Direction cosine diagrams for four orientations of a grating with period d=3λ illuminated with an obliquely incident beam (αi=−0.3 and βi=−0.4): (a) ψ=90  deg, (b) ψ=60  deg, ψ=30  deg, ψ=0  deg.

I have ghosting/double vision of illuminated objects like the moon, lights, reflecting street signs at night, and subtitles on the tv, to name a few examples. The second image appears hazy, overlapping below the true object. It is constant, not transient. It’s more obvious at night. Some signs have multiple images. Like while most street signs appear as described above, these little vertical yellow rectangular signs reflect and create three images not touching the true object in the center. The ghosted images appear one below, one to the right, and one to the left of it.

Polarization of light is a property that applies to turning waves that shows the geometrical blooming of the oscillations. In a turning wave, the way of the ...

The spatial frequencies ξ and η are the reciprocal variables in Fourier transform space. Also the Fresnel diffraction integral is given by the Fourier transform of the product of the aperture function with a quadratic phase factor.27,28 Implicit in both the Fresnel and the Fraunhofer approximation is a “paraxial limitation” that restricts their use to small diffraction angles and small angles of incidence.27,28 This paraxial limitation severely restricts the conditions under which this conventional linear systems formulation of scalar diffraction theory adequately describes real diffraction phenomena.

I am an hour north of Pittsburgh. PA. I went to 4 eye specialist for this and finally googled my symptoms. Ghost vision. The fifth dr fitted me with glasses that helped. I am concerned that it can be a serious condition. Where are you located please?

Vision is fine during the day, these issues occur when starts to get darker, or if I walk into wardrobe during day, I notice odd area in my left eye, to the right.

Illustration of diffraction orders for a transmission grating with λ/d=0.25 and the +1 diffracted order satisfying the Littrow condition.

A whole new era of spectral analysis opened up with Rowland’s famous paper in 1882. He constructed sophisticated ruling engines and invented the “concave grating,” a device of spectacular value to modern spectroscopists.10

Figure 29 schematically illustrates the dispersive behavior over the visible spectrum of a grating blazed for the first order at a wavelength 500 nm. The seven classical discrete colors: red (λ1=650  nm), orange (λ2=600  nm), yellow (λ3=550  nm), green (λ4=500  nm), blue (λ5=450  nm), indigo (λ6=400  nm), and violet (λ7=350  nm) are obtained by replacing the integral in the above equation by a discrete summation: Eq. (63)E2(x2)∝∑m=−∞∞∑λ=λ1λ7sinc2[m−(nBλB/λ+θ0d/λ)λz/dλz/d]⁢δ[x2−(θ0d/λ)λz/d].

They tell me it can take longer for people to get accustomed to these things after cataract surgery, but it does not feel right to me, it has been over 3 months. I feel I have more issues now than when I had the cataracts, and it should not really be this way. Looking forward to reply with any helpful information.

The optical power contained in the m’th diffraction order is obtained by integrating the above irradiance distribution over all space in the x2−y2 plane; however, due to Eq. (20), the integral of the quantity in curly brackets is just unity, and we simply obtain Eq. (29)Pm(x2,y2)=E0w2b2d2 sinc2(mbd).

We have specifically chosen the form of Eq. (3) not only to maintain the sign convention for directional angles used almost exclusively in geometrical optics and optical design ray trace codes (positive if counterclockwise and negative if clockwise), but also to be consistent with the sign convention for labeling diffraction grating order numbers used by the popular Diffraction Grating Handbook published and distributed free by the Newport Corporation (formerly Richardson Grating Laboratory).19

Figure 19 compares the predicted diffraction efficiency of this approximation with the results of Eq. (34) for a perfectly conducting surface (R=1) and illustrates how shallow the grating must be to satisfy various error tolerances. Note that the above approximation exhibits only a 1% error in the prediction of diffraction efficiency of the +1 diffracted order at h=0.0318λ, a 5% error at h=0.0702λ, and a 10% error at h=0.098λ.

A troublesome aspect of the multiple order behavior of diffraction gratings is that adjacent higher order spectra frequently overlap. In fact, in Fig. 3, one can see the third-order principle maximum for blue light almost overlapping the second-order red principle maximum. One can readily show that the second order for wavelengths 100, 200, and 300 nm is diffracted into the same directions as the first order for wavelengths 200, 400, and 600 nm.

We have demonstrated that when the grating equation is expressed in terms of the direction cosines of the propagation vectors of the incident beam and the diffracted orders, even wide-angle diffraction phenomena (including conical diffraction from arbitrarily oriented gratings) is shift invariant with respect to variations in the incident angle. New insight and an intuitive understanding of diffraction behavior for arbitrary grating orientation were then shown to result from the use of a simple direction cosine diagram.

Here Uo+(x1,y1)=Uo−(x1,y1)t1(x1,y1) is the complex amplitude distribution emerging from the diffracting aperture of complex amplitude transmittance t1(x1,y1), and Uo−(x1,y1) is the complex amplitude incident upon the lens.

One doctor mentioned it could be the thining retina membrane of my right eye, my right eye has myopic degeneration, for which I had shots of avastin.

Diffraction gratings can be categorized according to several different criteria: their geometry, material, their efficiency behavior, their method of manufacture, or their intended application. We thus talk about:

Hello I started to see ghost images recently Letters with shadow When I use optical glasses it comes to normal What us the reason for that And what are the treatment for it

One example of a diffraction grating would be a periodic array of a large number of very narrow slits. This would be a binary amplitude grating (completely opaque or completely transparent). Consider the cylindrical Huygens’ wavelet produced at each narrow slit when the grating is illuminated by a normally incident plane wave as shown in Fig. 1. It is clear to see that there will be constructive interference only in those discrete directions where the optical path difference from adjacent slits is an integral number of wavelengths (i.e., phase differences in multiples of 2π). Every point P in the focal plane of the lens that satisfies this condition will exhibit a primary maximum. The angular width of this interference maximum depends upon the number of slits making up the grating. Figure 2 illustrates the one-dimensional profile of the Fraunhofer diffraction pattern of an array of slits as we progress from two slits (Young’s interference pattern) to three slits, to five slits, and to eleven slits.

Illustration of the 100% efficiency achieved by a perfectly reflecting blazed grating satisfying the Littrow condition for the second diffracted order.

I have double vision and droopy eyelid. When I close one of my eyes, my double vision is gone. Same with the other eye. I really don’t know what I have.

Diffracted intensity distribution as predicted by the above paraxial model for a sinusoidal reflection grating of period d=20λ operating at normal incidence.

The equation for a reflection grating can be obtained by setting n′=−n, just as we do when tracing rays from a reflecting surface:4 Eq. (2)sin θm+sin θi=mλ/nd,m=0,±1,±2,±3.

What is aghostimage in radiography

Please visit the Vision Specialist of Michigan website to find a doctor near you. There are a few doctors in Texas who may be able to help you. The Neuro Visual Center

So my neck popped and I have something double vision in both eyes and when a car goes by it’s like and tracing image. Did he say the brain can heal From it?

There are two different types of double vision (diplopia): monocular and binocular. While monocular double vision can be caused by corneal disease or cataracts, binocular double vision can be caused by serious neurologic conditions.

Table 2 lists the efficiency for the first several orders for b/d=0.5. Note that the efficiency of all even orders is identically zero because the zeros of the envelope function in Eq. (28) fall exactly upon the even diffracted orders. We can also see from Fig. 17 that the maximum efficiency that can be achieved for the second order is 0.025 for b/d=0.25 or 0.75.

For a rectangular phase grating with an arbitrary phase step, the complex amplitude transmittance can be written as Eq. (42)tA(x1,y1)=exp[iϕ(x1)],where the phase variation is given by Eq. (43)φ(x1)=a rect(x1b)(1)**1dcomb(x1d)δ(y1).

The diffraction efficiency of the zero order and the +1 order which is maintained in the Littrow condition for a square-wave amplitude diffraction grating is plotted versus λ/d in Fig. 35.

How common is it to develop ghosting after cataract surgery? Does getting a multi focal IOL greatly increase one’s chances of developing ghosting?

Restasis is not a steroid, your eye doctor is correct. I am sorry that you are having such problems. If your ghost images go away with one eye covered, then you have a binocular problem that we could help. If the symptoms do not go away when one eye is covered, then you have monocular diplopia. Prismatic correction will not help monocular diplopia. All the best. The Neuro Visual Center

The left eye has never felt right from the get go, and still feels on some days uncomfortable like a hair in eye or similar.

I have shadowing with both eyes. Almost 2 years bow. Glasses dont help. My right eye does not keep up with my head movements, it lags. Quick dizzy sensation, slight quick nausea. Seen an optometrist, and opthalmologist. They see nothing. Neurologist says cervical dystonia and wants an mri of whole head. I think its cevical instability. Any thoughts?

Efficiency of the first few diffracted orders produced by a rectangular phase grating with a duty cycle of 0.5 as a function of the phase step a.

The complex amplitude transmittance of a thin sinusoidal amplitude grating can be written as Eq. (16)tA(x1,y1)=[12+a2 cos(2πx1/d)]rect(x1w,y1w).

Diffraction efficiency of the first several orders produced by a square-wave amplitude grating as a function of the width of the transparent slits relative to the grating period.

