Interferometric Optics |
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These are references on the optics derived from the application of Dirac's *bra-ket* notation to *N*-slit interferometry. As explained in the references, Dirac's quantum approach is applicable to the propagation of a *single photon* or to the propagation of *ensembles of indistinguishable photons*.

The nexus between the quantum probability interference equation and the measurable intensity is described and explained in
*Fundamentals of Quantum Entanglement*, 2nd Edn (Institute of Physics, Bristol, 2022) where the difference between the physics of quantum interference equations and the physics of classical interference, is also explained. This book also the generalized *N*-slit interference equations in 2 dimensions and 3 dimensions and provides in detail the probability amplitude equations describing:

- Hambury Brown-Twist Interferometers.
- HOM Interferometers.
- Mach-Zehnder Interferometers.
- Michelson Interferometers.
- Sagnac Interferometers

The following terminology, and concepts, are discussed in the given references:

- Granularity measurements.
- Interferometric Characters.
- Interferometric Communications.
- Interferometric Imaging.
- Interferometric Microdensitometry.
- Interferometric Optics.
- Interferometry in Textiles.
- MTF measurements.
*N*-Slit Interference.*N*-Slit Interferometer.*N*-slit Interferometric Microscopy.*N*-slit Interferometric Nanoscopy.*N*-Slit Interferometry.- Optical Metrology.
- Quantum Entanglement.
- Quantum Interference.

Optical architecture for the multiple-prism beam expander microscope/nanoscope (MPBEM/N). The beam incident on the object can be, for example, 25-60 mm wide X 20 µm high. This is an extremely elongated beam (in the plane of propagation) with a width to height ratio in the range of 1000:1 to 3000:1. This type of illumination has also become known as *light sheet illumination*. Comparisons between theory and experiments, in the *N*SLI configuration, have been performed for even values (*N* = 2, 4, 6...) and odd values (*N* = 3, 5, 7...) of *N*. This includes the cases of two-slit interference, three-slit interference, four-slit interference, etc. (Duarte, 1991, 2002, 2005). For reviews see *Tunable Laser Optics* and *Tunable Laser Applications*).

The 1991 and 1993 papers also reported, for the first time, on the use of quantum mechanics techniques, via Dirac's notation, in the field of imaging. In addition, these papers illustrated the prediction of measured interferograms using interferometric equations derived using Dirac's quantum notation.

- F. J. Duarte, Interferometric imaging, in
*Tunable Laser Applications*, Third Edition, F. J. Duarte (Ed.) (CRC, New York, 2016) Chapter 10.

- F. J. Duarte, Tunable laser microscopy, in
*Tunable Laser Applications*, Third Edition, F. J. Duarte (Ed.) (CRC, New York, 2016) Chapter 9.

- F. J. Duarte,
*Tunable Laser Optics*, 2nd Edition (CRC, New York, 2015).

- F. J. Duarte, Tunable laser optics: applications to optics and quantum optics,
*Progress in Quantum Electronics*37, 326-347 (2013) (*Invited*).

- F. J. Duarte, T.S. Taylor, A.M. Black, I.E. Olivares, Diffractive patterns superimposed over
propagating
*N*-slit interferograms,*J. Mod. Opt.*60, 136-140 (2013).

- F. J. Duarte, T. S. Taylor, A. M. Black, W. E. Davenport, and P. G. Varmette,
*N*-slit interferometer for secure free-space optical communications: 527 m intra interferometric path length ,*J. Opt.*13, 035710 (2011).

- F. J. Duarte, Triple-slit experiment,
*Optics & Photonics News*22(12), 7 (2010).

- F. J. Duarte, T. S. Taylor, A. B. Clark, and W. E. Davenport, The N-slit interferometer: an extended configuration,
*J. Opt.*12, 015705 (2010).

- F. J. Duarte, Interferometric imaging, in
*Tunable Laser Applications*, 2nd Ed., F. J. Duarte (Ed.) (CRC, New York, 2009) Chapter 12.

- F. J. Duarte, Secure interferometric communications in free space: enhanced sensitivity for propagation in the metre range,
*J. Opt. A: Pure Appl. Opt.*7, 73-75 (2005).

