scholarly journals Ray pencils of general divergency

2009 ◽  
Vol 68 (2) ◽  
Author(s):  
W. F. Harris
Keyword(s):  
Homogeneous Medium ◽  
Axiomatic Approach ◽  
Transverse Plane ◽  
Optical Systems ◽  
Linear Optics ◽  
Straight Line ◽  
Asymmetric Power ◽  
Dioptric Power ◽  
Homogeneous Media ◽  
Line Focus

That a thin refracting element can have a dioptric power which is asymmetric immediately raises questions at the fundamentals of linear optics.  In optometry the important concept of vergence, in particular, depends on the concept of a pencil of rays which in turn depends on the existence of a focus.  But systems that contain refracting elements of asymmetric power may have no focus at all.  Thus the existence of thin systems with asym-metric power forces one to go back to basics and redevelop a linear optics from scratch that is sufficiently general to be able to accommodate suchsystems.  This paper offers an axiomatic approach to such a generalized linear optics.  The paper makes use of two axioms: (i) a ray in a homogeneous medium is a segment of a straight line, and (ii) at an interface between two homogeneous media a ray refracts according to Snell’s equation.  The familiar paraxial assumption of linear optics is also made.  From the axioms a pencil of rays at a transverse plane T in a homogeneous medium is defined formally (Definition 1) as an equivalence relation with no necessary association with a focus.  At T the reduced inclination of a ray in a pencil is an af-fine function of its transverse position.  If the pencilis centred the function is linear.  The multiplying factor M, called the divergency of the pencil at T, is a real  2 2×  matrix.  Equations are derived for the change of divergency across thin systems and homogeneous gaps.  Although divergency is un-defined at refracting surfaces and focal planes the pencil of rays is defined at every transverse plane ina system (Definition 2).  The eigenstructure gives aprincipal meridional representation of divergency;and divergency can be decomposed into four natural components.  Depending on its divergency a pencil in a homogeneous gap may have exactly one point focus, one line focus, two line foci or no foci.Equations are presented for the position of a focusand of its orientation in the case of a line focus.  All possible cases are examined.  The equations allow matrix step-along procedures for optical systems in general including those with elements that haveasymmetric power.  The negative of the divergencyis the (generalized) vergence of the pencil.

scholarly journals Foci in ray pencils of general divergency

2009 ◽  
Vol 68 (3) ◽  
Author(s):  
W. F. Harris ◽  
R. D. Van Gool
Keyword(s):  
Optical Systems ◽  
Single Line ◽  
Second Line ◽  
Generalized Systems ◽  
Antisymmetric Component ◽  
Dioptric Power ◽  
Finite Distance ◽  
Line Focus

In generalized optical systems, that is, in systems which may contain thin refracting elements of asymmetric dioptric power, pencils of rays may exhibit phenomena that cannot occur in conventional optical systems.  In conventional optical systems astigmatic pencils have two principal meridians that are necessarily orthogonal; in generalized systems the principal meridians can be at any angle.  In fact in generalized systems a pencil may have only one principal meridian or even none at all.  In contrast to the line foci in the conventional interval of Sturm line foci in generalized systems may be at any angle and there may be only one line focus or no line foci.  A conventional cylindrical pencil has a single line focus at a finite distance but it can be regarded as having a second line focus at infinity.  Only in generalized systems is a single line focus possible without a second at infinity or anywhere else.  The purpose of this paper is to illustrate the types of pencils possible in generalized systems.  Particular attention is paid to the effect of including an antisymmetric component in the divergency of the pencil.

scholarly journals Thin lenses of asymmetric power

2009 ◽  
Vol 68 (2) ◽  
Author(s):  
W. F. Harris ◽  
R. D. Van Gool
Keyword(s):  
Thin Lens ◽  
Asymmetric Power ◽  
Dioptric Power ◽  
Power Matrix