In this section, we discuss the paraxial predictions of diffraction efficiency for five basic types of diffraction gratings: sinusoidal amplitude gratings, square-wave amplitude gratings, sinusoidal phase gratings, square-wave phase gratings, and the classical blazed grating (sawtooth groove profile). The paraxial diffraction efficiencies of various diffracted orders will then be tabulated and compared for these five elementary grating types. For all cases, transverse electric (TE) polarization for the incident beam has been assumed.

Before we proceed to discuss the classical blazed grating, we want to derive the general solution for the diffraction behavior of an “arbitrary rectangular phase grating.” This derivation will lay the groundwork for studying the behavior of diffraction gratings with “arbitrary groove shapes.”

If the grating is placed immediately behind an aberration-free positive lens of focal length f that is uniformly illuminated by a normally incident plane wave as illustrated in Fig. 12, the Fraunhofer diffraction pattern produced in the back focal plane of the lens is given by27,28 Eq. (13)E2(x2,y2)=E0λ2f2|F{tA(x1,y1)}|ξ=x2/λfη=y2/λf|2,where E0 is the irradiance of the incident beam and F{} denotes the Fourier transform operation: Eq. (14)F{t(x1,y1)}=∫−∞∞∫−∞∞tA(x1,y1)exp[−i2π(x1ξ+y1η)]dx1 dy1.

The diffraction efficiency is defined as the fraction of the incident optical power that appears in a given diffracted order (usually the +1 order) of the grating. Integrating the irradiance distribution representing a given diffracted order and dividing by the incident optical power Po=E0w2 gives the diffracted efficiency for that order. Since, for any b and xo Eq. (20)∫−∞∞∫−∞∞1b2 sinc2(x−xob,yb)=1,it is clear that the efficiencies are just the coefficients of the three sinc2 terms in the curly brackets of Eq. (18). These efficiencies are tabulated in Table 1.

Also the product of a sinc function with a comb function can be written as an infinite sum of shifted and scaled delta functions,28 hence, Eq. (24)F{tA(x1,y1)}=w2bd{[∑m=−∞∞sinc(mbd)δ(ξ−m/d)]*sinc(wξ)}sinc(wη).

Illustration of the dispersion produced over the visible spectrum by a grating blazed for a wavelength of 500 nm in the first diffractive order.

The paraxial behavior described by Eq. (36) above leads to the common misconception that it is impossible to get more than 33.86% of the incident energy into the first diffracted order with a sinusoidal phase grating. “Nothing could be further from the truth!” In fact, if you decrease the grating period, the diffracted angles increase and the higher orders eventually go evanescent. When only the zero and ±1 diffracted orders remain, changing the incident angle will cause the −1 order to go evanescent. Then one can vary the groove depth to squelch the energy in the zero order. For a perfectly conducting sinusoidal reflectance grating, we can thus get 100% of the incident energy in the +1 diffracted order!33

According to Eq. (68), the renormalization constant K is equal to Eq. (73)K=η−1+η0+η1η0+η1=0.0625+0.25+0.06250.25+0.0625=1.2,where ηm is the diffraction efficiency of the m’th diffracted order. The diffraction efficiency of a sinusoidal amplitude diffraction grating is plotted versus λ/d in Fig. 32.

For a reflection grating, the undiffracted zero order always lies diametrically opposite the origin of the α−β coordinate system from the incident beam. As the incident angle is varied, the diffraction pattern (size, shape, separation, and orientation of diffracted orders) remains unchanged but merely shifts its position maintaining the above relationship between the zero order and the incident beam. Note also that when the plane of incidence is perpendicular to the grating grooves (ϕ0=0), Eq. (4) reduces to the familiar grating equation presented in Eq. (3).

If the incident angle is nonzero, there would be an additional linear phase variation over the entire grating (not each facet individually). Equation (54) describing the complex amplitude distribution emerging from the reflecting blazed grating would thus have to be modified as follows: Eq. (58)tA(x1)=[rect(x1d)exp(−i2πnBλBx1/λd)*1dcomb(x1d)]exp(−i2πθ0λx1),where the diffraction angle of the zero order (angle of reflection) is merely the negative of the incident angle, i.e., θ0=−θi. Again, using the scaling theorem and the convolution theorem of Fourier transform theory, we obtain Eq. (59)F{tA(x1)}={sinc[d(ξ−nBλB/λd)][dcomb(dξ)]}*δ(ξ−θ0/λ).

Thank you. FYI – The image is still there in my field of vision. I don’t know if Medicare covers eye problems or not, but I guess I need to find out. Thank you again.

Efficiency of the first few diffracted orders produced by a rectangular phase grating with a phase step of π as a function of the duty cycle (b/d).

Due to the replication property of convolution with delta functions, we can now write the quantity in the curly bracket as an infinite series of shifted and scaled sinc functions, thus eliminating the convolution operation from the above equation: Eq. (25)F{tA(x1,y1)}=w2bd[∑m=−∞∞sinc(mbd)⁢sinc(ξ−m/d1/w)]sinc(wη).

I had the new catarac surgery with adjustable lenses. I see great but after close work or just “whenever” I see shadows above and below letters, musical notes road signs, etc etc. This comes and goes a a whim and it’s very disturbing because I am a pianist. My eye surgeon and eye doctor who did the adjustments have done every test on my eyes to determine if there are serious problems and said it’s got to be dry eye so recommended Ristasis the RX. My regular doctor and daughter-in-law, Doctor suggested not taking it because it is a Steroid. I’ve used Teradrops and Soothe and other off the counter drops but they don’t seem to do the trick for very long in the day. Eye Surgeon claims Ristasis is NOT a steroid..also it’s VERY expensive and I’ve already spent $7000 on these special lenses. Glasses do NOT help….HELP PLEASE

A missing insight that we now take for granted was provided by John Anderson in 1916 while working at the Mt. Wilson Observatory. He demonstrated that superior gratings could be produced by “burnishing” (plastic deformation of the surface) rather than cutting the grooves into the substrate.37 The material thus had to be soft enough to accept local deformation and at the same time be highly polished.

I cannot find much of anything on my issue. I see double/ghosting both monocular and binocular, with glasses and without. In other words, I see it with both eyes, when I cover just my right eye, and when I cover just my left eye. I watched a video from an neuro-opthalmologist out of Houston who explained it as “bilateral monocular cerebral polyopia,” but I can’t find very much information on this.

Schematic illustration of diffraction orders for a 50% duty cycle square-wave amplitude grating. Note that all even orders are absent.

A companion paper, Understanding Diffraction Grating Behavior, Part II is currently in progress and will discuss in detail the limits of applicability of nonparaxial scalar diffraction theory to sinusoidal reflection (holographic) gratings as a function of the grating period to wavelength ratio.

The conservation of energy is easily shown for this perfectly conducting paraxial (d≫λ) reflection grating at normal incidence because the sum over m from −∞ to ∞ of the squared Bessel function in Eq. (33) is equal to unity.

Dat. fehlerlosem. Akk. fehlerloses. Plural. Nom. fehlerlose. Gen. fehlerloser. Dat. fehlerlosen.

As previously shown in Table 1 of Sec. 5.1, for a narrow beam normally incident upon a paraxial sinusoidal amplitude grating with modulation of unity, five-eighths of the incident energy is absorbed and three-eights of it is transmitted. Twenty-five percent of the total incident energy is contained in the zero order and six and one-quarter percent is contained in both the +1 and the −1 orders.

For a thin diffraction grating in air, we thus have n=n′=1, and the two grating equations can be combined to yield Eq. (3)sin θm∓sin θi=∓mλ/d,m=0,±1,±2,±3.

Introducing an arbitrary incident angle will thus shift both the sinc2 envelope function and the diffracted orders by precisely the same amount. Therefore, under “paraxial” conditions, the diffraction efficiency does not change with incident angle. For example, if we illuminate the above grating blazed for the second order with an incident angle equal to the blaze angle (θi=θB), the incident beam will strike the individual facets at normal incidence and the second order will be retroreflected as illustrated in Fig. 27. This situation (θi=θ2) is referred to as the Littrow condition for the second order,19 and the efficiency will remain at 100% as shown in Fig. 28. The zero order will of course be specularly reflected from the plane of the grating, and the +1 order will be diffracted normal to the plane of the grating.

Making use of the Bessel function identity27 Eq. (32)exp[ia2 sin(2πx1/d)]=∑m=−∞∞Jm(a2)exp(i2πmx1/d),where Jm is a Bessel function of the first kind, order m, and the fact that the exponential Fourier transforms into a shifted delta function,28 it is readily shown that, within the paraxial limitation, the irradiance distribution in the back focal plane of the lens is given by Eq. (33)E(x2,y2)=E0w2{∑m=−∞∞Jm2(a2)[1(λf/w)2 ⁢sinc2(x2−mλf/dλf/w,y2λf/w)]},and the diffraction efficiency of the m’th diffracted order of a perfectly conducting sinusoidal phase grating is given by the following well-known expression:1,22,27,31,32 Eq. (34)efficiency≡Pm(x2,y2)Po=Jm2(a2),where a=4πh/λ and h is the peak to peak groove depth of the sinusoidal reflection grating.

Since the diffraction angle for a given order varies with wavelength, a diffraction grating produces angular dispersion. This angular dispersion is illustrated in Fig. 3 for a grating with a period d=10  μm. Diffracted orders for wavelengths 450, 550, and 650 nm are plotted versus angle.

