Diffraction Gratings

A grating consists of a series of equally spaced parallel grooves formed in a reflective coating deposited on a suitable substrate. The distance between adjacent grooves and the angle the grooves form with respect to the substrate influence both the dispersion and efficiency of a grating. If the wavelength of the incident radiation is much larger than the groove spacing, diffraction will not occur. If the wavelength is much smaller than the groove spacing, the facets of the groove will act as mirrors and, again, no diffraction will take place.

The way in which the grooves are formed separates gratings into two basic types, holographic and ruled. Physically forming grooves into a reflective surface with a diamond mounted on a “ruling engine” produces ruled gratings. Gratings produced from laser constructed interference patterns and a photolithographic process are known as interference or holographic gratings.

We are one of the few companies that produce both types of gratings in-house and has full replication facilities and expertise. Ruled and holographic gratings differ in their optical characteristics and each type has advantages for specific applications. To learn more about our ruling and holographic mastering processes, click here.

Grating Equation

The general grating equation is usually written as: nλ = d(sin i + sin i’) where n is the order of diffraction, is the diffracted wavelength, d is the grating constant (the distance between successive grooves), i is the angle of incidence measured from the normal and i’ is the angle of diffraction measured from the normal.

Diffraction grating angles and terms.

For a specific diffracted order (n) and angle of incidence (i), different wavelengths (λ) will have different diffraction angles (i’), separating polychromatic radiation incident on the grating into its constituent wavelengths.

Efficiency

Grating efficiency is a function of groove shape, angle of incidence and the reflectance of the coating.

The absolute efficiency of a grating is the percentage of incident monochromatic radiation that is diffracted into the desired order. In contrast, relative efficiency compares the energy diffracted into the desired order with that of a plane mirror coated with the same material as the grating. When comparing grating performance curves, it is important to keep this in mind. A relative efficiency curve will always show higher values than an absolute efficiency curve for the same grating. The efficiency curves in this brochure present absolute efficiency data.

Angle of incidence plays a role in grating performance. Because of the infinite number of configurations that a grating can be used in, a standard geometry is used in the measurement of the gratings. This is the Littrow (or autocollimation) mounting. In this mounting configuration, the diffracted order and wavelength of interest is directed back along the path of the incident light (i=i’). The blaze angle of a ruled grating is calculated based on this mounting. This mounting is practical and necessary for laser tuning applications, but most applications will require some deviation between the incident and diffracted beams. Small deviations from the Littrow mounting seldom have an appreciable effect on grating performance other than to limit the maximum wavelength achievable. Unless otherwise stated, all performance curves in this brochure present blazed first order Littrow data.

Blaze Angle and Wavelength

The grooves of a ruled grating have a sawtooth profile with one side longer than the other. The angle made by a groove’s longer side and the plane of the grating is the “blaze angle.” Changing the blaze angle concentrates diffracted radiation to a specific region of the spectrum, increasing the efficiency of the grating in that region. The wavelength at which maximum efficiency occurs is the “blaze Wavelength.”

Holographic gratings are generally less efficient than ruled gratings because they cannot be “blazed” in the classical sense. Their sinusoidal shape can, in some instances, be altered to approach the efficiency of a ruled grating. There are also special cases that should be noted, i.e. when the spacing to wavelength ratio is near one, a sinusoidal grating has virtually the same efficiency as a ruled grating. A holographic grating with 1800 g/mm can have the same efficiency at 500 nm as a blazed, ruled grating. In addition, a special process enables Optometrics’ holographic gratings to achieve a true sawtooth profile peaked at 250 nm, an ideal configuration for UV applications requiring good efficiency with low stray light.

Resolving Power

The resolving power of a grating is the product of the diffracted order in which it is used and the number of grooves intercepted by the incident radiation. It can also be expressed in terms of grating width, groove spacing and diffracted angles. The “theoretical resolving power” of a diffraction grating with N grooves is:

Resolving power equation for diffraction gratings.

The actual resolving power of a grating depends on the accuracy of the ruling, with 80-90% of theoretical being typical of a high quality ruling.

Resolving power is a property of the grating and is not, like resolution, dependent on the optical and mechanical characteristics of the system in which it is used.

System Resolution

The resolution of an optical system, usually determined by examination of closely spaced absorption or emission lines for adherence to the Raleigh criteria (R = λ/Δλ), depends not only on the grating resolving power but on focal length, slit size, f number, the optical quality of all components and system alignment. The resolution of an optical system is usually much less than the resolving power of the grating.

Dispersion

Angular dispersion of a grating is a product of the angle of incidence and groove spacing. Angular dispersion can be increased by increasing the angle of incidence or by decreasing the distance between successive grooves. A grating with a large angular dispersion can produce good resolution in a compact optical system.

Angular dispersion is the slope of the curve given by λ = f(i). In autocollimation, the equation for dispersion is given by:

Dispersion equation for diffraction gratings.

This formula may be used to determine the angular separation of two spectral lines or the bandwidth that will be passed by a slit subtending a given angle at the grating.

Diffracted Orders

For a given set of angles (i,i´) and groove spacing, the grating equation is valid at more than one wavelength, giving rise to several “orders” of diffracted radiation. The reinforcement (constructive interference) of diffracted radiation from adjacent grooves occurs when a ray is in phase but retarded by a whole integer. The number of orders produced is limited by the groove spacing and the angle of incidence, which obviously cannot exceed 90 degrees. At higher orders, efficiency and free spectral range decrease while angular dispersion increases. Order overlap can be compensated for by the judicious use of sources, detectors and filters and is not a major problem in gratings used in low orders.

