include diffraction gratings, binary optics, and holographic optical elements. these optical...

$: include diffraction gratings, binary optics, and holographic optical elements. These optical elements diffract light by producing periodic changes in the amplitude, phase, or both$
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASAP

Technical Guide

DIFFRACTION GRATINGS AND DOES

Breaul t Research Organizat ion, Inc.

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This Technical Guide is for use with ASAP®.

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brotg0915_diffraction (April 9, 2007)

ASAP Technical Guide 3

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Contents

Fundamentals of Diffractive Optical Elements 9Huygens-Fresnel principle 9Diffraction gratings 11Blazed gratings 14Phase function 16Binary optics 23

Transition Points 27Etch Depths 28

Holographic optical elements 28Multiple exposure holograms 32Volume holograms 32

Simulating Diffractive Optical Elements in ASAP 34MULTIPLE command 35INTERFACE command for DOEs 40DOE examples 46

Linear Phase Grating 46Sinusoidal Phase Grating 53Circular diffraction grating 59DOE Lens 69

References 78


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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .DIFFRACTION GRATINGS AND DOES

In this technical guide, we discuss how to model diffractive optical elements in the Advanced Systems Analysis Program (ASAP®) from Breault Research Organization (BRO). These elements diffract light by producing periodic changes in the phase of the incident wave.

The ASAP Primer discusses how the laws of reflection and refraction are used to transform a ray at a specular interface into reflected and refracted rays. In this technical guide, we discuss a theory for diffracting light from phase gratings according to the grating equation law.

Diffractive optical elements (DOEs) are a general class of optical elements that include diffraction gratings, binary optics, and holographic optical elements. These optical elements diffract light by producing periodic changes in the amplitude, phase, or both the amplitude and phase of an incident wave.

Before you learn how to perform DOE simulations, we must first introduce certain diffractive optical element definitions, nomenclature, and concepts. For some readers, this will be a review and for others it may be the first time that you have seen these definitions and concepts. In either case, you must first have a basic understanding of the physical behavior of diffractive optical elements to associate those definitions and concepts with ASAP procedures and commands for performing DOE simulations.

The remainder of this technical guide is divided into two primary sections:

• “Fundamentals of Diffractive Optical Elements” on page 9 introduces definitions and concepts. You may choose to skip this section if you already have a background in the subject. Introductory topics include the grating equation, transmission and reflection phase diffraction gratings, binary optics, and holographic optical elements.

• “Simulating Diffractive Optical Elements in ASAP” on page 34 presents DOE definitions and concepts with the equivalent ASAP procedures and commands for simulating DOE systems. We discuss how to set up a variety of phase gratings, binary optics, and holograms.

7

D I F F R A C T I O N G R A T I N G S A N D D O E S

Many of the scripts included in this document are available as INR files on the Quick Start toolbar in ASAP: Example Files> Scripts by Keyword> Diffraction.

8 ASAP Technical Guide

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. .D I F F R A C T I O N G R A T I N G S A N D D O E S

Fundamentals of Diffractive Optical Elements

F U N D A M E N T A L S O F D I F F R A C T I V E O P T I C A L

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Two common approaches exist for calculating the diffraction patterns from DOEs: vector calculation and scalar calculation.

Using vector diffraction theory, we can calculate the directions of propagation of the various reflected and transmitted orders of a beam after interacting with the DOE as well as the fractional power contained in each order. The fractional power diffracted into each order is called the diffraction efficiency, and is determined by the diffractive surface profile. This complicated calculation involves not only the DOE structural data, but also the physical and electromagnetic properties of the DOE. ASAP does not perform this type of calculation.

Using scalar diffraction theory, we can calculate the idealized directions of propagation of the various reflected and transmitted orders of a beam after interacting with a diffractive element. The scalar diffraction theory technique is valid if the grating period is larger than the wavelength of the diffracting light. Swanson notes that the scalar approximation can be used in Maxwell’s equations if the grating period is approximately five times the wavelength of the diffracting light. The diffraction orders are selected by adjusting the direction cosines of the exiting beams according to the grating equation. The diffraction efficiencies cannot be generally or easily calculated with this method since this is a scalar technique. Most optical design and analysis software programs, including ASAP, use this technique since it is adaptable to ray tracing.

The grating equation can also be derived from the Huygens-Fresnel principle. This is, perhaps, a more physically intuitive derivation than one from scalar diffraction theory, yet mathematically less rigorous. Therefore, we use Huygens-Fresnel principle to derive the grating equation.

Huygens-Fresnel pr incipleChristian Huygens proposed a theory of diffraction based upon primary and secondary wavefronts. A wavefront is a mathematical surface of constant phase. In a contemporary, geometrical sense, wavefronts are mathematical surfaces over which the optical path lengths from rays of a point source have the same length. Huygens proposed that every point on a primary wavefront can be considered a source of secondary wavelets (or sources), each with the same frequency and velocity, such that at a later time, the primary wavefront is the envelope of the




wavefronts from the secondary wavelets (or sources). Huygens’ principle is illustrated in Figure 1.

Figure 1 Huygens’ wavelet construction

Augustin Jean Fresnel added to Huygens’ intuitive ideas by including Thomas Young’s principle of interference. In doing so, Fresnel assumed under certain conditions that the amplitudes and phases of Huygens’ secondary wavefronts could interfere which each other.

The amplitude and phase of a source are a manifest part of its oscillatory behavior. The amplitude is a scalar number or function describing the maximum extent of the electric field vibration. The squared modulus of the amplitude is a measure of the energy density or power in the electric field. The phase is the optical path length of a ray, or the fraction of an oscillatory cycle measured from a specific reference or fixed origin, such as a point source.

Secondary Wavelets

Envelope of Secondary Wavelets

Primary Wavefront

Secondary Wavelets

Envelope of Secondary Wavelets

Primary Wavefront


. . .



NOTE For more information regarding amplitudes and phases, see the introductory material in the ASAP technical guide, Wave Optics.

The Huygens-Fresnel principle is a superposition principle using spherical wavefronts as the basis functions in the superposition. The principle of wavefront superposition states that the resultant perturbations on a wavefront are the algebraic summations of the individual waves comprising the wavefront. Superposition is also used to algebraically sum amplitudes.

The Gaussian beam superposition algorithm in ASAP is similar to Huygens-Fresnel. However, ASAP uses quadratic phase fronts and Gaussian beam amplitudes as the basis functions to superpose individual Gaussian beams to simulate arbitrary optical fields.

Gustav Kirchhoff developed a mathematical theory of the Huygens-Fresnel principle, which eventually became the Fresnel-Kirchhoff scalar wave diffraction theory. Arnold Johannes Wilhelm Sommerfeld later developed the Rayleigh-Sommerfeld scalar diffraction theory, which did away with certain nonphysical assumptions of the Fresnel-Kirchhoff scalar wave diffraction theory.

Joseph W. Goodman’s book, An Introduction to Fourier Optics, is an excellent introductory resource for learning about the Huygens-Fresnel principal and scalar wave diffraction theory.

Several papers describing the Gaussian beam superposition algorithm in ASAP are available in the Knowledge Base at http://www.breault.com/k-base.php.

Dif f ract ion grat ingsDiffraction gratings are regular arrays of apertures or phase structures that, by nature of their periodic structures, produce periodic perturbations in the amplitude, phase, or both amplitude and phase of a wavefront incident on the structure. If apertures are used, the diffraction grating is an amplitude grating and, specifically, an amplitude transmission grating if the light passes through and is not reflected by the grating. Young’s double slit is an example of a simple amplitude transmission grating.

If some type of phase altering material is used, the diffraction grating is a phase grating. If light transmits through the diffraction grating, the grating is called a phase transmission grating. If light reflects off the grating, it is called a phase reflection grating.


http://www.breault.com/k-base.php



Generally, phase gratings are more efficient at diffracting monochromatic light than amplitude gratings, and surface relief phase gratings are more efficient at diffracting broadband light than volume phase gratings. Theoretically, a phase grating can diffract 100% of incident monochromatic light into a single diffraction order. A surface relief phase grating has higher diffraction efficiencies than a volume phase grating operating over the same bandwidth. An amplitude grating cannot diffract 100% of the incident monochromatic light into a single diffraction order, because some of the incident light is reflected, absorbed, and so on, by the grating mask.