If the +1 diffracted order is in the Littrow condition (θ1=−θi) as shown in Fig. 31, the grating equation expressed in Eq. (3) results in the following expression for the incident angle Eq. (69)θi=sin−1(0.5λ/d).

Applying the scaling theorem and the convolution theorem of Fourier transform theory,28 we can write the Fourier transform of Eq. (16) as Eq. (17)F{tA(x1,y1)}=[12δ(ξ,η)+a4δ(ξ+1d,η)+a4δ(ξ−1d,η)]**w2sinc(wξ,wη),where ** is the symbolic notation for the two-dimensional convolution operation.28

The first reported observation of diffraction grating effects was made in 1785 when Francis Hopkinson (one of the signers of the declaration of independence and George Washington’s first Secretary of the Navy) observed a distant street lamp through a fine silk handkerchief. He noticed that this produced multiple images, which to his astonishment did not change location with motion of the handkerchief. He mentioned his discovery to the astronomer David Rittenhouse. Rittenhouse recognized the observed phenomenon as a diffraction effect and promptly made a diffraction grating by wrapping fine wire around the threads of a pair of fine pitch screws. Knowing the pitch of his screws in terms of the Paris inch, he determined the approximate wavelength of light.7

You should ask your doctor if you can try a drop to constrict your pupil. This may help with your symptoms. All the best

Making use of the replication properties of convolution with a comb function, the complex amplitude transmittance (or reflectance in this case) of a grating blazed for the n’th order and operating at the blaze wavelength can thus be written as Eq. (54)tA(x1)=rect(x1d)exp(−i2πnBλBx1/λd)*1dcomb(x1d).

Note the 20% increase in diffraction efficiency of both the zero and the +1 diffracted order at λ/d>0.667.41 It is thus possible to get a maximum diffraction efficiency of 0.075 for the +1 order with a sinusoidal amplitude grating. In spite of this increase over the paraxial prediction of Sec. 5.1, this low diffraction efficiency combined with the fact that precision sinusoidal amplitude gratings are difficult to fabricate explains why they are rarely used for practical applications.

Since much of the grating community erroneously believes that scalar diffraction theory is only valid in the paraxial regime (d≫λ), it was emphasized that this limitation is due to an “unnecessary” paraxial approximation in the traditional Fourier treatment of scalar diffraction theory, not a limitation of scalar theory itself. The development of a linear systems formulation of “nonparaxial scalar diffraction theory”20–23 was thus briefly reviewed, then used to predict the nonparaxial behavior of both the sinusoidal and the square-wave amplitude transmission gratings when the +1 diffracted order is maintained in the Littrow condition. This nonparaxial behavior included the well-known Rayleigh anomaly effects that are usually thought to require rigorous (vector) electromagnetic theory.

The fundamental diffraction problem consists of two parts: (i) determining the effects of introducing the diffracting aperture (or grating) upon the field immediately behind the screen and (ii) determining how it affects the field downstream from the diffracting screen (i.e., what is the field immediately behind the grating and how does it propagate).

When they ask “which is better A or B?” the ghost images sometimes rotate, but never pull in to a single image. When I report that the images are the same just rotated, they have never said anything. Once I asked what the shadow images come from and was told that’s my astigmatism.

Illustration of the 100% efficiency achieved by a perfectly reflecting blazed grating designed to operate at normal incidence in the second diffracted order.

First it needs to be determined if your problem is monocular or binocular. Monocular means it is coming from one eye only. Prismatic correction only works for binocular ghosting.

A “diffraction grating” is an optical element that imposes a “periodic” variation in the amplitude and/or phase of an incident electromagnetic wave.1 It thus produces, through constructive interference, a number of discrete diffracted orders (or waves) which exhibit dispersion upon propagation. Diffraction gratings are thus widely used as dispersive elements in spectrographic instruments,2–5 although they can also be used as beam splitters or beam combiners in various laser devices or interferometers. Other applications include acousto-optic modulators or scanners.6

The arrangement of the diffracted orders is the same for the two gratings except they are reflected about the plane of the reflection grating. Note also that the algebraic signs of two directional angles are different if they are measured on different sides of the grating normal. A final useful observation is that for both the transmission and the reflection grating, the positive diffracted orders lie on the same side of the grating normal as the incident beam; whereas the negative diffracted orders lie on the opposite side of the grating normal from the incident beam. A “plus” sign has thus been placed on the lower side of the grating normal in Fig. 5 and a “minus” sign has been placed on the upper side of the grating normal as an indicator of our sign convention. Some authors absorb the minus sign on the right side of Eq. (3) into the m, thus achieving a seemingly simpler equation. However, this results in a different sign convention for labeling the diffracted orders.

Figure 9 illustrates the propagating diffracted orders that would exist if a beam were normally incident upon a transmission diffraction grating with λ/d=0.08333. There would be precisely 25 propagating diffracted orders including the two at ±90  deg. The uniform diffracted order spacing in direction cosine space Δβ is contrasted with the increasing angular spacing Δθ , and the even more rapidly increasing linear spacing Δx, when the diffracted orders are projected upon a plane observation screen.

Making use of the Bessel function identity34 Eq. (49)exp[iz cos(θ)]=∑m=−∞∞imJm(z)exp(imθ),we have an infinite product of infinite sums, which upon Fourier transforming results in an infinite array of convolutions of infinite sums of delta functions: Eq. (50)F{tA(x1,y1)}={[∑m=−∞∞Jm(cn)δ(ξ−n m/d)]n=1*[∑m=−∞∞Jm(cn)δ(ξ−n m/d)]n=2*[∑m=−∞∞Jm(cn)δ(ξ−n m/d)]n=3*⋯*[∑m=−∞∞Jm(cn)δ(ξ−n m/d)]n=∞}.

: Dust-Off Compressed Gas Duster. : 3.5 oz., 7 oz ... First-aid measures after inhalation. : If breathing is difficult, remove victim to fresh air and keep at ...

What isghostimage in Computer

The classical definition of a paraxial ray is that the ray must lie close to, and make a small angle with, the optical axis, i.e., 29,30 Eq. (15)sin θ∼θ,tan θ∼θ,andcos θ∼1.

Again, applying the scaling theorem and the convolution theorem of Fourier transform theory,28 we can write Eq. (22)F{tA(x1,y1)}=[b sinc(bξ)δ(η)][comb(dξ)(1)]**w2 sinc(wξ,wη).

Phase variation for a special case of a rectangular phase grating with a peak-to-peak phase step of π and a 50% duty cycle.

Note that setting m=0 in Eq. (1) results in θ0 having the same sign as θi. Likewise, setting m=0 in Eq. (2) results in θ0 having the opposite sign as θi . We have thus adopted a sign convention that conforms to that used in geometrical optics whereby all angles are directional quantities measured from optical axes or surface normals to refracted or reflected rays. These directional angles are “positive if counterclockwise,” and “negative if clockwise.” An “angle” here is the smaller of the two angles that a ray forms with the axis or surface normal.

We will find later in Sec. 6 that a nonparaxial analysis indicates somewhat better performance for certain combinations of grating period and incident angle.

Hi Ethan This can be coming from a number of problems. Please schedule a routine eye exam with your doctor to start. All the best, The Neuro Visual Center

The classical blazed grating is thus a reflection grating with a sawtooth groove profile as shown in Fig. 25. Such gratings have been manufactured for over 150 years by scribing, or burnishing, a series of grooves upon a good optical surface. Originally, this surface was one of highly polished speculum metal.

In addition to being a paraxial (d≫λ) grating, if the sinusoidal reflection grating is also shallow (i.e., the groove depth is much less than a wavelength of the incident light), then the diffraction efficiency of the first orders of the sinusoidal reflection grating can be approximated by Eq. (37)efficiency≡J12(a/2)≈a2/16.

Note that in all cases, the zero order is diametrically opposite to the origin from the incident beam and the diffracted orders remain equally spaced in a straight line. However, this line is rotated about the zero order such that it is always perpendicular to the grating grooves. This simple behavior of conical diffraction from linear gratings when expressed in direction cosine space provides understanding and insight not provided by most textbook treatments. It is interesting to note that Rowland expressed the grating equation in terms of direction cosines in a paper published over 125 years ago.26

This can happen after cataract surgery with a patient who had a high refractive error. Please visit the Vision Specialist of Michigan website to find a doctor closest to you for an evaluation. All the best.

For large incident and/or diffracted angles, the diffracted radiance distribution function will be truncated by the unit circle in direction cosine space. Evanescent waves are then produced and the equation for diffracted radiance must be renormalized. The renormalization factor in Eq. (67) is given by20–23 Eq. (68)K=∫α=−∞∞∫β=−∞∞L(α,β−β0)dα dβ∫α=−11∫β=−1−α21−α2L(α,β−β0)dα dβand only differs from unity if the diffracted radiance distribution function extends beyond the unit circle in direction cosine space (i.e., only if evanescent waves are produced).

The Fraunhofer diffraction pattern of an array of equally spaced narrow slits is illustrated as the number of slits is increased: (a) two slits, (b) three slits, (c) five slits, and (d) eleven slits.

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However, since the sinc function is separable and the two-dimensional convolution of two separable functions can be written as the product of two one-dimensional convolutions, the above equation can be written as Eq. (23)F{tA(x1,y1)}=w2bd{[sinc(bξ)][dcomb(dξ)]*sinc(wξ)}sinc(wη).

