In 1704 the great Isaac Newton described in Opticks, using drawings and words, multiple-prism dispersion [1]. In 1813 Brewster introduced compensating prism pairs for beam expansion [2].

In 1982 physicists working on narrow-linewidth tunable lasers, incorporating multiple-prism grating assemblies, derived a generalized multiple-prism grating dispersion equation [3]. The generalized multiple-prism dispersion equation is the first derivative of the generalized prismatic refraction:

A pedagogical description of these equations is also available [4]. In 1987 the second generalized derivative of the multiple-prism refraction (first derivative of the dispersion) was introduced as a design tool for multiple-prism pulse compressors used in femtosecond lasers [5]. Over the years these equations have been successfully used in pulse compression calculations [6] and as a tool in the design of narrow-linewidth prismatic laser oscillators [7]. Additional applications include laser microscopy.

Multiple-prism beam expanders and arrays were described in matrix form in 1989 [8]. The multiple-prism dispersion theory was given in 4 X 4 matrix form in 1992 [9]. These matrix equations are applicable either to prism pulse compressors or multiple-pism beam expanders. These matrix results are nicely reviewed and summarized in Tunable Laser Optics [10]. Also, in 1992 the cavity linewidth equation was derived using quantum principles [11]. This 1992 paper [11], reported for the first time on the nexus between quantum mechanics principles and classical angular dispersion as expressed via the generalized multiple-prism grating dispersion theory. In 1997, Snell's law (or the law of rafraction) was derived using quantum mechanical interferometric principles [12] and in 2006 this derivation was extended to negative refraction [13]. Certainly, these principles also apply to the generalized description of angular dispersion as shown in [12, 13].

Recently, Duarte has elegantly constructed a theoretical framework that allows the evaluation of higher generalized derivatives, of multiple-prism dispersion, at will [14-16]. These higher derivatives can be used to refine the design of multiple-prism pulse compressors and should find further applications in nonlinear optics [15, 16]. On the other hand, this theoretical framework, provides an elegant, and complete, mathematical description of the prismatic arrays first outlined by Newton’s vision more than 300 years ago. An updated review on this subject is given in [15].

The multiple-prism dispersion theory has also been appled to quantify the angular dispersion in Amici prisms, also known as compound prisms [16]. A review of applicatons of the multiple-prism dispersion theory to laser optics, pulse compression, astronomy, metrology, microscopy, and scientic instrumentation is given in [17].

Multiple-prism array as described by Newton in his Opticks.

Review describing Newton's contribution to prism refraction and angular dispersion plus the link between quantum principles and the cavity linewidth equation (Duarte, (2000) [18]).

Multiple-prism grating tunable laser oscillator (from Duarte et al. (1998) [19]).


  1. I. Newton, Opticks (Royal Society, London, 1704).
  2. D. Brewster, A Treatise on New Philosophical Instruments for Various Purposes in the Arts and Sciences with Experiments with Light and Colours (Murray and Blackwood, Edinburgh, 1813).
  3. F. J. Duarte and J. A. Piper, Dispersion theory of multiple-prism beam expander for pulsed dye lasers, Opt. Commun. 43, 303-307 (1982).
  4. F. J. Duarte and J. A. Piper, Generalized prism dispersion theory, Am. J. Phys. 51, 1132-1134 (1983).
  5. F. J. Duarte, Generalized multiple-prism dispersion theory for pulse compression in ultrafast dye lasers, Opt. Quantum Electron. 19, 223-229 (1987).
  6. J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd Ed. (Elsevier Academic, New York, 2006).
  7. F. J. Duarte, Tunable Laser Optics (Elsevier-Academic, New York, 2003).
  8. F. J. Duarte, Ray transfer matrix analysis of multiple-prism dye laser oscillators, Opt. Quantum Electron. 21, 47-54 (1989).
  9. F. J. Duarte, Multiple-prism dispersion and 4 x 4 ray transfer matrices, Opt. Quantum Electron. 24, 49-53 (1992).
  10. F. J. Duarte, Tunable Laser Optics (Elsevier-Academic, New York, 2003).
  11. F. J. Duarte, Cavity dispersion equation Δλ ≈ Δθ(∂θ/∂λ) – 1: a note on its origin, Appl. Opt. 31, 6979-6982 (1992).
  12. F. J. Duarte, Interference, diffraction, and refraction, via Dirac's notation, Am. J. Phys. 65, 637-640 (1997).
  13. F. J. Duarte, Multiple-prism dispersion equations for positive and negative refraction, Appl. Phys. B 82, 35-38 (2006).
  14. F. J. Duarte, Generalized multiple-prism dispersion theory for laser pulse compression: higher order phase derivatives, Appl. Phys. B 96, 809-814 (2009).
  15. F. J. Duarte, Tunable organic dye lasers: physics and technology of high-performance liquid and solid-state narrow-linewidth oscillators, Progress in Quantum Electronics 36, 29-50 (2012).
  16. F. J. Duarte, Tunable laser optics: applications to optics and quantum optics, Progress in Quantum Electronics 37, 326-347 (2013).
  17. F. J. Duarte, Multiple-prism arrays and multiple-prism beam expanders: laser optics and scientific applications, in Tunable Laser Applications (CRC, New York, 2016) Chapter 13.
  18. F. J. Duarte, Newton, prisms, and the opticks of tunable lasers, Optics & Photonics News 11 (5), 24-28 (2000).
  19. F. J. Duarte, T. S. Taylor, A. Costela, I. Garcia-Moreno, and R. Sastre, Long-pulse narrow-linewidth dispersive solid-state dye laser oscillator, Appl. Opt. 37, 3987-3989 (1998).

Complete Set of References on Multiple-Prism Theory

Multiple-Prism Pulse Compressors: Tutorial

Laser Tutorial

Page published on the 18th of August, 2009

Updated on the 30th of September, 2016