The simplest type of surface relief—phase transmission diffraction grating—is a linear phase transmission grating embossed on a flat, slab substrate. We can gain an intuitive understanding of more complicated diffraction gratings from examining the linear grating. A linear grating is a series of regularly arrayed lines or scratches embossed in a transmissive material known as the substrate. Phase transmission gratings are usually replicated from a master on a plastic substrate. The master is quite often a series of regularly arrayed lines or scratches carved or etched onto a glass substrate.

As a simple physical model, the regularly arrayed lines can be considered scratches, troughs, or humps in the substrate. In the case of scratches, each line acts like a linear scattering center. In the case of troughs or humps, each line acts like a very short radius of curvature, highly divergent cylindrical lens. In both cases, the effect is to produce equivalent linear or line sources.

Each effective line source is illuminated by the same, or more correctly, different parts of the same wavefront. Therefore, each line source resembles the secondary sources or wavelets of the Huygens-Fresnel principle. See Figure 1

Their superposition at a later time or, equivalently, a later position, will determine the optical field at that point in time or position.

A constructive interference of all the wavelets in a particular direction leads to diffraction orders. Some of the light is not scattered or diverged in transmitting through the diffraction grating. This occurs in between the grating lines. This light, basically unperturbed from the specular direction, is called the zeroth order. In other words, the zeroth order of the diffraction grating is specular direction of the incident light.

We have constructed a simple graphical model of the Huygens-Fresnel principle as applied to linear phase transmission and reflection gratings in Figure 2.


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Figure 2 Transmission and reflection gratings

For the linear phase transmission illustration, note that the trigonometric sine of the incident angle is,

(EQ 1)

Equation 1

Here d is the grating line spacing. Similarly, note that the trigonometric sine of the diffracted order angle is,

(EQ 2)

Equation 2

Phase Transmission Grating

Phase Reflection Grating

Diffraction Orders m

Specular - Zeroth Order



d

d

P1

P1

P4

P4

m

m

m

i

i

P3

P3

P2

P2 i

iPhase Transmission Grating

Phase Reflection Grating





d

d

P1

P1

P4

P4

m

m

m

i

i

P3

P3

P2

P2 i

i




If the phase difference between lines P1 P2 and P3 P4 is an integer number of multiple wavelengths , all the waves from the secondary wavelets are in phase in the direction corresponding to m. Operationally, this is,

(EQ 3)

Equation 3

The above equation is called the grating equation, and m is called the grating or diffraction order. Note that the grating equation is independent of the refractive index of the slab substrate. This can be easily validated with Snell’s law. The grating equation is the same, even in the case of the phase reflection grating.

The grating equation is the same as the equation that describes the maximum orders for Young’s double slit experiment. However, there are some fundamental physical differences in the diffraction patterns for each situation:

• The diffraction or interference pattern from Young’s double-slit experiment is due to an amplitude transmission grating. It is also more diffuse or less sharp than that from the phase diffraction grating. This is primarily due to the larger size of the apertures in Young’s double slit experiment as compared to the relatively small line sources in the diffraction grating. In Young’s experiment, different points across the two apertures, other than the center points, can be in phase with each other and constructively interfere at slightly different locations than the center points causing a broadening of the diffraction pattern in that order.

• The linear phase transmission grating behaves more like a Fabry-Perot etalon or filter.

Blazed grat ingsIn many applications that use linear diffraction gratings, we are not interested in the light in the zeroth order. This light is often wasted. It is useful to transfer light out of the zeroth order and into another order or arbitrary direction other than the specular direction. Many modern diffraction gratings, especially phase reflection gratings are blazed to accomplish this task.

John William Strutt Rayleigh, or Lord Rayleigh suggested a clever way to transfer energy out of the zeroth order into other orders. This involved changing the angle


. . .



of or tilting the surfaces that the specular light sees in between the grating lines. Such a grating is called a blazed grating.

The key to a blazed grating is that the angular direction of the non-zero diffraction orders are a function of the grating spacing d, the wavelength of the incident light , and the angle of incidence i. However, m and i are measured from the grating plane and not the blazed (tilted) surface. If we tilt the specular surfaces in between the grating lines, we change the direction of the specular output, but not the higher diffraction grating orders. Figure 3.

Figure 3 Blazed transmission grating

The diffraction efficiency of a given order for regular and blazed linear phase gratings can be calculated using scalar diffraction theory. Swanson, for example, gives the result for diffraction efficiencies as,

Blazed Grating d

Grating Period

Blaze Angle

Grating has Refractive Index n

Blazed Grating d

Grating Period

Blaze Angle

Grating has Refractive Index n




(EQ 4)

Equation 4

Here, m is the diffraction order, T is the grating period, and :

(EQ 5)

Equation 5

where n is the refractive index of the grating, d is the blaze thickness, and is the wavelength of light. From the grating equation and these equations, we determine that the diffraction angles and efficiencies are highly dependent on wavelength and blaze thickness.

Light from large bandwidth sources will be diffracted into many orders. This is a type of scatter, but it is deterministic scatter since its direction and diffraction or scatter efficiency can be exactly computed with the above equations.

NOTE Deterministic scatter is different from the surface scatter due to random surface variations that are covered in the ASAP technical guide, Scattering.

The concept of the linear grating in this context can be used to develop a simple but useful physical model of random scatter. You can consider a random scattering surface as being composed of many randomly oriented linear gratings, whose combined grating spacing and orientation yield the surface scatter pattern.

Phase funct ionBlazed gratings—and, in fact, other types of optical elements such as apertures, prisms, lenses and mirrors—can be described in terms of a transmittance function. The transmittance function describes how the optical element changes the


. . .



amplitude and phase of a wavefront propagating through the component. We learn later that we must know the phase function of a diffractive optical element to model it in ASAP.

Generally, the transmittance function is defined as,

(EQ 6)

Equation 6

A(x,y) is the amplitude function and (x,y) is the phase function of the element.

The simplest transmittance function is perhaps that of a one-dimensional aperture whose transmittance function is one inside the aperture and zero outside of the aperture. Mathematically,

(EQ 7)

Equation 7

A prism has a transmittance function equal to,

(EQ 8)

Equation 8

Here, we assumed that phase is changed, but the amplitude was unaltered propagating through the prism. k is the wave number whose physical effect is to convert the linear dimension of the phase function into a fraction of an oscillatory cycle measured in radians. ois a constant term of the phase function.

A blazed grating has a transmittance function equal to,

(EQ 9)

Equation 9




Here

is a modulo 2 operation on the phase function of the blazed grating. Mathematically, a modulo operation keeps the remainder after a division. Physically, it produces a kinoform whose incremental depth is 2 radians or, equivalently, one wavelength. A kinoform is a surface-relief profile of the modulated phase. The common, constant-depth Fresnel lens is an example of a type of kinoform whose modulo operation is much larger than a wavelength of light. Its kinoform is a surface relief profile of the actual physical surface or surface sag, and not the phase change (transmittance function) through the element. See Figure 4.

Figure 4 Phase functions of a prism and blazed grating

Phase Function of a Prism

(x)

x

x

(x) Modulo or

Phase Function of a Blazed Grating

Phase Function of a Prism

(x)

x

x

(x) Modulo or

Phase Function of a Blazed Grating


. . .



The transmittance function of the prism looks similar to the blazed grating. Mathematically, the difference is manifest in the modulo operation. Physically, the blazed grating is the diffractive counterpart of the refractive prism. In general, a modulo 2 operation on the phase function of a refractive element yields its diffractive counterpart’s phase function.

We are now in a position to examine a more complicated phase behavior; namely, that of a simple thin lens. A thin lens is one whose thickness is very small compared to its diameter. A ray intersecting one side of a thin lens essentially exits at the same point on the other side of the lens.

A thin lens is a phase transformer in the sense that it delays, in time, the phase of the wavefront in proportion to the thickness of the lens. A positive lens, such as a biconvex lens whose outer edge thickness is less than its center thickness, delays collimated light, for example, more at the center of the lens than at the edge. The edge of the wavefront has propagated further in the same amount of time than the center of the wavefront when it exits the center of the lens, because it is not “slowed” down by the additional refractive lens material. Since the wavefront is continuous and is a surface of constant phase, the wavefront must converge toward a point in space, which in this particular case, is the focal point of the lens.

