Rather than measuring aberrations across the field to quantify the alignment of an optical system, we show how a single, on-axis measurement of the pupil mapping can be used to measure the off-axis performance of the system and determine the state of alignment. In this paper we show how the Abbe sine condition can be used to relate the mapping between the entrance and exit pupils to image aberrations that have linear field dependence. This mapping error then can be used to measure the linear astigmatism caused by the misalignment. Additionally, we present experimental results from the sine condition test on a simple system.

By taking a new look at an old concept, we have shown in our previous work how the Abbe sine condition can be used to measure linearly field-dependent aberrations in order to verify the alignment of optical systems. In this paper, we expand on this method and discuss the design choices involved in implementing the sine condition test (SCTest). Specifically, we discuss the two illumination options for the test: point source with a grating or flat-panel display, and we discuss the tradeoffs of the two approaches. Additionally, experimental results are shown using a flat-panel display to measure linearly field-dependent aberrations. Last, we elaborate on how to implement the SCTest on more complex optical systems, such as a three-mirror anastigmat and a double Gauss imaging lens system.

Imaging aberrations that have linear dependence on field angle are caused by pupil aberrations that can be described using the Abbe sine condition. This well-known relationship is frequently used to guide the design of optical imaging systems. For example, the aberration of coma is eliminated in the design of axisymmetric systems by controlling the pupil distortion, as defined by a standard implementation of the sine condition. An optical system with misalignments of surface irregularities will suffer pupil distortions that are quantified using a more generalized form of the sine condition. Such pupil aberrations create image aberrations that have linear dependence on field angle. While it is possible to infer the state of alignment by measuring multiple field points, it may be more straightforward to perform a single on-axis measurement of the sine condition violations. This paper summarizes the generalized sine condition and relationship between violations of this condition and aberrations with linear field dependence. An application is discussed for measuring sine condition violations of a 4-mirror system, which allows determination of the off-axis aberrations.