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This is the well-known Grating Equation. For a given angle of incidence, θ, it gives the angle of diffraction θm for each “order” m for which a solution to (1) exists. Often gratings are described by the frequency of grating lines instead of the period, where f (in lines/mm) is equal to 106/Λ (for Λ in nm). In terms of f the grating equation becomes

The dependence of the angles of diffraction on the angle of incidence can be more completely visualized on the graph shown in Figure 3. Here the specific case of θ = 10º illustrated in Figure 2 is indicated by the solid dots on the graph. From the graph it is apparent that different orders exist for different angles of incidence. For example, for θ smaller than about 5º only the +1st and –1st orders exist, while for θ larger than about 38º only the –1st, –2nd, and –3rd orders exist.

Diffraction gratingexperiment

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This constraint on the grating period is illustrated in Figure 7, which shows the cut-off ratio of wavelength to period, λ/Λ, for the lowest several diffracted orders as a function of the angle of incidence θ. For wavelength-period ratios above the m = –1 cut-off line, in the red-shaded region, no diffraction occurs at all; the period is simply too small. For ratios below the m = –1 cut-off line but above the m = +1 and m = –2 cut-off lines, in the yellow-shaded region, only a single –1st diffracted order exists. This is the region in which almost all PGL gratings fall.

In addition, it’s important that the team has a collective sense of purpose and shares similar values and business goals. Are you building a lifestyle business that you want to be involved in for the next 20 years, or a business that you want to sell for $2million five years from now?

Start-ups often differentiate on one of three different value propositions: a unique outcome, a reduced cost, or an environmental improvement. You must be able to articulate your value proposition.

A typical transmission grating is illustrated in Figure 6. Here the grating interface is shown as the front surface of a glass substrate. Since the incident and diffracted orders are all measured in air, the form of the Grating Equation in (1) should be used.

When Uber launched in San Francisco in 2009, it addressed people’s desire for price transparency, which stemmed from a distrust of strangers. By quoting the ride’s cost up front, Uber offered the promise of building driver-rider trust through a transparent review system. The problem was clearly defined, and visionary software engineers designed a product that solved it.

CycleTrack is a free system for tracking and visualizing medical school applications while crowdsourcing data to make admissions more transparent.

The first and most important question is, do you have a product that solves a real problem? If there isn’t a clear answer, you might have a product, but you don’t have a business. The success or failure of a start-up is not dependent on clever marketing or an excellent sales team: it rides on having a product that meets a well-defined customer need. To truly define and validate the problem, interview as many potential customers as you can, not just a handful. This will help identify whether the problem is pervasive and will yield a large enough target market.

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Once these four questions are answered, you’ll be better prepared to start down the path of a new startup. Remember that you’ll likely hear ‘No’ a thousand times before you hear ‘Yes!’ and that you will learn more from failure than success. Good luck on your journey!

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Insight into earlier failures can make all the difference in your own success. Look to the Israeli lidar company Oryx Vision, which shut down operations in August 2019, for an example.

But do you have the right team? Many great ideas based on solid science fail at this point just because the teams aren’t made up of the right people.

where ni and θi are the index of refraction and ray angle, respectively, in the incident medium, and nt and θt are the analogous quantities in the transmitted medium (see Figure 5). When nt is smaller than ni, light bends away from the normal to the interface (i.e., θt > θi), whereas when nt is larger than ni the opposite occurs.

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Sujatha Ramanujan, managing director of the Luminate Accelerator Programme in Rochester, New York, offers four key questions to consider before launching an optics start-up

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When you are confident that you know your customer, thoroughly scan the competition. Make sure you haven’t reinvented something that’s already out there. If you’re lucky, you might even hear of someone who tried something similar and failed.

The graph in Figure 3 also shows a dashed line called the “Littrow line.” When the curve associated with a particular m < 0 order intersects this line, the angle of diffraction is equal and opposite to the angle of incidence. In other words, light is diffracted exactly along the path of the incident light. This type of retroreflection is called Littrow diffraction.