The trend is evident. In the limit of a large number of very narrow slits, the primary interference maxima (diffraction orders) become narrower and narrower, with more and more (n−2) small secondary maxima in between them.

The paraxial behavior of the square-wave amplitude grating was discussed in detail in Sec. 5.2. Equation (28) indicated that there is a myriad of diffracted orders produced; however, they are rapidly attenuated by a sinc2 envelope function. For a 50% duty cycle square-wave amplitude grating (d=2b), the zeros of the envelope function fall precisely on the even diffraction orders as illustrated in Fig. 33. We see from Eq. (28) and Fig. 33 that the diffraction efficiency of the m’th diffracted order is given by Eq. (74)ηm=14 sinc2(m2).

John Strong, quoting G. R. Harrison, stated in a JOSA article in 1960—It is difficult to point to another single device that has brought more important experimental information to every field of science than the diffraction grating. The physicist, the astronomer, the chemist, the biologist, the metallurgist, all use it as a routine tool of unsurpassed accuracy and precision, as a detector of atomic species to determine the characteristics of heavenly bodies and the presence of atmospheres in the planets, to study the structures of molecules and atoms, and to obtain a thousand and one items of information without which modern science would be greatly handicapped.”11

When I see a patient who needs a lot of prismatic correct to align the eyes, I like to demonstrate to my staff how the patient sees without it. Putting prism in front of a person whose eyes are not misaligned gives them the same symptoms of the patient. One time, I gave a clip to my optician with prism on it for her to look out of. She put it on and said, “Wow! This is great! I see so clear!”

Here the minus signs describe a transmission grating and the plus signs describe a reflection grating as illustrated in Fig. 5. Note from this figure that the zero order corresponds to the directly transmitted or specularly reflected beam.

I just got new glasses on Friday and see ghost images with white words on a tv and stuff that lights up. I went to the dr yesterday they said to wait two weeks but didn’t check the physical glasses. I now have a headache and if I make a tiny hole with my thumb and pointer finger and bring it to my one eye with other closed while wearing glasses I don’t see the ghosting

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However, we can ignore the constant term resulting from the fact that ϕ(x1) as illustrated above does not have a zero mean. The rectangular phase variation is thus represented as a superposition of cosinusoidal phase variations: Eq. (46)ϕ(x1)=∑n=1∞cn cos(2πnx1/d).

The Fraunhofer diffraction pattern of an array of eleven equally spaced slits whose width is one-third of their spacing.

Since this is an even function, it can be decomposed into a discrete cosine Fourier series. The Fourier series coefficients for the above periodic function can be shown to be given by Eq. (44)cn=2abdsinc(nbd),thus Eq. (45)ϕ(x1)=a2+∑n=1∞cn cos(2πnx1/d).

Geometry for producing a Fraunhofer diffraction pattern of an aperture (or transmission grating) in the back focal plane of a lens.

Similarly, from Fig. 9, we can see that Eq. (9)xm=L tan θm,where L is the distance between the grating and the observation screen.

Thank you for this article! I have always had this ghosting, and not a single optometrist has asked me whether I have it when getting glasses for nearsightedness.

This renormalization process is also consistent with the law of conservation of energy. However, it is significant that this linear systems formulation of nonparaxial scalar diffraction theory has been derived by the application of Parseval’s theorem and not by merely heuristically imposing the law of conservation of energy.20–23

We have assumed that the grating is bounded by a square aperture of width w. The parameter a represents the peak-to-peak variation in amplitude transmittance and d is the spatial period of the grating. Figure 13(a) shows a two-dimensional image of the grating, and Fig. 13(b) illustrates a profile of the amplitude transmittance in the x direction.

Spectral resolution and diffraction efficiency are quantities of practical interest in many diffraction grating applications. The diffraction efficiency is defined as the fraction of the incident optical power that appears in a given diffracted order of the grating. Note from Fig. 3 that the zero order exhibits no dispersion, and there is twice as much dispersion in the second order as there is in the first order.

Following Goodman,27 a thin sinusoidal phase grating can be defined by the amplitude transmittance function: Eq. (31)tA(x1,y1)=exp[ia2 sin(2πx1/d)]rect(x1w,y1w),where we have ignored a factor representing the average phase delay through the grating. The parameter a represents the peak-to-peak excursion of the sinusoidal phase variation. The grating, bounded by a square aperture of width w, is again placed immediately behind an aberration-free lens that is illuminated with a normally incident plane wave of uniform irradiance E0 as shown in Fig. 13.

We have thus seen that the maximum efficiency of the +1 diffracted order (in the paraxial limit) increases from 0.0625 for a sinusoidal amplitude grating, to 0.1013 for a rectangular amplitude grating, to 0.3386 for a sinusoidal phase grating, and to 0.4053 for a rectangular phase grating.

Illustration of the position of the diffracted orders in real space and direction cosine space for an arbitrary (skew) obliquely incident beam.

Both Goodman27 and Gaskill28 discussed in some detail both the Fraunhofer and the Fresnel approximations and the geometrical criteria for each. Goodman, in particular, showed that the cosine obliquity factor in the more general Huygens–Fresnel principle must be approximately unity for both the Fraunhofer and the Fresnel approximations to be valid. It is this requirement that limits our diffraction angles to be paraxial angles.

I had high myopia(left eye power was -15 at the time of surgery) since childhood and decided to go for surgery. Surgeon performed phaco and put alcon iol.. after surgery my vision got really better but the problem is i started seeing larger image in one eye which causes ghost images and sometimes i feel burning sensations and pain around my eyes.. Please help.

I see used to see double. Then in my 50’ I began additionally seeing double out of just one eye. Then after cataract surgery I began seeing triple out of the other eye. My ability to read has been greatly reduced and impacted my ability to work. Recently I’ve had debridement in both eyes for dot dystrophy, but it has no impact on seeing 5 of everything. I’m miserable.

Diffraction configuration for a sinusoidal amplitude transmission grating with the +1 diffracted order satisfying the Littrow condition when λ/d=0.4.

I am not sure what kind of diplopia do I have. I have them in both eyes, which when I close one of the eyes, the other eye will still see the ghost image. I only see ghost images in low-light places. The ghost images will only appear when I open my eye wider. Is there a cure for this? D:

By incorporating sound radiometric principles into scalar diffraction theory, it becomes evident that the squared modulus of the Fourier transform of the complex amplitude distribution emerging from the diffracting aperture yields “diffracted radiance (not irradiance or intensity)20–23:” Eq. (67)L′(α,β−β0)=Kλ2As|F{Uo′(x^,y^;0)exp(i2πβ0y^)}|2for  α2+β2≤1L′(α,β−β0)=0for  α2+β2>1.

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Blazed gratings can be designed for a particular wavelength, incident angle, and diffracted order. The blaze angle θB of the grating is given by Eq. (51)θB=tan−1(h/d),where h is the groove depth and d is the grating period.

Ghost imageryliterature

Joseph von Fraunhofer began his detailed study of diffraction gratings about 1821. He built the first ruling engine for fabricating reflection gratings on metallic substrates. His insight into the diffraction process led him to predict that diffraction efficiency behavior would “strain even the cleverest of physicists,” which it did for the next 150 years. Many of Fraunhofer’s findings were written up in great detail, so we are entirely justified in calling him the father of diffraction grating technology.8,9

One of the disadvantages of amplitude gratings is that much of the incident optical power is lost through absorption, whereas phase gratings can be made with virtually no absorption losses. Transmission phase gratings can consist of periodic index of refraction variations, or of a periodic surface relief structure, in a thin transparent optical material. Reflection phase gratings are merely a surface relief grating covered with some highly reflective material.

When operating at the blaze wavelength λ=λB, the peak of the sinc2 function is centered on the nB’th diffracted order and all of the other delta functions (diffracted orders) fall on the zeros of the sinc2 function. All of the reflected energy is thus diffracted into the nB’th diffracted order. Figure 26 shows a plot of diffraction efficiency versus x2×λz/d for a coarse grating blazed to operate in the second order at normal incidence for a wavelength of 550 nm. If d≫nBλB, we can be assured, from the planar grating equation, Eq. (3), that the nB’th order will be diffracted at a paraxial angle and this predicted behavior will be accurate.

A linear systems approach to modeling nonparaxial scalar diffraction phenomena has been developed by normalizing the spatial variables by the wavelength of light:20–23 Eq. (65)x^=x/λ,y^=y/λ,z^=z/λ,etc.

This behavior is even more evident in Fig. 7, in which the location of the incident beam and the diffracted orders are displayed in direction cosine space for a reflection grating whose grooves are parallel to the y or β axis. The diffracted orders are always exactly equally spaced in direction cosine space and lie in a straight line perpendicular to the orientation of the grating grooves. From Eq. (4), this equidistant spacing of diffracted orders is readily shown to be equal to the nondimensional quantity λ/d. The diffracted orders that lie inside the unit circle are real and propagate, and the diffracted orders that lie outside the unit circle are evanescent (and thus do not propagate).

What could this be? Is it really neurological? Could astigmatism do this? And if so, could the opthalmologist really have missed that? Why is he so certain it’s my brain?

The diffraction efficiency of the m’th diffracted order is just the above optical power divided by the optical power in the incident beam, Po=E0w2, or Eq. (30)efficiency≡Pm(x2,y2)Po=b2d2 sinc2(mbd).

