be set at the angle of maximum shear factor, using a linear change in angle variable. This results in the cos-dependence of the shear factor on the shear angle. Then, density of fringes around the peak fringe factor decreases, and their inclination angle increases with increasing shear angle. This behavior agrees with experiment. Thus, theoretical development presented here affirms the suitability of the RSI as a promising planet detection instrument: this analysis reconciles the experimental results with the theoretical analysis of shearing.
Finally, the RSI transforms each aberration term in a MZ wavefront configuration into a complementary aberration only for the infinitesimally small shear angle. In all the other cases, transformations include the multiplicative shear factor which modifies the derivative. The modification depends on the magnitude of shear angle and the polynomial order.
Disclosures
The author declares no conflict of interest.
Data availability
Data is included in the MS.
References
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