Applications: Retrieval from through-focus intensities
It is widely recognized that the observation of the intensity distribution in the nominal image plane only is not sufficient for an unambiguous assessment of the quality or performance of an imaging system. The so-called through-focus intensity map is a collection of data that contains much more information on the optical system. It can be used for a reconstruction of the complex exit pupil function of the system, including its amplitude and phase defects. The Extended Nijboer-Zernike analysis provides us with an analytic tool that connects the properties of the complex pupil function of the lens (amplitude and phase) to the intensity distribution in the focal volume. This analytic relationship can be exploited by using the intensity data from a certain number of through-focus intensity 'maps' in a numerical inversion scheme. The analytic basis of the method has proven to lead to a robust inversion scheme that is not sensitive to measurement noise. The detection of the through-focus image intensity distribution can be done by measuring various exposure profiles in a developed resist layer (lithography) or by putting a detector with spatial resolution in image space and recording several defocused images. In the latter case, because of the finite size of the CCD-pixels, the intensity detection has to be done at the lower NA side of the imaging system:
Recording of the through-focus intensity distribution with the aid of a CCD-camera and a point source that can be translated in the axial focus direction
The through-focus behaviour of the intensity point-spread function can be visualized using the geometrical optics picture of rays:
A schematic illustration of the influence of wave-front distortion and transmission variation on the light distribution in the focal region (ray optics picture)
The geometrical rays represent a first order picture of the intensity distribution in the focal region. One observes that both wavefront distortion and amplitude variation in the exit pupil of the optical system produce a characteristic through-focus picture. Looking at the above picture on the relation between wave-front distortion and light distribution, one may wonder whether one can reconstruct the exit pupil function from the intensity distribution recorded in several defocused image planes. This is the so-called inversion problem in optics and the Extended Nijboer-Zernike theory is capable of solving this problem with great accuracy.
- An example of the experimental retrieval of lens aberration data
In the figure below we have plotted some measured intensity distributions (solid contour lines) in the focal volume (typically some 10 to 30 planes are used). After the numerical retrieval procedure based on the ENZ-theory, we have at our disposal the (complex) Zernike coefficients that represent the amplitude transmission and the aberration of the exit pupil function of the lens.
Some plots of the measured (solid lines) and reconstructed (dashed lines) intensity distribution in the 'best-focus' plane and in two defocused planes
The substitution of the retrieved coefficients in the forward calculation scheme yields the matched intensity distribution through focus (dashed contour lines in the figure). The match between the reconstructed and the measured intensity distribution is convincing (as it can also be seen in the figure below).
In the lithographic practice the recorded intensities are blurred versions of the true intensities. A common cause for this blurring in the horizontal plane is wafer stage noise and feature broadening due to diffusion during the development process in resist-based experiments. Stochastic focus variations in the vertical direction occur due to the non-zero bandwidth of the laser source used. The ENZ-method can be used to estimate the variances σr and σf due to these effects in the horizontal plane and the focus direction. Here we start with a lens having negligibly small phase errors. We subject the through-focus data obtained with this lens to the ENZ-retrieval calculation in which the retrieval algorithm is corrected for the unknown diffusion and focus variances. In the figure below we show the contour plot of the square of the retrieved phase error as a function of the supposed σr and σf. We estimate the variances as the minimum point of the squared phase error: this point yields the best match with the assumption that the lens has negligibly small phase errors.