Note in Fig. 35, the incremental increase in diffraction efficiency of both the zero and the +1 diffracted order as successive pairs of diffracted orders go evanescent.41 A major increase is observed at λ/d>0.667 when the −1 order goes evanescent, after which the renormalization factor has a value of Eq. (78)K=0.5η0+η1=0.50.25+0.1013=1.4233.

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With the wide-spread availability of rigorous electromagnetic (vector) analysis codes for describing the diffraction of electromagnetic waves by specific periodic grating structures, the insight and understanding of nonparaxial parametric diffraction grating behavior afforded by approximate methods (i.e., scalar diffraction theory) is being ignored in the education of most optical engineers today. Elementary diffraction grating behavior is reviewed, the importance of maintaining consistency in the sign convention for the planar diffraction grating equation is emphasized, and the advantages of discussing “conical” diffraction grating behavior in terms of the direction cosines of the incident and diffracted angles are demonstrated. Paraxial grating behavior for coarse gratings (d  ≫  λ) is then derived and displayed graphically for five elementary grating types: sinusoidal amplitude gratings, square-wave amplitude gratings, sinusoidal phase gratings, square-wave phase gratings, and classical blazed gratings. Paraxial diffraction efficiencies are calculated, tabulated, and compared for these five elementary grating types. Since much of the grating community erroneously believes that scalar diffraction theory is only valid in the paraxial regime, the recently developed linear systems formulation of nonparaxial scalar diffraction theory is briefly reviewed, then used to predict the nonparaxial behavior (for transverse electric polarization) of both the sinusoidal and the square-wave amplitude gratings when the +1 diffracted order is maintained in the Littrow condition. This nonparaxial behavior includes the well-known Rayleigh (Wood’s) anomaly effects that are usually thought to only be predicted by rigorous (vector) electromagnetic theory.

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If there are many grating periods within the aperture, then w≫d, and there will be negligible overlap between the three sinc functions; hence, there will be no cross terms in the squared modulus of this sum. Substituting this into Eq. (13) thus yields the diffracted irradiance distribution in the focal plane of the lens: Eq. (19)E(x2,y2)=E0w4λ2f2 sinc2(y2λf/w)[14 sinc2(x2λf/w)⏟m=0+a216 sinc2(x2+λf/dλf/w)⏟m=+1+a216 sinc2(x2−λf/dλf/w)⏟m=−1].

Relative position of diffracted orders and incident beam in direction cosine space for a transmission grating. The zero order and incident beam are superposed.

The second generalization includes the situation where the light is incident upon the grating at an arbitrary angle θi rather than normal incidence. This situation will be taken care of by including the incident angle in the grating equation discussed in Sec. 2, where we will review the planar grating equation and the sign convention for numbering the various diffracted orders.

when i see an image or try to focus on a text , i see a shadow image , the shadow isn’t too much , just along the curvature of the image , its a bit uncomfortable to focus on the image /text . I am also using glasses to correct my farsightedness. With the glasses the shadow gets reduced but it does not completely goes away . Am having trouble in having a sharp image , i can read and see but its a bit uncomfortable with the shadow , also am having frontal , temporal and pain around and behind the eyes

A thin rectangular phase grating can thus be defined by the amplitude transmittance function: Eq. (47)tA(x1,y1)=exp[i∑n=1∞cn cos(2πnx1/d)].

I did have my right eye done 4 weeks later, my specialist assured me that left eye will get better and lense was in correct position. The right eye felt quite different to left eye after op, far-away vision good as in left eye, I do get slight halo effects etc in this eye but nothing that really bothers me and can’t live with, although I do have quite a few floaters in both eyes.

In spite of the fact that it is almost universally believed that—“in no way can scalar theory deal with cut-off anomalies,”40 the renormalization factor K in Eq. (67) and defined by Eq. (68) enables this linear systems formulation of nonparaxial scalar diffraction theory to predict and model the well-known Wood’s (Rayleigh) anomalies16 that occur in diffraction efficiency behavior for simple cases of amplitude transmission gratings discussed in the following two sections of this paper.

The paraxial diffraction efficiencies of the first 19 diffracted orders of a square-wave amplitude grating with a 50% duty cycle are listed in Table 6. Note that 25% of the incident energy is contained in the zero diffracted order, all even orders are identically zero, and the remaining diffracted orders contain another 25%. The remaining 50% of the energy in the incident beam is absorbed by the opaque strips making up the square-wave amplitude grating.

There is thus a myriad of diffracted orders produced by the square-wave amplitude grating as shown in Fig. 16. However, they are rapidly attenuated by the sinc2 envelope function. The irradiance distribution representing the m’th diffracted order is thus given by Eq. (28)Em(x2,y2)=E0w2b2d2 sinc2(mbd)[1(λf/w)2 ⁢sinc2(x2−mλf/dλf/w,y2λf/w)].

When my eyes don’t focus, the image separates into 3 copies slightly offset, 2 slightly distorted. Both eyes in turn, monocular. It’s like having astigmatism but in 3 axises. My accommodation range (lens in each eye) is reduced. So glasses correct the myopia but the astigmatism corrections are a compromise that never quite work. I pass the snellen chart in the room but clarity breaks up past about 8m. My visual span for detail like reading is 1char in focus, then the 2 on either side are half clear. So I read with a window of 3 or 4 characters scanning across the whole line. If I deliberately defocus my eyes I can see circles blur into 3 rings. It’s probably worse with dry eyes. Most Optometrists don’t seem to understand the extent for me or other rarer cases. They get frustrated when trying to work out the astigmatism corrections but I keep saying the difs are equally blurry & can’t find the right one per eye. It always takes a lot longer than scheduled but I think results are still not ideal. I don’t have Keratoconus, no major damage to surface of eyes so it must be more to do with the crystalline lense inside each eye. Which you can’t see (only specialist ophthalmologists have the equipment for full mapping of surface & optics, and I’m in Australia not USA). I did go through atleast 4yrs of school (11yo to 15yo) needing glasses after they tested me wrongly.

We can now readily calculate the diffraction efficiencies for a paraxial rectangular phase grating with an arbitrary phase step and duty cycle. Figure 23 graphically illustrates the efficiency of the first few diffracted orders produced by a rectangular phase grating with a phase step of π as a function of the duty cycle (b/d). Note that when b/d equals either zero or unity, that no phase variations exist, and all of the diffracted energy remains in the undiffracted beam (zero order). Also note that for b/d=0.5, we obtain the same results as those tabulated in Table 4.

Hi I have monocular diplopia, ghost images in both eyes (clear overlapping images with glasses), example of what I see: https://www.allaboutvision.com/en-in/conditions/double-vision/

I had a concussion two weeks ago. Ever since then I have been having burred images mainly when I look at screens. It looks like the letters are popping of the screen sometimes and everything looks fake. Is it something I should get checked out or just ride it out?

Hi Teri, Does the shadowing go away if you close one eye? If it does then there is a problem with the way the two eyes are working together. I would cover the right eye and see if that improves things. All the best, The Neuro Visual Center

Quantumghostimaging

The reciprocal variables in Fourier transform space become the “direction cosines” of the propagation vectors of the plane wave components in the angular spectrum of plane waves discussed by Ratcliff,39 Goodman,27 and Gaskill:28 Eq. (66)α=x^/r^,β=y^/r^,andγ=z^/r^.

Evaluating this function at spatial frequencies ξ=x2/λf and η=y2/λf, and again writing as a two-dimensional function, we obtain Eq. (26)F{tA(x1,y1)}|ξ=x2/λfη=y2/λf=w2bd[∑m=−∞∞sinc(mbd)sinc(x2−mλf/dλf/w,y2λf/w)].

Following the discussion of the square-wave amplitude grating, we obtain a Fraunhofer diffraction pattern given by Eq. (40)E(x2,y2)=E0w4λ2f2[∑m=−∞∞sinc2(m2)⁢sinc2(x2−mλf/dλf/w,y2λf/w)],except that the zero diffracted order is absent. Continuing, we obtain Eq. (41)efficiency≡Pm(x2,y2)Po=sinc2(m2)for  m≠0.

The complex amplitude transmittance of a thin square-wave amplitude grating can be written as Eq. (21)tA(x1,y1)=[rect(x1b)(1)**1dcomb(x1d)δ(y1)]rect(x1w,y1w),where d is the period of the grating, and b

The specularly reflected plane wavefront segments will then be out of phase by precisely 2π, thus producing constructive interference for that wavelength and diffracted order. Stated another way, the reflected phase variation over one period of the above grating can be written as Eq. (53)ϕ(x1)=2πλOPD(x1)=2πλ2hx1d=2πnBλBx1/(λd).

The maximum value of J12(a/2) is 0.3386 and occurs for a=3.68, corresponding to a groove depth of h=0.293λ . The diffraction efficiency of the first few orders for this value of a is tabulated in Table 3. Note that the energy falls off rapidly, with 99.88% of the diffracted radiant power contained in diffracted orders |m|≤3. This paraxial model is accurate only for very coarse gratings (d≫λ).

A major advance in the development of diffraction gratings was the discovery by John Strong in 1936 that vacuum deposited aluminum on glass is a far superior medium into which to rule grating grooves than speculum metal, which had been almost universally used for nearly a century.38 Therefore, in recent times, diffraction gratings have been ruled in thin layers of aluminum or gold deposited upon a glass substrate.