It is generally supposed that thin systems, including refracting surfaces and thin lenses, have powers that are necessarily symmetric.  In other words they have powers which can be represented assymmetric dioptric power matrices and in the familar spherocylindrical form used in optometry and ophthalmology.  This paper shows that this is not correct and that it is indeed possible for a thin system to have a power that is not symmetric and which cannot be expressed in spherocylindrical form.  Thin systems of asymmetric power are illustratedby means of a thin lens that is modelled with small prisms and is chosen to have a dioptric power ma-trix that is antisymmetric.  Similar models can be devised for a thin system whose dioptric power matrix is any  2 2 ×  matrix.  Thus any power, symmetric, asymmetric or antisymmetric, is possible for a thin system.  In this sense our understanding of the power of thin systems is now complete.

scholarly journals Optical axes of catadioptric systems including visual, Purkinje and other nonvisual systems of a heterocentric astigmatic eye

2010 ◽  
Vol 69 (3) ◽  
Author(s):  
W. F. Harris
Keyword(s):  
Visual System ◽  
Optical Axis ◽  
The Other ◽  
Optical Systems ◽  
Numerical Examples ◽  
Straight Line ◽  
Optical Axes ◽  
Catadioptric Systems ◽  
Reflecting Surfaces ◽  
Special Case

For a dioptric system with elements which may be heterocentric and astigmatic an optical axis has been defined to be a straight line along which a ray both enters and emerges from the system.  Previous work shows that the dioptric system may or may not have an optical axis and that, if it does have one, then that optical axis may or may not be unique.  Formulae were derived for the locations of any optical axes.  The purpose of this paper is to extend those results to allow for reflecting surfaces in the system in addition to refracting elements.  Thus the paper locates any optical axes in catadioptric systems (including dioptric systems as a special case).  The reflecting surfaces may be astigmatic and decentred or tilted.  The theory is illustrated by means of numerical examples.  The locations of the optical axes are calculated for seven optical systems associated with a particular heterocentric astigmatic model eye.  The optical systems are the visual system, the four Purkinje systems and two other nonvisual systems of the eye.  The Purkinje systems each have an infinity of optical axes whereas the other nonvisual systems, and the visual system, each have a unique optical axis. (S Afr Optom 2010 69(3) 152-160)

scholarly journals Pupillary axis of a heterocentric astigmatic eye

2013 ◽  
Vol 72 (1) ◽  
Author(s):  
W. F. Harris
Keyword(s):  
Anterior Chamber ◽  
Contact Lens ◽  
Line Of Sight ◽  
Compound System ◽  
Linear Optics ◽  
Optical Instrument ◽  
Entrance Pupil ◽  
Naked Eye ◽  
Straight Line

The pupillary axis of the eye is a clinically useful concept usually defined as the line through the centre of the entrance pupil that is perpendicular to the cornea. However if the cornea is astigmaticthen, strictly speaking, the entrance pupil is blurred and the pupillary axis is not well defined.  A modified definition is offered in this paper: the pupillary axis is the infinite straight line containing the incident segment of the ray that passes through the centre of the (actual) pupil and is perpendicular to the first surface of the eye.  The definition holds for the naked eye and for an eye with an implant in the anterior chamber.  It also holds for the com-pound system of eye and optical instrument such as a contact lens in front of it if the first surface is interpreted as the first surface of the compound system and the pupil as the limiting aperture of the compound system.  Linear optics is applied to obtain a formula for the position and inclination of the pupillary axis at incidence onto the system; the refracting surfaces may be heterocentric and astigmatic.  The formula allows one to examine the sensitivity of the pupillary axis to displacement of the pupil and any other changes in the anterior eye.  Strictly the pupillary axis depends on the frequency of light but examples show that the dependence is probably negligible.  The vectorized generalization of what is sometimes called angle lambda is easily calculated from the inclination of the pupillary axis and the line of sight. (S Afr Optom 2013 72(1) 3-10)