We can equivalently understand this phenomena by examining a ray of light. Since the wavefront is delayed more at the center of the lens than at its edge, the ray is also delayed by the same amount because it is related to the wavefront and in fact is perpendicular to the wavefront. The magnitude of the optical path length of the ray at the edge of the lens is less than at the center. The optical path length of a ray is the distance that light travels in a vacuum in the same time it travels a given distance in a material. Therefore, the edge ray is bent toward the focal point of the lens.

The geometry for a radially symmetric convex-plano lens is illustrated in Equation 10.Figure 5 The wavefront experiences a total phase delay due to the lens and the remaining region around the lens. Mathematically, this is,




(EQ 10)

Equation 10


. . .



Figure 5 Phase-transforming lens, resulting quadratic phase function, and diffractive

modulo 2 phase function

By invoking the paraxial approximation with the binomial theorem, we can replace the spherical surface with a quadratic approximation. This leads to,

Equation 11

R

d

()

Quadratic (Parabolic) Phase Function from Lens

Modulo Phase of the Quadratic Phase Function

Plano-Convex Lens with a Spherical Surface

R

d

()

Quadratic (Parabolic) Phase Function from Lens

Modulo Phase of the Quadratic Phase Function

Plano-Convex Lens with a Spherical Surface




Our phase function then becomes,

Equation 12

We recognize the last term of the previous equations as a form of the lens maker’s equation for a convex-plano lens with focal length f.

(EQ 13)

Equation 13

Therefore, the phase function becomes,

(EQ 14)

Equation 14

The first term is a constant phase term of little interest, and the second term is a quadratic approximation to a spherical wavefront. It is a quadratic phase function of a focused beam. The lens transforms the planar phase (wavefront) of the incident collimated beam into a converging quadratic phase (wavefront) focusing at the focal point of the lens. If modulo 2 operation is carried out on the phase function, we obtain the diffractive counterpart to the refractive lens.

Arbitrary phase functions and their diffractive counterparts are obtained in similar manners, usually with the aid of lens design software. However, the wavefronts and, subsequently, the phase functions of these elements can be much more complicated than the plane waves and quadratic waves we examined for the linear grating and a diffractive lens.


. . .



Even with more complicated phase functions, the grating equation can still be used to determine, in a local sense, the direction of the orders of the diffracted light. The grating spacing used in the grating equation is the local grating spacing where a small pencil beam of light, or perhaps a ray, intersects the diffractive element. The diffraction efficiencies are still governed by the wavelength and depth relationships developed for the simple linear grating. However, the fundamental problem is that with more complicated phase surfaces, the overall depth and grating spacing can vary radically across the element, making it impractical and usually impossible to properly apply these equations. We recommend resorting to vector diffraction theory to properly account for the canonical diffraction efficiencies.

Binary opt icsThe diffractive phase kinoforms discussed in the previous section are continuous in profile over the 2 phase depth. However, these profiles cannot be manufactured in a continuous profile over such small intervals. Even most Fresnel lenses, whose kinoform depths and intervals are many orders of magnitudes larger than those of diffractive optical elements, typically use linear approximations to the continuous surface profiles. Fortunately, in the last decade the techniques of the integrated circuit (IC) industry were adapted to produce approximations to a diffractive optical element’s kinoform. The manufacturing technique has led to the term and field called binary optics.

The IC manufacturing process used to generate binary optics is similar to chiseling or carving stair-step structures into a substrate that, on a microscopic scale, approximate the continuous profile of the modulo 2 phase function.

This concept is illustrated in Figure 6 The diffraction efficiency for a given order is a function of the number of levels. The more levels, the higher the diffraction efficiency. Two levels produce diffraction efficiencies on the order of 40%. Eight levels and above produce diffraction efficiencies of 95% and higher.




Figure 6 Four-level binary optic representation of some phase functions

A series of lithographic masks are made whose total number is determined by the number of etch levels and, therefore, the desired diffraction efficiency. The steps are listed below.

The masks are like zero order amplitude gratings, which transmit and reflect a certain percentage and pattern of the light incident on the substrate.

1 The masks are placed one at a time over a substrate coated with a photoresist.

2 The transmitting portion of the mask allows light to expose the photoresist. If the

photoresist is exposed to light, it is developed and washed away from the

substrate.

Phase function of a blazed grating

Multi-level phase structure or binary optic equivalent

A more complicated phase function


Phase function of a blazed grating


A more complicated phase function



. . .



3 The substrate areas without photoresist are then etched to a special depth

leaving a binary step. The unexposed photoresist remaining on the substrate is

impervious to the etching process.

4 The rest of the photoresist is removed and the entire substrate is again coated

with photoresist.

5 The next mask is positioned above the coated substrate and the combination, in

turn, is illuminated.

6 The exposed photoresist is removed and that portion of the substrate is etched

to a special depth.

7 This process is repeated until you have etched the desired number of levels into

the substrate. See Figure 7.




Figure 7 Binary optic fabrication process

Substrate recoated with photoresist

Amplitude mask for second level

Remove exposed photoresist

Etch bare substrate to required depth

Remove remaining photoresist

Resulting Binary Optic (only 4 levels)

Substrate recoated with photoresist

Amplitude mask for second level

Remove exposed photoresist

Etch bare substrate to required depth

Remove remaining photoresist

Resulting Binary Optic (only 4 levels)


. . .



The width of the masks for each step is a function of the type of phase surface. For the blazed grating, the width of the mask on the first etch is half of the modulo 2 phase interval. The width in subsequent mask intervals is half that of the previous mask. For a circularly symmetric case, like the thin lens, the width of the mask on the first etch, step, or transition begins at the radial value corresponding to a modulo phase depth. Subsequent radial transition points occur at multiples of the modulo phase depth. On the second mask, the transitions occur at integer multiple modulo /2 phase depths, and so on.

The entire binary optics process as well as required transition points and etch depths are discussed in great detail in Swanson’s report from the Massachusetts Institute of Technology (MIT) titled “Binary Optics Technology: The Theory and Design of Multi-level Diffractive Optical Elements”. The transition points and etch depths are shown in Equation 15 and Equation 15.

T R A N S I T I O N P O I N T S

(EQ 15)

Equation 15




E T C H D E P T H S

Equation 16

Even though binary optics are manufactured in discrete steps, their diffractive behavior in ASAP is still simulated by defining the grating lines and using the grating equation. The actual stair-step phase structures are not modeled. The resulting diffraction efficiencies are adjusted according to the number of etch levels or steps.

Holographic opt ical e lementsHolographic optical elements are actually the result of a diffractive application of holography. Holography is a process where an interference pattern of a three-dimensional object is produced that contains not only amplitude information about the object but also the phase information. The two-dimensional pictures that we view, including this page and these words, are irradiance maps of objects. An irradiance map contains information only about the amplitude of the object and not its phase. In this sense, we get only half the picture, as the phase information about the object is missing. Perhaps this lead Dennis Gabor to name the phenomena he discovered in 1947 “holography”, and the pictures created with this process: “holograms”, after the Greek word “holos” meaning whole.

The interference pattern of a hologram is produced by interfering a reference wavefront with a split portion of itself reflected from an object. The hologram is typically recorded on a photographic emulsion. The recording is “played back” by re-illuminating the hologram with the reference beam. The hologram can be a transmission or reflection hologram.

We can generate a simple hologram by interfering two plane waves together. Imagine a linearly polarized plane wave incident on a photographic plate at an angle 1 with respect to the plate normal, and another plane wave—a split portion of the first plane wave—incident on the same plate but at an angle 2. The electric field of the two waves is,


. . .



(EQ 17)

Equation 17

If the phase difference 1-2 is a constant, the two waves are mutually coherent. See Figure 8.

Figure 8 The k vectors and fringes of two interfering plane waves

The k Vectors of 2 Interfering Plane Waves

Y

Y

Z

Z

k 1

k 2 12

Bright Fringes

The k Vectors of 2 Interfering Plane Waves

Y

Y

Z

Z

k 1

k 2 12

Bright Fringes




We can write the phase relationships in Equation 18at the recording plane, based upon the propagation vectors of the two interfering plane waves.

Equation 18

The last equation is of a line. This implies that the phase function of the two interfering beams is linear. Recall that the phase function of a prism was linear and a modulo 2 operation on it resulted in its diffractive counterpart, a blazed grating.