Diffracted orders

 

Free Spectral Range

Free spectral range is the maximum spectral bandwidth that can be obtained in a specified order without spectral interference (overlap) from adjacent orders. As grating spacing decreases, the free spectral range increases. It decreases with higher orders. If 1, 2 are lower and upper limits, respectively, of the band of interest, then:

Free spectral range = 2 — 1 = 1/n

Higher orders and free spectral range for diffraction gratings.

 

Ghosts and Stray Light

Ghosts are defined as spurious spectral lines arising from periodic errors in groove spacing. Interferometrically controlled ruling engines minimize ghosts, while the holographic process eliminates them.

On ruled gratings, stray light originates from random errors and irregularities of the reflecting surfaces. Holographic gratings generate less stray light because the optical process which transfers the interference pattern to the photoresist is not subject to mechanical irregularities or inconsistencies.

Coatings

Gratings used in the ultraviolet, visible and infrared are normally replicated with an aluminum coating. Aluminum is used rather than silver because it is more resistant to oxidation and has superior reflectance in the ultraviolet. Aluminum averages over 90% reflectance from 200 nm to the far infrared, except in the 750 to 900 nm region where it drops to approximately 85%. When maximum reflectance is required in the near infrared, as is the case with some fiber optic applications, the aluminum coating may be overcoated with gold. Though gold is soft, it is resistant to oxidation and has a reflectance of over 96% in the near infrared and over 98% above 2.0 μ. The reflectance of gold drops substantially below 600 nm and is not recommended for use in the visible or ultraviolet regions.

Reflectance versus wavelength of aluminum and gold coated optics.

Dielectric overcoatings such as aluminum magnesium fluoride (AlMgF2) protect aluminum from oxidation, maintaining the original high reflectance of aluminum in the visible and ultraviolet. Gold overcoatings and aluminum magnesium fluoride dielectric coatings must be specified separately when ordering.

While gold overcoating can increase reflectivity, any overcoating may reduce the damage threshold by a factor of two or more.

Tunable Laser Gratings – Molecular Lasers

The output wavelength of a molecular or dye laser can be tuned by rotating a Littrow mounted grating around an axis parallel to the grooves. The grating equation:

nλ = d(sin i + sin i’)

where n is the order of diffraction, l is the diffracted wavelength, d is the grating constant (the distance between successive grooves), i is the angle of incidence measured from the normal and i’ is the angle of diffraction measured from the normal, reduces to nl= 2d sin i for the Littrow configuration.

The angle of incidence (i) is adjusted to select the output wavelength while creating a narrow gain profile.

Blaze Angle and Alignment

Because the ML series of gratings are designed for peak polarized efficiency, the groove angle is not equivalent to the Littrow blaze angle of the grating. As a result, when using a He-Ne laser for preliminary grating alignment, the brightest He-Ne order will not correspond to the blaze wavelength of the grating. The grating must be aligned using the calculated He-Ne order that corresponds to the wavelength of interest, regardless of its relative intensity.

The blaze arrow marked on the side or back of the grating should be oriented as shown below.

Blazed grating in Littrow.

Polarization

Typical efficiency curves illustrate that, in all cases, orienting the polarization of the E vector (P-Plane) perpendicular to the grooves (E ^) increases the efficiency over a specific wavelength region. This should be considered when optimizing the figure of merit (Q) of a cavity, particularly when it is polarized by auxiliary components such as Brewster angle windows.

Polarization dependent efficiencies for diffraction gratings.

Coating

ML gratings can be overcoated with gold, increasing the reflectivity at 10.6 microns by approximately 1%, (single pass) but the damage threshold in high power applications may be reduced. No damage threshold minimums apply for overcoated gratings.

For Dye laser gratings

The equation relating wavelength to angle of incidence on the grating is: nλ = 2d sin i

Using this relationship, one can calibrate the mirror mount micrometer to i and, using a grating as one of the reflectors in a dye laser, cause the spectral line width of the output to be reduced to a narrow region around the Littrow wavelength. Maximum reduction is attained by increasing dispersion to the practical limit. The equation for angular dispersion is:

Dispersion equation for diffraction gratings.

Even greater selectivity can be attained by adding a Fabry-Perot etalon to the grating-mirror cavity. The efficiency of the spectral condensation in organic dye lasers is quite high, with figures of 70% being Typical.

Gratings are normally used in one of two configurations: Littrow or grazing incidence.

Filtered Gratings

patent markBlock Undesired Spectra Before Dispersion
Reduce Number of Optical Components
Reduce System Size
Designs for Reflection and Transmission Gratings

Our Filtered Gratings are custom solutions for OEM design applications which require isolation of the desired spectral band prior to diffraction, and are design limited in doing so prior to the grating component.

Example: An application requires 450 – 750 nm data collection, although the raw signal from the sample includes ambient energy covering 300 – 1,000nm. As shown in the grating efficiency curve, energy from 300 – 450 nm, and 750 – 1,000 nm will diffract, generating unanticipated 2nd order radiation, and cause significant ray scattering within the optics chamber as well. The QE of a Si detector cannot differentiate l. Filtering the undesired spectral band prior to diffraction minimizes these second order and ray scattering effects, maximizing spectral dynamic range.

View the Optometrics Corporate Capabilities Brochure

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