- F. J. Duarte, Comment on "Reflection, refraction, and multislit interference,"
*Eur. J. Phys.*25, L57-L58 (2004).^{*****}

- F. J. Duarte,
*Tunable Laser Optics*(Elsevier-Academic, New York, 2003) Chapter 2.

- F. J. Duarte, Secure interferometric communications in free space,
*Opt. Commun.*205, 313-319 (2002).^{**}

- F. J. Duarte, Answer to question #60. Interference of two independent sources,
*Am. J. Phys.*66, 662-663 (1998).^{***}

- F. J. Duarte, Interference, diffraction, and refraction, via
Dirac's notation,
*Am. J. Phys.*65, 637-640 (1997).

- F. J. Duarte, Generalized interference and optical processing,
in
*Proceedings of the International Conference on Lasers '95*, V. J. Corcoran and T. A. Goldman (Eds.) (STS, McLean, Va, 1996) pp. 615-617.

- F. J. Duarte, Interferometric imaging, in
*Tunable Laser Applications*, F. J. Duarte (Ed.) (Marcel-Dekker, New York, 1995) Chapter 5.

- F. J. Duarte, On a generalized interference equation and
interferometric measurements,
*Opt. Commun.*103, 8-14 (1993).^{**}

- F. J. Duarte, Electro-optical interferometric microdensitometer system,
*US Patent*5255069 (1993).^{**}

- F. J. Duarte, Cavity dispersion equation Δ
*λ*≈ Δ*θ*(*∂θ/∂λ*)^{– 1}: a note on its origin,*Appl. Opt.*31, 6979-6982 (1992).^{*}

- F. J. Duarte, Dispersive dye lasers, in
*High Power Dye Lasers*, F. J. Duarte (Ed.) (Springer-Verlag, Berlin, 1991) pp. 7-43.

- F. J. Duarte and D. J. Paine, Quantum mechanical description of
N-slit interference phenomena, in
*Proceedings of the International Conference on Lasers '88*, R. C. Sze and F. J. Duarte (Eds.) (STS, McLean, Va, 1989) pp. 42-47.

- F. J. Duarte, Beam shaping with telescopes and multiple-prism beam expanders,
*J. Opt. Soc. Am. A*4, P30 (1987).

The first description of interference directly applicable to the interference of narrow-linewidth high-power laser beams was prophetically given by Dirac in 1930 [1]. Dirac's description of interference was not always understood, or accepted, and is referred to by some as "Dirac's dictum." Dirac's description of interference was explained, using practical laser terminology:

"... interference can be analyzed via the interaction of probability amplitudes. These probability amplitudes are said to be represented by wave functions. Hence, interference can be described via the multiplication of an addition of complex wave functions, with its corresponding complex conjugate. Dirac writes about *a [monochromatic] beam of light consisting of a large number of photons*... In the case of a large number of indistinguishable photons his words are just fine" [2].

"The Dirac discussion... begins with reference to *a beam of roughly monochromatic light*; then prior to his dictum on interference, he writes about *a beam of light having a large number of photons*... In present terms this is no different than the description of interference due to the interaction of a high-power narrow-linewidth laser beam with a two-beam interferometer" [1]

In the previous description a key concept is that a beam of monochromatic light is a beam is *indistinguishable photons* which is equivalent to a beam of narrow-linewidth laser emission [1].

1. F. J. Duarte, *Tunable Laser Optics* (Elsevier-Academic, New York, 2003) Chapter 2.

2. F. J. Duarte, Interference of two independent sources, *Am. J. Phys.* 66, 662-663 (1998).

- F. J. Duarte, K. M. Vaeth, and L. S. Liao, Electrically excited organic light-emitting diodes with spatial and spectral coherence, US Patent 7667391 (February 23, 2010).

- F. J. Duarte, Laser sensitometer using multiple-prism beam expansion and a polarizer,
*US Patent 6236461*(22nd of May, 2001).

- F. J. Duarte, Electro-optical interferometric microdensitometer
system,
*US Patent 5255069*(19th of October, 1993).

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