A bright fringe occurs when m. In other words, the relative phase between the two waves varies by 2 in between adjacent bright fringes. Furthermore, the fringe spacing in the special case of z and 1-2 = 0 is,

(EQ 19)

Equation 19

The irradiance pattern from this superposition is,


. . .



(EQ 20)

Equation 20

The dot product of the polarization component vectors results in an additional cosine term whose argument is the angle between the linear polarization components.

The important results are that the irradiance pattern varies cosinusoidally, but the fringes are straight-lined and equally spaced. The fringes are also non-localized in that a fringe pattern exists at every point where the two plane waves overlap. When this hologram is illuminated with the reference beam, we see part of the original reference beam passing through the hologram, just like the zeroth order from a diffraction grating and two additional first-order beams.

The interference pattern, or hologram, when re-illuminated with the reference beam, behaves just like a sinusoidal grating. The phase information of the hologram is manifest in the local and global fringe spacing, and determines the diffraction angles of the orders. The amplitude information of the hologram is manifest in the visibility or contrast of the fringes, and determines the diffraction efficiency of the order.

NOTE In this technical guide, we are interested only in the phase portion of the hologram, as this is needed to model its diffractive behavior in ASAP.

Our results above imply that holograms can be simulated with the same grating equation that we used to model the behavior of diffraction gratings and binary optics. However, in the case of two plane waves, we simulate only the zeroth and first orders, because the hologram behaves like a sinusoidal grating and does not contain phase and amplitude information for higher orders. We again need the phase information of the hologram, which is manifest in the fringe spacing.

Our fringe/phase theory can be extended to more complicated fringe patterns. If a plane wave and a spherical wave interfere, circular fringes result. The circular fringes are spaced just like those found in a Fresnel phase plate, which behaves like a diffractive lens. If two spherical waves interfere, hyperbolic fringes result.

In general, the more complicated the return wavefront, the more complicated the interference pattern. In fact, some holograms may not look like a conventional fringe pattern at all, and resemble a speckle pattern (a coherent scatter pattern).




Lens design codes commonly use two point sources to generate holographic optical elements, which in turn are used as diffractive lenses or corrective elements.

The first-order beams of a hologram are also referred to as side bands. In complicated holographic pictures, one of the side bands contains the original amplitude distribution, but a negative phase term. This results in the famous inside-out image. The other side band contains the original amplitude distribution and the normal phase term. Its image accurately reproduces the object as it appeared during recording.

Holographic optical elements are found in a large number of different types of optical systems, such as scanners and heads-up displays (HUDs). In one sense, interference coatings might be considered as holographic optical elements operating only in the zeroth order.

Mult iple exposure hologramsMultiple exposure holograms are a single hologram whose recording media contain more than one exposed holographic fringe pattern. The most common type of multiple exposure hologram is the double-exposure hologram.

One type of double-exposure hologram is a composite hologram made of an unperturbed object and its resulting fringe pattern, and a perturbed object and its resulting fringe pattern. The composite fringe pattern actually represents the difference in perturbation between the unperturbed object and perturbed object. The perturbation might be, for example, the displacement (vibration) or distortion of an object from its normal state.

A multiple exposure hologram exposed over a long period of time is called a time-averaged hologram. Its interference pattern is actually a superposition of many interference patterns. The resultant interference pattern is really a standing wave whose contours represent areas of constant perturbation. In most instances, the perturbation is a vibration.

Multiple exposure holograms are not volume holograms.

Volume hologramsThe interference pattern produced by a hologram is non-localized. This means that it exists everywhere in space. Our previous example of two interfering plane waves is an example of non-localized fringes. In other words, the interference pattern produced via holography is three-dimensional. In this respect, it is a three-


. . .



dimensional diffraction grating. It can also be regarded as a three-dimensional interference filter.

A volume hologram has a thickness that is much greater than the thickness of its thin holographic counterpart. Even then, it is still very thin, perhaps on the order of tens of microns. The larger thickness is needed to record the three-dimensional fringe pattern. The thicker the volume hologram, the more light is diffracted into the first order. In fact, efficiencies of 100% are achievable.

Volume holograms de-couple wavelength and the diffraction angle. In other words, only a specific wavelength is diffracted at a specific angle in the hologram. Because of this property, you can store many holograms at one time. Just change the incident wavelength. Similarly, you can change the angle of incidence to achieve the same effect.

Volume holograms are highly susceptible to shrinkage. Therefore, if shrinkage is not accounted for, the replay wavelength must be shorter than the recording wavelength. Only at special wavelengths is light diffracted at a particular angle. In this sense, the volume hologram acts like a spectral filter.

ASAP can simulate volume holograms, but it does so using the grating equation. However, we must set up appropriate diffraction efficiency models with the COATING MODELS option in ASAP, and specifically use the interface diffract option with this coating model.

ASAP simulates volume holograms as thin holograms with appropriately adjusted diffraction efficiencies. Most other lens design codes also use this technique. However, they sometimes include approximated diffraction efficiency calculations from, for example, Kogelnik’s theory (Bell Systems’ Technical Journal, 48, 2909, 1969). This simulation technique assumes that the primary difference between volume and surface holograms is the behavior of the diffraction efficiencies.



Simulating Diffractive Optical Elements in ASAP

S I M U L A T I N G D I F F R A C T I V E O P T I C A L

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .E L E M E N T S I N A S A P

ASAP simulates diffractive optical elements (DOEs) by using the phase function of the DOE to generate a series of grating lines. Therefore, as a first step in simulating DOEs in ASAP, you must know the mathematical form of the phase function of the DOE or the grating line function. This information is easily obtained by knowing the line spacing for simple DOEs, such as linear and circular phase diffraction gratings. More complicated phase functions are usually obtained from lens design codes in the form of polynomial equations.

ASAP automatically replicates grating lines from information it obtains from the phase function. In effect, ASAP automatically performs the modulo 2 phase operation on the phase function. From the preceding section, we know that a grating line occurs whenever the phase function goes through 2 radians.

The resultant replication is then associated with the base entity of an object. After intersecting the base object, ASAP computes the local grating spacing, and applies the grating equation to compute the direction of propagation for defined orders. Since ASAP can split rays, it can also propagate several diffracted orders, including transmitted and reflected, simultaneously after the intersection.

The phase functions in ASAP can be described by any of the ASAP SURFACES. However, the most common way is to describe it with the GENERAL or USERFUNC surface commands. ASAP surfaces must be modified to convert the phase function into a modulo 2 phase function and, ultimately, the grating line representation. This is done with the surface modifier, MULTIPLE (see the section, “MULTIPLE command”). MULTIPLE creates multiple sheets of a phase function to simulate the grating lines.

The replicated grating line spacing, originally obtained from the phase function, is then assigned to an object’s interface with the INTERFACE command.

The phase surfaces of an optical field exiting a DOE may exhibit 2 step discontinuities. Although the wavefront shows the phase steps, SPREAD and FIELD removes the discontinuities in the process of synthesizing the optical field.


. . .



MULTIPLE commandThe MULTIPLE command is a SURFACE modifier. Therefore, it must come after an appropriate surface definition. The MULTIPLE command converts a designated surface into multiple parallel surfaces, which may be used to create repetitive objects. It creates multiple surfaces, but in a fundamentally and mathematically different manner than the ASAP surface modifier, ARRAY. We will also use the MULTIPLE surface modifier to create arbitrary grating lines of a diffractive optical element. MULTIPLE is more often used for modeling diffractive optical elements than creating for multiple parallel surfaces that are used purely for optical system geometries. The syntax is,

where

n is the number of sheets to be generated,

f' is an additive constant to the original function,

d is the distance between original and first sheets,

x y z is an arbitrary point on the original surface, and

EXPONENT p is the exponent to which sheet number is raised.

The surface is converted into multiple parallel surfaces in the following mathematical manner. If the original surface is given by f(x,y,z), MULTIPLE replaces it with,




If MULTIPLE is used to define a diffraction grating, the value of n is irrelevant, and can even be zero. ASAP automatically knows how to use the multiple surface in the case of a diffractive optical element because of its special reference on the INTERFACE command where j specifies the diffractive order numbers. Depending upon the nature of the surface, the value of the exponent p, and the additive constant f' to the original surface, the sheets may not be equally spaced. The exponent p is defaulted to 1, but can be used, for example, to get evenly spaced cylinders or spheres (p=2). The zeroth sheet is just the original surface.

