Abbe’s theory of image formation: The object AB (which acts as a diffraction grating) is illuminated with coherent light. The light is diffracted into orders, shown with different colors (some orders are collected by the lens). In the back focal plane, the orders are separated; in the image plane, diffracted orders interfere to form the image B'A'.
Abbe’s theory of image formation
In the 1840s, the Italian microscopist Giovan Battista Amici invented the water immersion microscope objective. Since the microscope’s apertures were still small, the image quality was not optimal. One problem was that only a thin bundle of rays illuminated the objective; thus, the illuminating light did not fill the back aperture of the microscope objective.
At first, Abbe produced microscope objectives with very small angular apertures. While these objectives had improved spherical and chromatic corrections (less of these two aberrations), they were less bright and had a lower resolution than similar microscope objectives made with large angular apertures. Abbe realized that he could explain this result by the diffraction of the illumination light by the objects in the specimen whose dimensions are similar to the wavelength of the illumination light. The angles of the diffracted light then completely fill the numerical aperture of the objective.
In 1873, Abbe published his landmark paper in a biological journal that explained the role of wavelength and aperture of the microscope objective on microscope resolution—the ability to resolve two closely positioned points on an object as separate points.
Abbe’s article comprised 55 pages without any equations or diagrams. Abbe first considered the diffraction of light by the object and then the effect of the aperture. In 1896, Lord Rayleigh published a paper in which he first considered the aperture and then the object, but he reached similar conclusions to Abbe’s. Rayleigh used Lagrange’s theorem and Fourier analysis in order to calculate the diffraction pattern of apertures with various shapes, as well as the diffraction pattern from gratings.
Abbe described several experiments using a sinusoidal grating as an object. He observed the diffraction pattern of the grating with the ocular removed and with various stops placed in the microscope to alter the number of diffraction orders that produced the diffraction pattern. Each of the diffracted beams was called a “spectra” due to their appearance with white light illumination. It was the combination of all the diffraction orders that formed the image.
In other words, with more diffraction orders from the object entering the microscope objective, the observer could resolve more detail. Abbe also noted that oblique illumination increased the resolution of the microscope; this was because a higher order of diffraction entered the objective when the central illumination beam was shifted to one edge of the objective by tilting the illumination with respect to the optical axis. In the back focal plane of the objective, each Airy disk is a source that forms a spherical wave; the spherical waves interfere in the image plane to form the image of the object.
In 1876, Abbe traveled to London with his gratings, apertures and microscope in order to demonstrate his diffraction theory in front of the members of the Royal Microscopical Society. The theme that Abbe promoted with his experiments was that microscopic imaging of structures whose dimensions are similar to the wavelength of light cannot be explained on the basis of geometrical optics; rather, the phenomena required diffraction and interference effects.
Abbe was able to set the limit of the resolution (later called the Abbe limit of resolution) of the optical microscope from his requirement that both the central (0th order of diffraction) and at least one of the diffraction order maxima must enter the objective to achieve maximum resolution. The nondiffracted 0th order rays and the nth order rays are separated in the back focal plane (diffraction plane) and combined in the image plane. Abbe calculated this for an object that consisted of a periodic structure (lines) for a non-immersion microscope objective, and a circular aperture (the microscope objective lenses) using direct illumination as: d=λ/(n sin α), and as d=λ/(2n sin α) for oblique illumination, where d is the smallest separation that can be resolved, λ is the wavelength of the illumination light in vacuum, and α is one-half of the angular aperture of the microscope objective.
Abbe is also credited with the formulation of the term “numerical aperture.” In microscopy, the numerical aperture, or NA, of a microscope objective is given as: NA=n sin θ, where n is the refractive index of the medium between the object and the microscope objective, and θ is one-half of the angle of the cone of light that can enter the objective. The angular aperture of the lens is twice the angle θ.
When other microscopists objected to Abbe’s theory claiming that they could observe smaller details, Abbe replied that they were observing false detail—artifacts that were not actually part of the object. The presence of the refractive index in the denominator offers another way to increase the resolution of the microscope (reduce the separation distance that could be resolved)—the concept of oil immersion. If we use oil (n≈1.5), the resolution will be increased as compared to air (n≈1).
In 1874, Helmholtz published a paper in a physics journal in which he calculated the maximum resolution of an optical microscope. Helmholtz used ray tracing methods that were long used in telescope design and concluded that the smallest separation of two distinct points in the object that could be resolved was equal to one-half of the wavelength of the illumination light (for light of 500 nm, the resolution is 250 nm). After writing his paper, Helmholtz came across Abbe’s prior article on the same topic. He attached a note to his paper that acknowledged the priority of Abbe.
Within a decade, others published articles explaining the Abbe diffraction of image formation in a microscope and included illustrations of the patterns described in the earlier Abbe experiments.
In 1906, Porter described the same theory and Abbe’s experiments in only 12 pages because he invoked Fourier’s theorem. Thus, he demonstrated the power of Fourier optics. These experiments were repeated and published in 1906 by Porter. When the ocular is removed, the spectral bands of the grating (object) are observed in the back focal plane of the objective lens. By placing different obstructions in the back focal plane (diffraction plane), the image could be altered (spatial filtering).
We can summarize the Abbe theory as follows: In an optical system, there are two causes of aberration; the first is spherical and chromatic aberration, which have their origins in properties of lenses (i.e. refractive indices, curvatures, etc.), and the second is the effect of light diffraction. It is the latter phenomena that is the foundation of the Abbe theory of image formation in the microscope.
Barry R. Masters is with the Biological Engineering Division at the Massachusetts Institute of Technology in Cambridge, Mass.
References and Resources
>> E. Abbe. Beiträge zur Theorie des Mikroskops und der Mikroskopischen Wahrnehmung. Archiv für Mikroskopische Anatomie, IX, 413-68 (1873).
>> H. Helmholtz. Die Theoretische Grenze für die Leistungsfähigkeit der Mikroskope. Annalen der Physik Jubelband, 557-584, Leipzig, 1874. Translated as: “On the limits of the optical capacity of the microscope,” Monthly Microscopical Journal, 16, 15-39 (1876).
>> Lord Rayleigh. On the Theory of Optical Images, with Special Reference to the Microscope. Philosophical Magazine and Journal of Science, London, XLII, 167-95 (1896).
>> A.B. Porter. On the diffraction theory of microscope vision. Phil Mag. 6(11):154-6 (1906).
>> H. Hartinger. Zum fünfundzwanzigsten Todestage von Ernst Abbe, Die Naturwissenschaften, Heft 3, 49-63, 1930.
>> H. Volkmann. Ernst Abbe and His Work, Appl. Opt. 5(11), 1720-31 (1966).
>> E. Abbe. Gesammelte Abhandlungen, I-IV, Hildesheim, Georg Olms Verlag (1989).
>> M. Born and E.Wolf. Principles of Optics, 7th (expanded) edition. Cambridge, Cambridge University Press, 1999.
>> K. Gerth. Ernst Abbe, Scientist, Entrepreneur, Social Reformer, Jena: Verlag Dr. Buseert & Stadeler (2005).
>> B.R. Masters. Confocal Microscopy and Multiphoton Excitation Microscopy: The Genesis of Live Cell Imaging, Bellingham, Wash., SPIE Press (2006).