Finite Element Analysis of Elastoplastic Indentation: Part I—Homogeneous Media

1993 ◽  
Vol 115 (1) ◽  
pp. 10-14 ◽  
Author(s):  
P. Montmitonnet ◽  
M. L. Edlinger ◽  
E. Felder
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Finite Element Model ◽  
Homogeneous Medium ◽  
Element Model ◽  
Elastoplastic Analysis ◽  
Element Analysis ◽  
Friction Law ◽  
Indentation Analysis ◽  
Homogeneous Media

Indentation analysis is performed using a finite element model. The material is supposed elastoplastic, the spherical indenter is elastic. Elastic indentation with friction is first analyzed. It leads to a discussion of the formulation of the friction law and the effect of a sliding/sticking threshold. Elastoplastic analysis of the indentation of a homogeneous medium is conducted till unloading, suggesting the occurrence of replastification during unloading.

scholarly journals Retention of Nanoparticles: From Laboratory Cores to Outcrop Scale

Geofluids ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-16 ◽  
Author(s):  
Harpreet Singh ◽  
Farzam Javadpour
Keyword(s):  
Porous Media ◽  
Laboratory Experiments ◽  
Multiple Scales ◽  
Homogeneous Medium ◽  
Laboratory Scale ◽  
Small Scale ◽  
Length Scale ◽  
The Media ◽  
Small Change ◽  
Homogeneous Media

Laboratory experiments on small scale core plugs have shown controlled nanoparticles (NPs) retention. The length scale of subsurface media where NPs must be transported is an important factor that should be accounted for in a comprehensive manner when translating laboratory results to field scale. This study investigates the fraction of NPs retained inside porous media as a function of length scale of the media. A two-dimensional numerical model was used to simulate the retention of NPs at multiple scales of porous media, starting from laboratory scale cores to heterogeneous outcrop scales. Retention of NPs is modeled based on the concept of reversible and irreversible retention, by using the laboratory scale determined parameters. Our results show that the fraction of retained NPs increases nonlinearly with the length scale of the homogeneous media. The results also show that if the heterogeneity of the medium is consistent across scales, the fraction of retained NPs would behave just like homogeneous medium. In this study, small change in heterogeneity at two outcrop scales affects the retention of NPs, suggesting that heterogeneity may significantly impact the retention behavior of NPs that may not necessarily follow the behavior predicted from homogeneous cores (or periodically heterogeneous medium).

Quantitative analysis of eyes and other optical systems in linear optics

2017 ◽  
Vol 37 (3) ◽  
pp. 347-352 ◽  
Author(s):  
William F. Harris ◽  
Tanya Evans ◽  
Radboud D. van Gool
Keyword(s):  
Quantitative Analysis ◽  
Optical Systems ◽  
Linear Optics

scholarly journals Yves Le Grand on matrices in optics with application to vision: Translation and critical analysis

2013 ◽  
Vol 72 (4) ◽  
Author(s):  
William Frith Harris
Keyword(s):  
English Translation ◽  
Critical Analysis ◽  
Careful Analysis ◽  
Linear Optics ◽  
Physiological Optics ◽  
Dioptric Power ◽  
Power Matrix

An appendix to Le Grand’s 1945 book, Optique Physiologique: Tome Premier: La Dioptrique de l’Œil et Sa Correction, briefly dealt with the application of matrices in optics.  However the appendix was omitted from the well-known English translation, Physiological Optics, which appeared in 1980.  Consequently the material is all but forgotten.  This is unfortunate in view of the importance of the dioptric power matrix and the ray transference which entered the optometricliterature many years later.  Motivated by the perception that there has not been enough care in optometry to attribute concepts appropriately this paper attempts a careful analysis of Le Grand’s thinking as reflected in his appendix.  A translation into English is provided in the appendix to this paper.  The paper opens with a summary of the basics of Gaussian and linear optics sufficient for the interpretation of Le Grand’s appendix which follows.  The paper looks more particularly at what Le Grand says in relation to the transference and the dioptric power matrix though many other issues are also touched on including the conditions under which distant objects will map to clear images on the retina and, more particularly, to clear images that are undistorted.  Detailed annotations of Le Grand’s translated appendix are provided. (S Afr Optom 2013 72(4) 145-166)