What is the function f(x,y,z) when used with the first syntax of MULTIPLE to simulate a diffractive optical element? It is an equation describing the grating line spacing. You can derive this equation from the phase function, or you can compute it directly. To derive f(x,y,z) from the phase function, we must understand that it is a normalized form of the phase function that we described in the section, “Fundamentals of Diffractive Optical Elements” on page 9.

Recall that the phase function, in general, describes how the phase of an incident wave is altered by an optical element. A modulo 2 phase function is the phase function of a diffractive optical element.

NOTE The function f(x,y,z) is the function obtained when the phase function of a given order m is divided by 2m uniquely specifies the grating line equation.

The grating line spacing is a physical spacing, which does not change for other orders. Other orders are diffracted according to this spacing. In other words, a physical grating line is obtained whenever a specific phase goes though 2. In general, the function f(x,y,z) on the MULTIPLE command, as well as its other terms, can be obtained from an arbitrary phase function by dividing the phase function by 2m or where m is the order number corresponding to that phase function.

We can mathematically demonstrate this relationship by re-examining Figure 2 “Transmission and reflection gratings” on page 13. The local phase change within a specific period in one dimension is,


. . .



(EQ 21)

Equation 21

But the difference in sine terms is equal to

(EQ 22)

Equation 22

Note here that d(y) indicates that the local spatial period may, in general, change as a function of position, and n(y) is a spatial frequency denoting the number of local grating lines per unit length.




We can equate the above equations, rearrange, and integrate to yield,

(EQ 23)

Equation 23

Our results can be extended to rotationally symmetric systems or to three dimensions without loss of generality. If f(y) and f’ are determined in this manner, f’ is commonly set equal to one, and the exponent term of the MULITIPLE command is not needed normally.

If you choose to, you can derive the grating line equation directly. In this case, you must derive f(x,y,z) and determine f’ and the exponent p of the j term.

Alternatively, ASAP can calculate f’ such that the distance from a point (x,y,z) on the original surface to the first sheet is d. It does this by evaluating the expression,

(EQ 24)

Equation 24

This second expression is most useful for diffractive optical elements, such as linear gratings that have equally spaced grating lines. You are not specifying the phase function, but rather the direction in which parallel replications of the original surface are generated. At each distance d along the original surface’s normal,


. . .



computed from the gradient of f(x,y,z), you get a replicated version of the original surface. In the case of a plane, resulting planes are stacked on top of each other but spaced at a distance d apart. In the case of cylinders, resulting concentric cylinders are spaced a distance d apart. This special spacing is the grating line spacing.

Note the similarity of the above equation to the special case of the MULTIPLE command, when the grating lines are equally spaced.

(EQ 25)

Equation 25

Before we examine several DOE examples, we must learn about the necessary options of the INTERFACE command for assigning a diffractive optical interface to an object.




INTERFACE command for DOEsThe INTERFACE command assigns specular reflection and transmission properties to the interface of an object. We also use the INTERFACE command to assign diffractive optical properties to it.

The INTERFACE command has several different syntaxes. We are most interested in those that allow us to assign diffractive properties to an interface, which are listed below.

Here,

DIFFRACT is a flag to assign a diffractive interface to an object,

i is MULTIPLE surface number representing the grating line spacing,

j j' ... are diffraction order number(s),

e e' ... are relative efficiencies of the corresponding orders, and

coat coat' are the names of a given coating property.

The first part of the INTERFACE command, preceding any of the DIFFRACT options, is exactly the same as you have seen before. You can reference a coating from the coating database and assign it to the object, as well as refractive indices on either side of the geometry of the object.

If the object is an optical boundary through which rays are to be traced, the optical properties of the interface must be specified using the INTERFACE command after the definition command for that object. If an INTERFACE command does not follow an edge or surface object definition, the surface is assumed to be perfectly absorbing, and all rays reaching the surface are trapped there. If your interface has non-zero reflection and transmission coefficients and in addition is a diffractive interface, you will see reflected and transmitted diffraction orders.

When you use the first form of the INTERFACE DIFFRACT syntax, grating lines are created by the intersection of the object surface with the different sheets of a MULTIPLE surface i. For example, a ruled linear phase grating is created if i is a plane, a zone phase plate is created if the surface is a cylinder, etc. If i is positive, the multiple sheet spacing is taken to be the grating spacing in system units. If i is


. . .



negative, the spacing is assumed to be in the same units as the last WAVELENGTH specification. The number of sheets entered on the MULTIPLE surface command has no bearing on this application. Specification of the sign of i must be consistent with the units in ASAP and the definition of f(x,y,z), which is calculated from your phase function.

ASAP generates diffracted rays/beams for the diffraction order numbers given by the j's with relative efficiencies given by the corresponding positive e's. ASAP does not compute the diffraction efficiencies as a function of order. You must enter the diffraction efficiencies as a function of order. ASAP uses the scalar grating equation and does not have the vector capability to compute diffraction efficiencies. If a diffraction efficiency, e, is entered as a negative number or as a name “coat”, it is a COATING PROPERTY that possibly contains polychromatic complex amplitudes.

The simplest coating model is the COATING PROPERTIES. This model allows you to specify a table of reflection and transmission coefficients or diffraction efficiencies as a function of wavelength. ASAP linearly interpolates between these values to obtain the values at intermediate wavelengths. In our current case, the reflection and transmission coefficients are diffraction efficiencies of specific orders.

ASAP cannot compute diffraction efficiencies on the fly. Furthermore, the diffraction efficiency of a given order can change as a function of incident angle. However, with the second form of INTERFACE DIFFRACT, a named COATING MODEL can be used to specify the angular variation of the diffraction order efficiencies; that is,

COATING MODELS allows you to specify the angular, dispersion, and polarization properties for S and P reflection and transmission coefficients or diffraction efficiencies at an interface. In our current case, we use it to specify the diffraction efficiency envelope or discrete diffraction efficiencies. It uses models from the ASAP scatter model set. COATING MODELS has two syntax forms. Its syntax is,




Here,

k is the starting coating number,

r r' r" ... are the real energy (or complex amplitude) reflectance efficiencies, and

t t' t" ... are the real energy (or complex amplitude) transmittance efficiencies.

Starting with coating number k, coatings are entered with real energy (or complex amplitude) reflectance r and transmittance t diffraction efficiencies for the zeroth order. The default value for k is one more than the largest coating number defined, and is set to one at the start of program execution.

In the first syntax, separate angular properties of the coating are specified by using previously defined scatter MODELS where

i is the model for reflected S polarization,

j is the model for reflected P polarization,

m is the model for transmitted S polarization, and

n is the model for transmitted P polarization.

We can specify different reflection and transmission diffraction efficiencies for the zeroth order for different wavelengths, as specified on the last WAVELENGTH command. ASAP uses the normalized scatter model data and the reflection and transmission diffraction efficiencies for the zeroth order to account for the angularly dependent nature of the diffraction grating. If you are not using a functional scatter model such as USERBSDF, but rather data from BSDFDATA, ASAP linearly interpolates in logarithmic amplitude space to determine reflection and transmission diffraction efficiencies at other orders than those specified in the data set. However, with this syntax the same angularly dependent polarization models are used for all sources at different wavelengths. Therefore, this form of COATING MODELS should be used for diffraction gratings that are non-dispersive.

Alternatively, in the second syntax, groups of six numbers can be entered to account for grating dispersion and volume holograms. Each group corresponds to a wavelength entered on the last multiple WAVELENGTH(S) command. The first number is the reflection diffraction efficiency of the zeroth order corresponding to the first wavelength on the last multiple WAVELENGTH command. The next two numbers are the angularly dependent S and P reflection diffraction efficiency models, as discussed previously, at that wavelength. The fourth number is the


. . .



transmission diffraction efficiency followed by its angularly dependent S and P transmission diffraction efficiency models, and so on for different wavelengths. Again, angular data is linearly interpolated in logarithmic amplitude space when a function is unavailable.