I have seen an Optometrist twice who expressed confusion by my vision and an Opthalmologist who said my eyes are healthy and it’s my brain. He referred me to a Neurologist. I am waiting for my appointment, but it’s far out.

I just had my eyes examined 2 months ago and I was told my eyes looked really good and healthy. I will be 66 years old in 9 days. I have worn glasses since 4th grade and I am very near sighted and my vision prescription requires that prism be built into to my prescription glassess.

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For obliquely incident beams and arbitrarily oriented gratings, a complicated three-dimensional diagram is required to depict the diffraction behavior in real space.25 However, the direction cosine diagram provides a simple and intuitive means of determining the diffraction grating behavior even for these general cases. The general grating equation for a reflection grating with arbitrarily oriented lines (grooves) is given by24 Eq. (12)αm+αi=(mλd)sin ψβm+βi=(mλd)cos ψ,where ψ is the angle between the direction of the grating grooves and the α axis. Note that Eq. (12) still reduces to Eq. (3) when ψ=90  deg. Figure 11 illustrates the direction cosine diagram for a beam obliquely incident (αi=−0.3 and βi=−0.4) upon the same reflection grating discussed above (d=3λ) for different orientations of the grating.

It is thus possible to get a maximum diffraction efficiency of 0.1442 for the +1 order with a square-wave amplitude grating. This is a 42.3% increase over the paraxial value of 0.1013.

You can see ghost images with uncorrected refractive error. We you put your glasses on the refractive error is corrected.

Monocular double vision in one eye often presents as a “ghost” image, a shadow that overlaps with the primary image. It can affect the left eye, the right eye or both eyes at once. Monocular double vision can be caused by dry eye syndrome. When your eyes dry out, the surface of your eye becomes rough and irregular as the tear film along the surface of your eyes dries out. People with dry eye syndrome often suffer from itching, scratching, and burning sensations in their eyes, or they may feel like something has become lodged in their eye. These symptoms can get markedly worse if the patient does a lot of reading or computer work, as we unconsciously blink less while focusing on reading. The first step of treatment for dry eye syndrome is typically the use of artificial tears.

As the grating is rotated to increase λ/d, both the angle of incidence and the diffraction angles increase. If we use Eq. (71) to calculate at what value of λ/d the −1 diffracted order goes evanescent, θ−1=π/2, we obtain Eq. (72)λ/d=2/3=0.667.

For a paraxial grating designed to operate at normal incidence, the groove depth must be equal to Eq. (52)h=nBλB/2,where nB is the blaze (or design) order and λB is the blaze (or design) wavelength.

In spite of the fact that increasing the parameter b/d reduces the absorption of the grating, we see that for b/d>0.5, all of the additional transmitted power, plus some, goes into the zero order, with the efficiency of the +1 order actually diminishing with increasing b/d.

About 10 years ago I began having “ghosting”, which was a partial double image would appear rising above a line of print on a page. This began the day afer a chiropractor did a severe pop to my neck. I talked to him about this later, and he swore the pop could not have caused the ghosting. I later went to a neurologist, who said he had encountered this thing before, and said the neck pop had caused a mini sroke in my brain, which affected the vision. The probem has never gone away, but the brain is usually able to ignore it, except in especially stressful times. So never let anyone pop your neck!

We are located outside of New York City in Garden City, New York. Please call the office at 516-244-4888 for more information.

Using the scaling theorem and the convolution theorem of Fourier transform theory, we can write Eq. (55)F{tA(x1)}=sinc[d(ξ−nBλB/λd)][d comb(dξ)].

Consider diffraction from a conventional linear reflection grating. However, suppose the incident light strikes the grating at a large oblique angle (represented by direction cosines αi and βi) as illustrated in Fig. 6. The resulting diffraction behavior is described by the following grating equation written in terms of the direction cosines of the propagation vectors of the incident beam and the diffracted orders (the grooves are assumed to be parallel to the y axis):24 Eq. (4)αm+αi=mλ/d,βm+βi=0,where Eq. (5)αm=sin θm cos ϕo,αi=−sin θo cos ϕo,βi=−sin ϕo.

Can this cause damage to cornea or lens i wore glassess with such error for 4 months from that time ghost image is still present but lesser degree i want to know what happened and the secondaly lasik would still be an option for me or should i take any consideration

And last, but not least, two weeks ago I had an episode where I lost the entire left field of vision in both eyes. I was unable to read words because the left side of each word was missing. It was extremely bizarre. It lasted 45 minutes.

Illustration of Rayleigh anomalies from a square-wave amplitude transmission grating with the +1 order satisfying the Littrow condition.

Since w≫d, there is again negligible overlap between the discrete diffracted orders, and there will be no cross terms in the squared modulus of this sum. The Fraunhofer diffraction pattern predicted by Eq. (13) for a square-wave amplitude grating is thus given by Eq. (27)E(x2,y2)=E0w4λ2f2b2d2[∑m=−∞∞sinc2(mbd)⁢sinc2(x2−mλf/dλf/w,y2λf/w)].

Although the above expression might at first appear to be rather unwieldy, it is rather easily solved numerically with the array operations provided with the MATLAB software package. In fact, the above operation results in an array of delta functions that represents the diffracted orders produced by the rectangular phase grating. The squared moduli of the coefficients of those terms are the efficiencies of the diffracted orders.

The above technique can also be used to calculate the paraxial diffraction efficiencies of a reflection grating with arbitrary groove shape by merely supplying the appropriate Fourier coefficients in Eq. (44).

About two hours later, I decided that I need to get up and continue with my preparations for Halloween. This is when I realized that my floating tiny orange eye “image” had changed into another image that I had been looking at just a few minutes earlier. This time it was piece of black thread that was all knotted up as I had just been working on a piece jewelry that I weaving together using a KumiHimo disc and the thread somehow got had gotten all tangled up into a knot. The same thing was occurring only this time I was seeing the black knotted thread float around in front of my left eye. This has been going on all day and night and it is now 1:00 AM in the morning and I am getting ready to head to bed.

Ghostpic girl

Relative position of diffracted orders and incident beam in direction cosine space for a reflection grating. Diffracted orders outside the unit circle are evanescent.

This optician had always complained to me of not seeing clear enough. Of course, she didn’t: she had a ghost image. She described it as things always appearing “smeary” when trying to focus. Interestingly, she did not complain of headache or dizziness at all, but she did have vertical heterophoria (VH), a type of binocular vision disorder caused by a misalignment of the eyes. With the prism glasses, her world is so much easier. Aligning her eyes gave her the binocular vision and depth perception that she was missing!

Plotting the expressions provided by Eqs. (8) and (11) provides a graphical comparison of the relative spacing between adjacent diffracted orders Δx, Δθ, and Δβ.

Table 4 thus lists the efficiency for the first several orders for this special case of a rectangular phase grating. Note that the π phase step has eliminated the zero order, and the efficiency of all other even orders is identically zero because the zeros in the envelope function in Eq. (40) fall exactly upon the even diffracted orders. This thus maximizes the efficiency of the remaining orders.

Elementary diffraction grating behavior (including diffraction efficiency and dispersion) was reviewed and early challenges in the development of diffraction grating fabrication technology were discussed. The importance of maintaining consistency in the sign convention for the planar diffraction grating equation was emphasized. The advantages of discussing conical diffraction grating behavior in terms of the direction cosines of the incident and diffracted angles were demonstrated, particularly for oblique incident angles and arbitrary grating orientation.

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Setting Δm equal to unity, we obtain the following expression for the angular spacing of “adjacent” diffracted orders as a function of diffracted angle: Eq. (8)Δθm=λd cos θm.

Note that the sign of these two angles are consistent with the sign convention previously illustrated in Fig. 5. Figure 31 illustrates this situation for λ/d=0.4.

The +1 diffracted order thus contains at most (if the quantity a is equal to unity) 6.25% of the optical power incident upon a sinusoidal amplitude grating. This very low diffraction efficiency is not adequate for many applications. As seen in Table 1, the sum of the efficiencies of all three orders is only equal to 1/4+a2/8. The rest of the incident optical power is lost through absorption by the grating.

Figure 10 indicates that both Δx and Δθ asymptotically approaches infinity for diffracted angles of 90 deg, whereas Δβ remains constant for all diffracted angles. When projected upon a plane screen, the spacing of adjacent diffracted orders increases by a factor of two (100% increase) at a diffraction angle of merely 38 deg. The angular spacing of adjacent diffracted orders increases by a factor of two at a diffraction angle of 60 deg. If only a 5% increase in Δx were allowed, the diffraction angle would have to be held below 10 deg. For Δθ, a 5% increase is observed at a diffraction angle of 18 deg.

As with the case of the reflection grating, the diffracted orders remain equally spaced and in a straight line as the incident angle is changed, i.e., the size, shape, separation, and orientation of diffracted orders again remains unchanged, merely shifting its position such that the zero order remains superposed upon the incident beam.

Illustration of Rayleigh anomalies from a sinusoidal amplitude transmission grating with the +1 diffracted order satisfying the Littrow condition.

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Monochromatic light of wavelength λ incident upon a refractive transmission grating (interface between two dielectric media exhibiting a periodic surface relief pattern) of spatial period d at an angle of incidence θi will be diffracted into the discrete angles θm according to the following (planar) grating equation:3,4,16–18 Eq. (1)n′ sin θm−n sin θi=−mλ/d,m=0,±1,±2,±3,where n is the refractive index of the media on the incident side of the diffracting surface, n′ is the refractive index of the media containing transmitted diffracted light, and m is an integer called the order of diffraction. The sign of m is arbitrary and determines the sign convention for labeling diffracted orders.