Generalized two‐pass three‐dimensional migration for imaging steep dips in vertically inhomogeneous media

Geophysics ◽  
1989 ◽  
Vol 54 (5) ◽  
pp. 544-554 ◽  
Author(s):  
Naide Pan ◽  
William S. French
Keyword(s):  
Finite Difference ◽  
Three Dimensional ◽  
Homogeneous Medium ◽  
Synthetic Data ◽  
Inhomogeneous Media ◽  
Velocity Error ◽  
Straightforward Application ◽  
Time Migration ◽  
First Pass ◽  
Homogeneous Media

Conventional two‐pass 3-D time migration is exactly equivalent to full 3-D time migration in a homogeneous medium. For vertically inhomogeneous media representing typical earth velocities, however, conventional two‐pass 3-D migration fails to correctly image dips beyond about 45 degrees. This failure is the result of an inherent velocity error incurred during the first pass of a two‐pass 3-D migration. For a vertically inhomogeneous medium, the theory of residual migration can be combined with the results for homogeneous media to derive a series of successive two‐pass migration stages which are equivalent to a full 3-D migration. Each stage of this generalized two‐pass 3-D migration is implemented using an appropriate constant migration velocity. In practice, the required number of two‐pass stages can be reduced to a computationally manageable few; and the I/O can be reduced by one‐third to one‐half of that required using a straightforward application of repeated two‐pass migrations. This procedure allows existing 2-D migration programs to be upgraded to steep‐dip 3-D migration programs by use of a simple I/O structure. Any of the basic 2-D migration algorithms can be used, but we have employed a 50-degree finite‐difference algorithm. In addition, generalized two‐pass 3-D migration overcomes the dip limitations of the underlying 2-D finite‐difference migration algorithm for the same reasons that cascaded 2-D migration extends the dip range of 2-D migration algorithms. Synthetic data examples clearly show the success of this method in imaging steep dips in vertically inhomogeneous media.

scholarly journals Quantum verification of NP problems with single photons and linear optics

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Aonan Zhang ◽  
Hao Zhan ◽  
Junjie Liao ◽  
Kaimin Zheng ◽  
Tao Jiang ◽  
...  
Keyword(s):  
Quantum Computing ◽  
Large Scale ◽  
Complete Solution ◽  
Optical Systems ◽  
Single Photons ◽  
Linear Optics ◽  
Computational Capability ◽  
Intractable Problems ◽  
Information Manipulation ◽  
Np Problems

AbstractQuantum computing is seeking to realize hardware-optimized algorithms for application-related computational tasks. NP (nondeterministic-polynomial-time) is a complexity class containing many important but intractable problems like the satisfiability of potentially conflict constraints (SAT). According to the well-founded exponential time hypothesis, verifying an SAT instance of size n requires generally the complete solution in an O(n)-bit proof. In contrast, quantum verification algorithms, which encode the solution into quantum bits rather than classical bit strings, can perform the verification task with quadratically reduced information about the solution in $$\tilde O(\sqrt n )$$ O ̃ ( n ) qubits. Here we realize the quantum verification machine of SAT with single photons and linear optics. By using tunable optical setups, we efficiently verify satisfiable and unsatisfiable SAT instances and achieve a clear completeness-soundness gap even in the presence of experimental imperfections. The protocol requires only unentangled photons, linear operations on multiple modes and at most two-photon joint measurements. These features make the protocol suitable for photonic realization and scalable to large problem sizes with the advances in high-dimensional quantum information manipulation and large scale linear-optical systems. Our results open an essentially new route toward quantum advantages and extend the computational capability of optical quantum computing.

Sign in / Sign up

Export Citation Format

Share Document