The reflection and transmission diffraction efficiencies for sources with wavelengths between those defined in COATING MODELS are interpolated, for example, as in Equation 26. The actual diffraction efficiencies at an incidence angle a and a wavelength b between the first two WAVELENGTH(S) w w' would be,

(EQ 26)

Equation 26

In these equations, r, r’, t, t’ are the complex amplitudes or square roots of the real energy coefficients of the diffraction efficiencies. f(i,a) is the normalized angular amplitude,

(EQ 27)

Equation 27

where BSDF is the bi-directional scattering distribution function.




TIP The key to implementing a COATING MODEL lies in understanding how to set up the data. If your data is not in a functional form, such as that which can be used with USERBSDF or another functional ASAP scatter model, it will be in the form of reflection and transmission diffraction efficiencies as a function of incident angle.

The most commonly used format for entering data in this form is the BSDFDATA scatter model format, since it allows you to enter this type of data directly. ASAP linearly interpolates in logarithmic amplitude space to determine coefficients between those entered in BSDFDATA. Regardless of the scatter model you choose, you must first run the MODELS keyword command, just like you would with COATING or MEDIA.

The BSDFDATA syntax for a coating model is then as follows,

Here,

k is the model data base number;

ANGLES specifies spherical angle coordinates in degrees;

ao bo are the first specular direction, polar and spherical angles

a b [a' b' ...] are spherical ANGLES from and around normal for other orders;

f [f' ...] are the diffraction efficiencies; and

ao' bo' are the second specular direction.

The data entered on the lines with two entries only, indicated by o’s following the letters a, b and so on, defines one incident specular direction for the sets of triplets to follow on the next lines. In the case of a COATING MODEL, the first two numbers of the first set of triplets are the incident specular angles repeated again. For in-plane data, you need enter only the ao, ao', and accompanying a 's as they specify the angle from the normal. The bo 's and b 's can be set to 0 as they are the angles around the normal in plane data. The f parameters are the diffraction


. . .



efficiencies. The next line that has only two numbers defines the next specular direction with its associated triplets, and so on.

What then are the additional a' b' f', a" b" b" terms, and so on? They are the diffraction angles and efficiencies corresponding to a diffraction envelope or, specifically, to the other orders of a diffraction grating. In one sense, a coating is like a diffraction grating operating only in the zeroth order (a, b, f), which is specified by a single set of triplets. However, a diffraction grating can have multiple sets of triplets simulating other diffraction order efficiencies. The numbers you enter are the discrete numbers of an envelope of the diffraction efficiency function at this wavelength and angle of incidence. The numbers can correspond directly to specific diffraction order efficiencies if you have only this data.

The COATING MODELS/ USERBSDF technique is an appropriate way to model diffraction efficiencies. USERBSDF is a general scatter model and diffraction gratings produce discrete, deterministic scattering functions.

Finally, multiple exposure holograms can be modeled by using the third form of the INTERFACE DIFFRACTive syntax. However, you specify only the zeroth order once. For example,




DOE examplesThe following examples illustrate how to set up various diffraction gratings, binary optical elements, and holograms. The examples move from the simple to the complex and demonstrate various forms of the MULTIPLE command and the INTERFACE…DIFFRACT command.

L I N E A R P H A S E G R A T I N G

In our first example, we will simulate a simple linear phase grating with both forms of the MULTIPLE command. When we use the first form of the multiple command, we must know the mathematical description of the phase surface of our linear grating. We know that the grating spacing is d and that the grating frequency is 1/d. But we also have, from our previous work, the relationship between the grating frequency, the phase function, and the grating line equation,

(EQ 28)

Equation 28

Substituting 1/d in the above equation for the grating frequency yields the following relationships,

(EQ 29)

Equation 29

Clearly, f(y)=y and f’=d. We also could have specified the coefficient of y as 1/d and set f’ equal to 1. ASAP now has all the necessary information to compute the local grating line spacing for a ray incident on any point of the base surface. It then uses the grating equation to generated diffracted rays according to the orders specified on the INTERFACE...DIFFRACT command.

ASAP is splitting rays at the diffractive interface. Some of these rays will become child rays. If there are more diffraction gratings in the optical system, you may have to set SPLIT to more than one to generate these other diffracted rays. Example 1 illustrates the ASAP syntax for configuring this grating. Only the zero,


. . .



and plus/minus first orders are simulated, even though more orders are physically present.

Note how the GENERAL command is used to define f(x,y,z), and how the base surface of intersection is a surface perpendicular to the z axis. Its geometry is independent from the MULTIPLE GENERAL surface. Finally, note that the base surface is both reflecting and transmitting, as defined in the COATING PROPERTIES database. See Example 1.




!!ASAP EXAMPLE OF A PHASE DIFFRACTION GRATING!!USING THE FIRST FORM OF THE MULTIPLE COMMANDSYSTEM NEWRESET

$DATIM OFF OFF

PI=4*ATAN(1)LAMBDA=0.025D=0.1

COATING PROPERTIES 0.4 0.6 'DOE'

SURFACE GENERAL 0 0 0 Y 1 MULTIPLE 1 (D) PLANE Z 0 RECTANGLE 1 grating base surfaceOBJECT 'LINEAR_GRATING'INTERFACE COATING DOE AIR AIR DIFFRACT 0.2 -1st order 0.25, 0th 0.5, 1st 0.25

SURFACE PLANE Z 10 RECT 1 detector planeOBJECT 'DETECTOR' ROTATE X ASIN[.25] 0 0 BEAMS INCOHERENT GEOMETRICWAVELENGTH (LAMBDA)

GRID ELLIPTIC Z -1 -4@1 1 11 SOURCE DIRECTION 0 0 1

WINDOW Y ZWINDOW 1.4TITLE 'GRATING W/-1,0,1st ORDERS IN REFLECTION AND TRANSMISSION'MISSED 10


. . .



PLOT FACETS 3 3 OVERLAYTRACE PLOT TEXT 0 -2.6 10, 0 0 .25, 0 .25 0, 1 '-1st order' 0 2.8 4.5, 0 0 .25, 0 .25 0, 1 '+1st order' 0 1 8.75, 0 0 .25, 0 .25 0, 1 '0th order' 0 1.75 -2.25, 0 0 .25, 0 .25 0, 1 '+1st order' 0 -2 -2.25, 0 0 .25, 0 .25 0, 1 '-1st order'

!!$VIEWRETURN

Example 1. Linear phase diffraction grating with the first form of MULTIPLE

Output from the Plot Viewer is shown in Figure 9.

Figure 9 Example 1: Diffraction orders from a reflection and transmission grating, using

the first form of the MULTIPLE command




We can also simulate the same diffraction grating with the second form of the MULTIPLE command. Recall that in this second form you specify a distance between the original function or surface and the first replica. ASAP automatically computes the f’ value from,

(EQ 30)

Equation 30

In our case, the gradient of the function f(x,y,z), obtained from the phase function, specifies the direction in which parallel replications of the original surface are generated. We know that the grating lines are perpendicular to the y axis, so we can use a plane whose normal is collinear with this axis to denote the gradient of f(x,y,z). The MULTIPLE terms then include the grating spacing, which is just the distance between the original function or surface and the first replica, and a point on the original function. Our plane, whose normal is collinear with the y axis, is replicated to form a series of parallel planes whose spacing is the grating line spacing.

The second form of the MULTIPLE syntax illustrates the ASAP syntax for this case. See Example 2.


. . .



!!ASAP EXAMPLE OF A PHASE DIFFRACTION GRATING!!USING THE SECOND FORM OF THE MULTIPLE COMMANDSYSTEM NEWRESET

$DATIM OFF OFF

COATING PROPERTIES 0.4 0.6 'DOE'

SURFACE PLANE Y 0 grating line generating surface MULTIPLE 1, grating spacing 0.1, point 0 0 0 PLANE Z 0 RECT 1 grating base surfaceOBJECT 'LINEAR_GRATING'INTERFACE COATING DOE AIR AIR DIFFRACT 0.2 -1st order 0.25, 0th 0.5, 1st 0.25

SURFACE PLANE Z 10 RECT 1 detector planeOBJECT 'DETECTOR' ROTATE X ASIN[.25] 0 0

BEAMS INCOHERENT GEOMETRICWAVELENGTH 0.025GRID ELLIPTIC Z -1 -4@1 1 11SOURCE DIRECTION 0 0 1

WINDOW Y ZWINDOW 1.4TITLE 'GRATING W/-1,0,1st ORDERS USING 2nd FORM OF MULTIPLE'MISSED 10

PLOT FACETS 3 3 OVERLAY




TRACE PLOT TEXT 0 -2.6 10, 0 0 .25, 0 .25 0, 1 '-1st order' 0 2.8 4.5, 0 0 .25, 0 .25 0, 1 '+1st order' 0 1 8.75, 0 0 .25, 0 .25 0, 1 '0th order' 0 1.75 -2.25, 0 0 .25, 0 .25 0, 1 '+1st order' 0 -2 -2.25, 0 0 .25, 0 .25 0, 1 '-1st order'

RETURNExample 2. Linear phase diffraction grating with the second form of MULTIPLE

Again, only the zero and plus/minus first orders are simulated, even though more orders are physically present. See Figure 10.