James E. Harvey received his PhD in optical sciences from the University of Arizona. He is a retired associate professor from CREOL at the College of Optics and Photonics of the University of Central Florida, and currently a senior optical engineer with Photon Engineering, LLC, Tucson, Arizona, USA. He is credited with more than 220 publications and conference presentations in diverse areas of applied optics. He is a member of OSA and a fellow and past board member of SPIE.

Like VH, binocular double vision is caused by eyes that are misaligned. This condition can be caused by a variety of serious factors, so it’s very important to contact a neurovisual professional as soon as possible to determine the type and cause of your double vision, and to get treatment started right away. If you are being plagued by ghost images or other double vision symptoms, give the Neuro Visual Center of New York a call today at (516) 224-4888 to schedule your evaluation.

For a transmission grating, with our sign convention, the diffraction angle for the zero order is equal to the incident angle (θ0=θi). Thus the coordinates of the location in the direction cosine diagram representing the zero order and the incident beam are superposed as illustrated as shown in Fig. 8.

The irradiance of the Fraunhofer diffraction pattern in the x2−y2 observation plane a distance z from the grating is proportional to the squared modulus of the Fourier transform of the complex amplitude distribution emerging from the grating: Eq. (56)E2(x2)∝1λz|F{tA(x1)}|ξ=x2/λz|2,Eq. (57)E2(x2)∝sinc2[x2−(nBλB/λ)λz/dλz/d]1λz/dcomb(x2λz/d).

Due to the replication property of convolution with a delta function, and since the two-dimensional function is separable into the product of two one-dimensional functions:28 Eq. (18)F{tA(x1,y1)}|ξ=x2/λfη=y2/λf=w2 sinc(y2λf/w)[12sinc(x2λf/w)+a4 sinc(x2+λf/dλf/w)+a4 sinc(x2−λf/dλf/w)].

Clearly, the total amount of energy transmitted through this thin grating does not vary as the angle of incidence of the narrow beam is increased. Thus when the −1 diffracted order goes evanescent, the energy that was contained in it (6.25% of the incident energy) is redistributed into the two remaining propagating orders (the Rayleigh anomaly phenomenon).

I see ghost images on things that light up, in the left eye , when wearing glasses but not when wearing contact lenses or with the naked eye. My optician can’t tell me why.

We likewise discover that the −2 and +3 diffracted orders go evanescent when λ/d=2/5, and the −3 and +4 diffracted orders go evanescent when λ/d=2/7, etc.

If this grating is placed immediately behind an aberration-free positive lens of focal length f that is uniformly illuminated by a normally incident plane wave as illustrated in Fig. 12, the Fraunhofer diffraction pattern produced in the back focal plane of the lens is given by Eq. (13).

Also, I have episodes of transient vision disturbances lasting in 1 second bursts over and over again for up to a couple of hours, maybe more.

The concept of a blazed grating is that each groove should be so formed that independently, by means of geometrical optics, it redirects the incident light in the direction of a chosen diffracted order, thus making it appear to “blaze” when viewed from that direction. Lord Rayleigh was first to describe the ideal groove shape in 1874.35 He wrote: “…the retardation should gradually alter by a wavelength in passing over each element of the grating and then fall back to its previous value, thus springing suddenly over a wavelength.” He was not very optimistic about achieving such geometry, but 36 years later, in 1910, Wood36 produced the first grating that we would call “blazed” for use in the infrared. He did this with a tool of carborundum, ruled into copper.

Since the Fraunhofer diffraction integral implicitly contains the paraxial approximation, the diffraction angle is proportional to displacement on the focal plane containing the Fraunhofer diffraction patterns Eq. (35)θx=tan−1(x2/f)≈x2/f,θy=tan−1(y2/f)≈y2/f.

Figure 18 illustrates the diffracted intensity distribution as a function of diffraction angle θx and groove depth h, for a sinusoidal “reflection” grating with period d=20λ operating at normal incidence.

As discussed briefly in Sec. 1–Sec. 5, it is well-known that the paraxial irradiance distribution on a plane in the far field (Fraunhofer region) of a diffracting aperture is given by the squared modulus of the Fourier transform of the complex amplitude distribution emerging from the diffracting aperture.27,28 A slight variation of Eq. (13), without the presence of the lens, can be written as Eq. (64)E(x2,y2)=E0λ2z2|F{Uo+(x1,y1)}|ξ=x2λz,η=y2λz|2.

Now, anytime a patient uses the word “smeary” or just isn’t seeing sharp enough, I immediately pull out my instruments and screen for VH. Sometimes a little prism is all you need.

The above grating equations are restricted to the special case where the grating grooves/lines are oriented perpendicular to the plane of incidence, i.e., the plane containing the incident beam and the normal to the grating surface. For this situation, all of the diffracted orders lie in the plane of incidence.

Hey! I have had a very slight double vision in both of my eyes and it is a slight shadow ontop of it and also my left eye is much blurrier than my right eye but just so you know my left eye has always been much blurrier than my right eye for years now so I doubt it’s anything very sudden and I don’t see my vision progressing getting worse but the double vision thing kind of goes up or down depending on how far or close I am and this never really bothered me but I can kind of curious on what’s going on.

Similarly, Fig. 24 graphically illustrates the efficiency of the first few diffracted orders produced by a rectangular phase grating with a duty cycle of 0.5 as a function of the phase step a. Note that the even orders are absent. Equation (50) and Figs. 23 and 24 combined constitute a rather unique and comprehensive graphical display of the parametric paraxial performance of square-wave phase gratings.

Illustration of paths of superposed optical disturbances that interfere constructively to produce discrete diffracted orders.

Recalling our definitions of radiometric quantities, it is clear that the diffracted intensity distribution (radiant power per unit solid angle) emanating from the grating is thus proportional to the diffracted irradiance distribution (radiant power per unit area) incident upon the focal plane as given by Eq. (33): Eq. (36)I(θx,θy)=I0∑m=−∞∞Jm2(a2)[1(λf/w)2 ⁢sinc2(x2−mλf/dλf/w,y2λf/w)].

Figure 30 illustrates that the dispersion is indeed doubled if the grating is blazed for the second diffracted order. Note also that the diffraction efficiency is substantially reduced for all wavelengths other than the blaze wavelength.

We are so sorry that you are still having difficulty. Please contact Dr. Andrew Taylor in Albany, Australia, he may be able to help you.

The product of a sinc2 function with a comb function can be written as an infinite sum of shifted and scaled delta functions,28 each of which represents a different diffracted order. Equation (60) can, therefore, be rewritten as Eq. (61)E2(x2)∝∑m=−∞∞sinc2[m−(nBλB/λ+θ0d/λ)λz/dλz/d]⁢δ(x2−(θ0d/λ)λz/d).

Lauren I just went to the eye Dr had my eyes checked she made glasses for me when I got them I seen double she checked eyes again and said I need cataract surgery I then went to a eye specialist he said eyes were okay need surgery in 2 years I went back to the eye Dr she checked my eyes again and changed the left lense now I see shadows can you please tell what you did to get rid of the ghosts. Thank you

Hence, when plotting diffraction efficiency versus λ/d, there can be at most only two propagating orders (the zero order and the +1 that is being maintained in the Littrow condition) for λ/d>2/3. All other orders are evanescent.

We thus have three discrete diffracted waves or “orders,” each of which is a scaled replica of the Fraunhofer diffraction pattern of the square aperture bounding the grating. The central diffraction lobe is called the “zero order,” and the two side lobes are called the plus and minus “first orders.” The spatial separation of the first orders from the zero order is λf/d, whereas the width of the main lobe of all orders is 2λf/w as shown in Fig. 14.

I had cataract surgery in July on left eye , around 12 days after I started noticing halos at night on ceiling lights, lights on heater, tv, quite a few floaters, then when I looked at my cellphone or I pad it was like it had a halo/glare all around it, if I watch tv or go to a movie, when screen gets darker I see multiple halos. I have to have screen brightness up high, then this problem goes. Also I have an area in the left eye where it looks like something is in my vision, a whiteish/ghostlike , I have mono lenses and my faraway vision is great, I need glasses for reading.

I had the oddest symptoms today and I am kind of concerned. Today is October 31st, 2023, and this afternoon I was preparing for the trick or treaters that would be coming later on this evening. I was getting ready to carve a pumpkin to set out near our front door. I printed out the pattern and taped it to the Pumpkin. The image I printed consisted of two very bright orange eyes that were outlined in black. When I was ready to sit down and carve the pumpkin, one of the bright orange eyes I had been looking was now in my field of vision, no matter where I looked. The image that was on the piece of paper that I printred out was now floating around in front of my left eye. I could see it perfectly, it was the left eye from the pumpkin carving pattern, but much, much smaller. This tiny bright orange eye outlined in black floated a round in front of my left eye all afternoon. Finally, I thought that I would try toI lay down and rest for a little bit to see if it would go away.