Figure 10 Example 2: Diffraction orders from a reflection and transmission grating, using

the second form of the MULTIPLE command


. . .



S I N U S O I D A L P H A S E G R A T I N G

A sinusoidal phase grating is like a linear phase grating, with the exception that its phase structure is a sinusoid instead of discrete steps. As a consequence, light is diffracted only into the zero and first orders. This can be understood by examining the Fourier aspects of diffraction at a sharp edge or phase structure compared to that of a smooth edge or phase structure. Higher frequency components of a sharp edge in Fourier space contribute to the higher-order diffraction angles and subsequent diffraction orders. These frequency components are missing in the sinusoidal phase structure and, therefore, so are the diffraction orders higher than the first order.

In a sinusoidal grating, the phase function is of the form,

(EQ 31)

Equation 31

where d is again the grating period. The order m is one. The function f(x,y,z) of the MULTIPLE command is then,

(EQ 32)

Equation 32

However, we now have a grating line function that is not a polynomial function, a functional form easily implemented with the GENERAL command. How can we represent this grating line function? The answer is with USERFUNC.




The USERFUNC command creates a surface specified by a user-defined function. The user-defined function can be one of the ASAP intrinsic functions like sine or cosine, or it can be one created with the internally defined macro $FCN. Its syntax is as follows,

Here

x y z are the global coordinates of the reference point;

fcn is a user-defined function; and

c c' c" ... are user-defined coefficients of the function.

USERFUNC specifies a user-defined function with reference point (x,y,z) and double-precision coefficients. You can define its value and gradient at any point in the macro, $FCN named fcn or the Fortran function USERFUNC. If the function is continuous in both value and gradient everywhere in space, there are no restrictions on the use of this function in ASAP, except possibly the application of non-orthogonal transformations to it; that is, SCALE or SKEW or non-isotropic SCALE.

If the fcn is specified, the local (x,y,z) coordinates are passed in the _1, _2, and _3 registers. You can also define other parameters of the function and set them, with up to 63 coefficients c c' c" ..., in the registers _4, _5,. . ._66. If four or more values are returned, the last four entries of the function that was run must be the functional value and its gradient vector. For example, a sphere of radius 10 centered about the reference point is done as follows,

Otherwise, the default USERFUNC is an aspheric conicoid. You can examine the other parameters of the default user function in the ASAP online help.


. . .



In our example, we use USERFUNC to define the sinusoidal phase function. On USERFUNC, we must enter the functional form of the grating line function and its gradient. The gradient of our grating line function is,

(EQ 33)

Equation 33

These values are entered on USERFUNC as illustrated in Example 3. The diffraction orders are shown in Figure 11 We chose f’ to be 1, but we could also have set it equal to 1/2.




!!ASAP EXAMPLE OF A PHASE DIFFRACTION GRATING!!USING THE FIRST FORM OF THE MULTIPLE COMMANDSYSTEM NEWRESET

$DATIM OFF OFF

PI=4*ATAN(1)LAMBDA=0.025D=0.1

COATING PROPERTIES 0.4 0.6 'DOE' $FCN GRATING Y=_2, SIN(2*PI*Y/D)/(2*PI) 0, (1/D)*COS(2*PI*Y/D) 0

SURFACE USERFUNC 0 0 0 GRATING MULTIPLE 1 1 PLANE Z 0 RECT 1 grating base surfaceOBJECT 'SINUSOIDAL_GRATING' INTERFACE COATING DOE AIR AIR DIFFRACT 0.2 -1st order 0.25, 0th 0.5, 1st 0.25

SURFACE PLANE Z 10 RECT 4 detector planeOBJECT 'DETECTOR'

BEAMS INCOHERENT GEOMETRICWAVELENGTH (LAMBDA)

GRID ELLIPTIC Z -1 -4@1 1001 1001 RANDOM 1SOURCE DIRECTION 0 0 1

WINDOW Y ZWINDOW 1.4TITLE 'SINUSODIAL GRATING W/-1,0,1st ORDERS IN REFLECTION AND TRANSMISSION'


. . .



MISSED 10PLOT FACETS 3 3 OVERLAYTRACE PLOT 11111 TEXT 0 -2.8 4.5, 0 0 .25, 0 .25 0, 1 '-1st order' 0 2.8 4.5, 0 0 .25, 0 .25 0, 1 '+1st order' 0 .2 10.1, 0 0 .25, 0 .25 0, 1 '0th order' 0 1.75 -2.25, 0 0 .25, 0 .25 0, 1 '+1st order' 0 -2 -2.25, 0 0 .25, 0 .25 0, 1 '-1st order'

WINDOW X YPIXELS 101CONSIDER ONLY DETECTOR

SPOTS POSITION EVERY 137DISPLAY AVERAGE PICTURE

RETURNExample 3. Sinusoidal phase diffraction grating with USERFUNC and the first form of the

MULTIPLE command




The following illustrations are all part of Figure 11.


. . .



Figure 11 Example 3: Diffraction orders from a sinusoidal reflection and transmission grating, using USERFUNC and the

first form of the MULTIPLE command

C I R C U L A R D I F F R A C T I O N G R A T I N G

The circular diffraction grating is a linear grating bent around an axis of symmetry. This grating, therefore, has circularly concentric grating lines. Its radial frequency is 1/d. Following a similar technique, we can integrate the radial frequency to obtain the phase surface and f(x,y,z).

Recall that the phase surface and the grating line equation are related, and specifically related to the radial frequency by

(EQ 34)

Equation 34




Substituting 1/d in the above equation for the grating frequency yields the following relationships.

(EQ 35)

Equation 35

(EQ 36)

Equation 36

and f’=d. However, our relationship for the grating line equation contains a square root. Therefore, it is best to use USERFUNC, since a square root cannot be directly entered on the GENERAL command, and then MULTIPLEd. Remember that when you use USERFUNC, you should configure it to return four values, the first being the value of the function and the last three its gradient. In our case, the function and its gradient are,


. . .



(EQ 37)

Equation 37

These values are entered in the USERFUNC command as shown in Example 4. The diffracted first order for the circular grating is illustrated in Figure 12.




!!ASAP EXAMPLE OF A CIRCULAR DIFFRACTION GRATING!!USING THE FIRST FORM OF THE MULTIPLE COMMANDSYSTEM NEWRESET

$DATIM OFF OFF

PI=4*ATAN(1)D=0.058476 grating line spacing

$FCN CIRC SQRT(_1^2+_2^2),_1/SQRT(_1^2+_2^2),_2/SQRT(_1^2+_2^2),0

COATING PROPERTIES 0 1 'TRANS'

BEAMS INCOHERENT GEOMETRICWAVELENGTH 0.02

SYSRAY { 2RAYS 0GRID ELLIPTIC Z -1 -4@1 #1 #2 SOURCE DIRECTION 0 0 1}ENTER NUMBER OF RAYS ALONG X-AXISENTER NUMBER OF RAYS ALONG Y-AXIS

SURFACE USERFUNC 0 0 0 CIRC MULTIPLE 1 (D) PLANE Z 0 ELLIP 1OBJECT 'CIRCULAR_GRATING' INTERFACE COATING TRANS AIR AIR DIFFRACT 0.2 -1

SURFACE PLANE Z 4 RECT 1.5


. . .



OBJECT 'DETECTOR'

TITLE 'CIRCULAR GRATING | 1st FORM OF MULTIPLE'MISSED ARROW 3.5

$SYSRAY 1 11WINDOW Y -2 2 Z -1 3PLOT FACETS 9 9 0 OVERLAYTRACE PLOT

RETURN

$IO VECTOR REWIND

$SYSRAY 5 11WINDOW Y -2 2 Z -1 3WINDOW 1.5 OBLIQUE

PLOT FACETS 3 3 OVERLAYTRACE PLOT

RETURN

!!$VIEWExample 4. Circular phase diffraction grating with the first form of MULTIPLE




Figure 12 Example 4: First diffraction order from a circular phase grating, using USERFUNC and the first form of

MULTIPLE


. . .



All our examples to this point were generated by integrating the radial frequency function or modifying the phase function to generate the grating line function. They all also used the default EXPONENT on the MULTIPLE command.

We now demonstrate a technique for deriving the grating line function directly, which is also a case where the EXPONENT on the MULTIPLE command is not equal to one, the default. We again return to the circular grating for this example.

A circular diffraction grating has concentric, equally spaced grating circles (lines). We can directly write the equation of these grating lines as,

(EQ 38)

Equation 38

d is again the grating circle spacing. This can be verified by computing the grating spacing between two adjacent grating circles as,

(EQ 39)

Equation 39

Therefore, is a constant.

We must now put this equation into a form that ASAP can simulate as a surface, and then we must identify the appropriate terms of the MULTIPLE command. By squaring both sides of the previous equation we get,

(EQ 40)

Equation 40




When we compare this with the form of the MULTIPLE command, we see that,

(EQ 41)

Equation 41

The MULTIPLE syntax with the EXPONENT option for this case is illustrated in Example 5. Note that the grating spacing d is 0.058476, and the wavelength is 0.02—both are in arbitrary units. We can compute the diffraction angle for this element by using the grating equation, sin(q)=ml/d, and we see that it is 20, the same as in the previous circular grating example.


. . .



!!ASAP EXAMPLE OF A CIRCULAR DIFFRACTION GRATING!!USING THE EXPONENTIAL OPTION OF THE MULTIPLE COMMANDSYSTEM NEWRESET

$DATIM OFF OFF

D=0.058476 grating line spacing


SURFACE GENERAL 0 0 0; X2 1; Y2 1 MULTIPLE 0 D*D EXPONENT 2 PLANE Z 0 ELLIP 1OBJECT 'CIRCULAR_GRATING' INTERFACE COATING TRANS AIR AIR DIFFRACT 0.2 -1

SYSRAY { 1RAYS 0BEAMS INCOHERENT GEOMETRICWAVELENGTH 0.02GRID ELL Z -1 -4@1 #1 11SOU DIR 0 0 1}ENTER NUMBER OF RAYS ALONG X-AXIS>

TITLE 'CIRCULAR GRATING | EXPONENTIAL OPTION OF MULTIPLE'MISSED ARROW 3.5

$SYSRAY 1WINDOW Y -2 2 Z -1 3

PLOT FACETS 9 9 0 OVERLAYTRACE PLOT$IO VECTOR REWIND




$SYSRAY 5WINDOW 1.5OBLIQUE

PLOT FACETS 9 9 0 OVERLAYTRACE PLOT

RETURNExample 5. Circular phase diffraction grating with the EXPONENT option of the

MULTIPLE command

The plots for the above script are shown in Figure 13.


. . .



Figure 13 Example 5: First diffraction order from a circular phase grating, using GENERAL and the EXPONENTIAL

options of the MULTIPLE command

D O E L E N S

Up to this point, all the examples have been simple linear and circular gratings. They were made deliberately simple to demonstrate the different forms of the MULTIPLE command and the techniques needed to simulate DOE elements.

We now examine a more complicated diffractive element, a DOE lens. With more complicated DOEs, we still use the same techniques we have developed so far. We must determine the grating line equation, which the MULTIPLE command uses to simulate DOEs. Even with more complicated phase structures, we will create multiple surfaces whose intersection with the base surface specifies the grating lines. These grating lines represent 2 phase steps. Finally, since these grating lines are created by the MULTIPLE command, they must be defined by the equation,


(EQ 42)

Equation 42

where j and p are integers and f’ is a constant. In general, the process requires knowing the phase function at a particular order m and converting that phase function into the grating line equation by dividing by 2m. These terms can then be related directly to the grating line equation and the constant and exponential terms that are needed in the MULTIPLE command.

We will model a DOE lens that produces a simple quadratic phase transformation on a collimated input beam. We have already determined the phase function of this lens in the previous section. Its phase function for the minus first order was,

(EQ 43)

Equation 43

Recall that the first term was a constant phase term of little interest, and the second term was a quadratic approximation to a spherical wavefront. The phase function is related to the grating line function,

(EQ 44)

Equation 44

Therefore, the grating line function is,

(EQ 45)

Equation 45

. . .



The paraxial focal length of the lens is f. We can write f(x,y,z) and the other MULTIPLE terms as,

(EQ 46)

Equation 46

Note that other equally valid equations describing the grating line equation and constant terms can be obtained by rearranging terms as follows.




(EQ 47)

Equation 47

ASAP syntax that is needed to simulate the DOE lens is shown in Example 6. The script is shown in two parts due to its length. Output from this example illustrate the geometric and diffractive output from the DOE lens for on- and off-axis beams. See Figure 14 , Figure 15, Figure 16, Figure 17, and Figure 18.


. . .



SYSTEM NEWRESET

$DATIM OFF OFF$GUI CHARTS_ON !!for Chart Viewer!!$GUI CHARTS_OFF !!for Plot Viewer

L=0.6328E-4 F=20


SURFACE GENERAL 0 0 0 X2 1 Y2 1 !!f(x,y,z) MULTIPLE n 1, f prime 2*L*F PLANE Z 0 ELLIPSE 2OBJECT 'DOE' INTERFACE COATING TRANS AIR AIR DIFFRACT surface 1, -1st order

SURFACE PLANE Z 20 ELLIPSE 3OBJECT 'DETECTOR'

BEAMS INCOHERENT GEOMETRICWAVELENGTH (L)

GRID ELLIPTIC Z -1 -4@2 301 301 RANDOM 1SOURCE DIRECTION 0 0 1

WINDOW Y ZPIXELS 255TITLE 'F/5 DOE LENS WITH SIMPLE QUADRATIC PHASE'

CONSIDER ONLY DETECTORWINDOW X Y




RAYS 0BEAMS COHERENT DIFFRACTWAVELENGTH (L)PARABASAL 4WIDTHS 1.6

GRID ELLIPTIC Z -1 [email protected] 101 101 RANDOM 1SOURCE DIRECTION 0 0 1, 0 SIN[1] COS[1]

CONSIDER ALLWINDOW Y Z!!PLOT FACETS 3 3 0 OVERLAYTRACE PLOT 71

CONSIDER ONLY DETECTORSELECT ONLY SOURCE 1

FOCUS MOVE

WINDOW Y -.005 .005 X -.005 .005WINDOW .7PIXELS 201

SPREAD NORMALDISPLAY ISOMETRIC 2 'IRRADIANCE DISTRIBUTION (ON-AXIS)' PICTURE 'ON-AXIS SOURCE'RETURN

SELECT ONLY SOURCE 2FOCUS MOVEWINDOW Y .344 .354 X -.005 .005WINDOW .7PIXELS 201

SPREAD NORMALDISPLAY ISOMETRIC 2 'IRRADIANCE DISTRIBUTION (OFF-AXIS)'


. . .



PICTURE 'OFF-AXIS SOURCE'

RETURNExample 6. DOE lens input (above)

Figure 14 .Example 6: DOE lens with simple quadratic phase

Figure 15 Example 6: Irradiance distribution (on-axis)




Figure 16 Example 6: On-axis source

Figure 17 Example 6: Irradiance distribution (off-axis)


. . .



Figure 18 Example 6: Display Viewer output, off-axis source



References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .R E F E R E N C E S

• Optics, Hecht, Second Addition.

• Introduction to Fourier Optics, Goodman.

• Binary Optics Technology: The Theory and Design of Multi-level Diffractive Optical Elements, Swanson.

• Fundamentals of Polarized Light A Statistical Approach, Brosseau.

• Polarized Light Production and Use, Shurcliff.

• Optical Waves in Crystals, Yariv and Yeh.

• Principles of Optics, Born and Wolf.

• Optical Thin Films Users Handbook, Rancourt.

• Optical Scattering Measurement and Analysis, Stover.


include diffraction gratings, binary optics, and holographic optical elements. these optical...

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