Hi Julie, I am not sure but it sounds like you were having a persistent afterimage or possibly a vitreal detachment. I would return to your eye doctor and them look at your retina. All the best, The Neuro Visual Center

I am concerned because this has never happened before, at least, not for 10 straight hours in row. Do you know I may have been experiencing

Richard N. Pfisterer received his bachelor’s and master’s degrees in optical engineering from the Institute of Optics at the University of Rochester in 1979 and 1980, respectively. He is a co-founder and a president of Photon Engineering at LLC. Previously, he was the head of optical design at TRW (now Northup-Grumman) and a senior optical engineer at Breault Research Organization. He is credited with more than 20 articles and conference presentations in the areas of optical design, stray light analysis, and phenomenology. He is a member of OSA and SPIE.

In this section, we have systematically described in detail the paraxial behavior of five different classical grating types: the sinusoidal amplitude grating, the square-wave amplitude grating, the sinusoidal phase grating, the square-wave phase grating, and the blazed reflection grating (sawtooth profile). The result of the paraxial diffraction efficiency analyses of these five grating types is summarized in Table 5.

This sounds like a corneal problem and not binocular double vision. We are so sorry that you are suffering. All the best, The Neuro Visual Center

This paraxial requirement obviously places strong limitations on the applicability of the results of this section concerning the grating period-to-wavelength ratio d/λ. The paraxial expressions in Eq. (15) are accurate to within 5% if the angle does not exceed about 18 deg. Although scalar diffraction theory is known to predict diffraction grating performance for TE-polarized light, not transverse magnetic (TM) or unpolarized light,22 at these paraxial angles there will be very little difference between the diffraction efficiency for the two orthogonal polarizations.

Its primary function is to further magnify the image produced by the objective lens. Ocular lenses are often interchangeable, allowing users to customize their ...

As with the sinusoidal amplitude grating, the total amount of energy transmitted through a square-wave amplitude grating does not vary as the angle of the incident beam is increased. Thus as each pair of diffracted orders goes evanescent, the energy that was contained by them is redistributed into the remaining propagating orders (again the Rayleigh grating anomaly phenomenon) according to the nonparaxial scalar diffraction theory summarized earlier in this section. The renormalization constant K is equal to Eq. (77)K=∑m=−∞∞ηm∑prop.ordersηm=0.5∑prop.ordersηm,where ηm is the diffraction efficiency of the m’th diffracted order.

Comparison of diffracted efficiency of a sinusoidal phase grating as predicted by Eq. (34) and the common approximation for shallow (smooth) gratings expressed in Eq. (37).

The ghost images are more blur than the actual object. And also whenever during the process i open my eyes from closed (like blinking), I sometimes see the ghost images rising like layers by layers D:

My eyes also have trouble reading or focusing in on text or on a lot of visual stimuli, such as many cereals in the cereal aisle at the grocery store or words on a page of many words or one store aisle in a sea of store aisles.

Let us first look at a special case of a rectangular phase grating where the peak-to-peak phase step is equal to π (this should result in zero efficiency for the zero diffracted order) and a duty cycle of b/d=0.5 as illustrated in Fig. 20. From Euler’s equation Eq. (38)exp(iϕ)=cos(ϕ)+i sin(ϕ),we readily see that exp(iϕ) is equal to −1 when ϕ=π and +1 when ϕ=0 as illustrated in Fig. 21. The complex amplitude transmittance of this rectangular phase grating bounded by a square aperture of width w thus can be written as Eq. (39)tA(x1,y1)={[2 rect(x1d/2)(1)**1dcomb(x1d)δ(y1)]−1}⁢rect(x1w,y1w).

The diffracted orders now propagate along the surface of a cone and will strike the observation hemisphere in a cross section that is not a great circle, but instead a latitude slice as illustrated for a reflection grating in Fig. 6. Note that the direction cosines are obtained by merely projecting the respective points on the hemisphere down onto the plane of the aperture and normalizing to a unit radius. Even for large angles of incidence and large diffracted angles, the various diffracted orders are equally spaced and lie on a straight line only in the direction cosine space.

Evaluating at ξ=x2/λz and substituting into Eq. (56) yields the following expression for the diffraction pattern projected onto a screen at a distance z from the grating: Eq. (60)E2(x2)∝sinc2[x2−(nBλBλ+θ0dλ)λzdλzd]1λzdcomb[x2−(θ0dλ)λzdλzd].

Using Eq. (69) to calculate at what value of λ/d the +2 diffracted order goes evanescent, we obtain Eq. (75)sin(−π/2)=−1=−(2−12)λdorλ/d=2/3.

One can readily calculate that a square-wave amplitude grating with transparent and opaque strips of equal width (b=d/2) results in only 10% of the incident optical power being diffracted into the +1 order. This is a little better than we achieved with the sinusoidal amplitude grating, but still not adequate for many applications. Figure 17 illustrates the diffraction efficiency of the first several orders as a function of the parameter b/d.

If you’ve recently started experiencing symptoms of double vision, there’s a quick way to determine which type you have. If you’re experiencing double vision, try closing one eye. If you see images as double out of one eye with the other eye closed, you have monocular double vision. If you have binocular double vision, the double vision will go away when you close either eye.

Dr. Cheryl specializes in the optometric treatment of dizziness and headaches utilizing the Feinberg method. She is the only practitioner in the state of New York trained by Dr. Debby Feinberg of Vision Specialists of Michigan. Dr. Cheryl is a graduate of Cornell University and SUNY College of Optometry.

Since many individual measurements are required to completely characterize the efficiency behavior of a given grating, it has become commonplace to make diffraction efficiency measurements with a given diffracted order in the Littrow condition.19 For transmission gratings, a given diffracted order satisfies the Littrow condition if θm=−θi. For reflection gratings, the Littrow condition is satisfied if the given diffracted order is antiparallel to the incident beam, i.e., θm=θi. This allows the experimenter to leave the detector and the source in a fixed location and merely rotate the grating between measurements.

For polychromatic light, we can represent the resulting diffracted orders with a summation over the discrete diffracted orders of an integral over some spectral band Δλ=λ2−λ1: Eq. (62)E2(x2)∝∑m=−∞∞∫λ1λ2sinc2[m−(nBλB/λ+θ0d/λ)λz/dλz/d]⁢δ[x2−(θ0d/λ)λz/d].

(a) Two-dimensional image of square-wave amplitude grating (b) and profile of amplitude transmittance in the x direction.

Two generalizations to the behavior of gratings must now be discussed. First, if the individual slits making up the grating have significant width (in order to transmit more light), the Fraunhofer diffraction pattern of an individual slit will form an envelope function modulating the strength of the discrete diffracted orders.12–15 For the case illustrated in Fig. 4, we have chosen the width of the slits to be one-third of the slit separation. You will note that every third diffracted order is absent. This is caused by the envelope function going to zero at those locations.

The more general phenomenon of “conical” diffraction that occurs with large obliquely incident angles will be discussed in Sec. 3 and the parametric behavior will be shown to be particularly simple and intuitive when formulated and displayed in terms of the direction cosines of the incident and diffracted angles. In Sec. 4, we will use the remarkably intuitive direction cosine diagram to portray the conical grating behavior exhibited in the presence of large obliquely incident beams and arbitrary orientation of the grating. Section 5 examines the paraxial diffraction efficiency behavior of several elementary grating types. Section 6 will review the underlying concepts of nonparaxial scalar diffraction theory and apply them to the sinusoidal and square-wave amplitude gratings when the +1 diffracted order is maintained in the “Littrow condition.” This nonparaxial behavior includes the well-known Rayleigh (Wood’s) anomaly effects that are usually thought to only be predicted by rigorous (vector) electromagnetic theory.16

The more general phenomenon of conical diffraction that occurs with large obliquely incident angles is rarely discussed in elementary optics or physics text books. However, the formulation of a nonparaxial scalar diffraction theory20–23 provides a simple and intuitive means of gaining additional insight into this nonparaxial diffraction grating behavior.

Again setting Δm equal to unity yields an expression for the linear spacing of adjacent diffracted orders projected upon a plane observation screen as a function of diffracted angle Eq. (11)Δxm=λdL(1cos θm+sin2 θmcos3 θm).

When operating in the Littrow condition, the diffracted orders are distributed symmetrically about the grating normal as shown in Fig. 34. For small λ/d, there are many diffracted orders, but they all have small diffraction angles. As λ/d is increased, both the angle of incidence and the diffraction angles increase, and the higher diffracted orders start going evanescent.

In addition, I failed the field of vision test three times since February. I cannot complete the test because my vision immediately goes black from the outside in, to a pinpoint, to whatever object is in the very middle of the screen.

The paraxial grating behavior for coarse gratings (d≫λ) was derived and displayed graphically for five elementary grating types: the sinusoidal amplitude grating, the square-wave amplitude grating, the sinusoidal phase grating, the square-wave phase grating, and the classical blazed grating (sawtooth groove profile). Paraxial diffraction efficiencies for various diffracted orders were calculated, tabulated, and compared for these five elementary grating types.

We acknowledge that this list of grating types is nonexhaustive and nonexclusive but none-the-less is useful for comparing and contrasting grating performance for different gratings types, characteristics, and manufacturing techniques.

Hence, the +1 and −1 diffracted orders produced by a sinusoidal amplitude grating propagate at angles: Eq. (71)θ1=−sin−1(12λd)andθ−1=sin−1(32λd).

Since the diffracted orders are distributed symmetrically about the grating normal, a positive and a negative order always go evanescent simultaneously. Figure 34 illustrates the situation for a transmission grating with λ/d=0.25 and the +1 diffracted order satisfying the Littrow condition (θ1=−θi).