For many gratings, especially those designed for use with lasers, it is desirable for all of the incident light to be diffracted into a single order to minimize loss in the overall system. The most straightforward way to realize a single order is to make the grating period Λ small enough to eliminate all other nonzero orders as solutions to the Grating Equation (1). How small Λ must be depends on the wavelength and the angle of incidence.

Fraunhoferdiffraction

Can you identify who, exactly, cares about what you’ve built? Resist the urge to make assumptions here. Put together a list of companies that you believe will want your product. This list should include those who actually make the purchasing decisions, in addition to those who will influence the decisions, such as the user of the technology.

The condition for Littrow diffraction can be understood using the simple ray picture in Figure 4. Following the logic used in analyzing Figure 1, constructive interference along the Littrow direction occurs when the path difference between adjacent green-blue ray paths is given by (AB = 0) – 2A’B’ = mλ, or

The larger the period Λ, or the lower the frequency f, the more orders there are. As an example, suppose a HeNe laser beam at 633 nm is incident on an 850 lines/mm grating. Referring to Figure 2, there will be three diffracted orders (m = –2, –1, and +1) along with the specular reflection (m = 0). This 0th order is typically not considered a diffracted order since it does not provide any angular dispersion (change in angle with change in wavelength).

Referring to Figure 1, imagine a beam of light represented by the two green rays incident on the binary (rectangular profile) grating shown. The light is diffracted in many directions, only one of which is indicated by the blue rays. If the difference between adjacent green-blue ray paths diffracted off of identical locations on adjacent periods is equal to a multiple of the wavelength of light, the two blue rays interfere constructively.

where θL is called the Littrow angle. The same relation can also be derived by simply setting θm = – θ in the Grating Equation (1). Most often we are interested in the Littrow condition for the –1st order, for which the equation is

Once your product and market are clearly defined, you must be able to articulate your advantage. Why is yoursthe best solution to the problem?

Echellegrating

Diffraction grating

If your product will cost $200 to manufacture, but your target customer is only willing to pay $50, then you need to either find a way to reduce your costs or find a new customer to target.

So, you know you’ve got a technology that solves a problem, you know exactly who you’re going to solve the problem for, and you know why your solution is the best.

Grating

Diffraction gratings can be understood using the optical principles of diffraction and interference. When light is incident on a surface with a profile that is irregular at length scales comparable to the wavelength of the light, it is reflected and refracted at a microscopic level in many different directions as described by the laws of diffraction. If the surface irregularity is periodic, such as a series of grooves etched into a surface, light diffracted from many periods in certain special directions constructively interferes, yielding replicas of the incident beam propagating in those directions.

Sujatha Ramanujan, PhD, was featured in the 2019 SPIE Women in Optics planner. Luminate is an accelerator programme for optics, photonics, and imaging-enabled applications. Each year, 10 teams from around the world receive $100,000 each to attend a six-month programme, providing business development training, technical facilities, mentoring, and access to potential partners and investors. The programme also invests $2million annually in follow-on funding to the most promising companies.

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They saw that the field of lidar was becoming a place for big companies and big players, and as a small company, they couldn’t justify the investments that would be needed to stay in the game until the technology took off commercially. They tried to sell the company but couldn’t. For an entrepreneur wanting to enter the lidar space, their story could be important.

For example, in addition to finding someone with the right skills to fill the CFO and CTO roles, look for a pessimist to balance an optimist. If the project leader is a big-picture person, make sure you’ve got someone who likes to get stuck in the weeds. If your product is an innovative new moisture sensor for baby diapers, someone on your team had better have an infant.

Blazedgrating

Note the sign conventions for the angles. Incident light is shown traveling left-to-right, for which the angle θ ≥ 0. For diffracted light traveling left-to-right, θm ≥ 0, whereas for diffracted light traveling right-to-left, θm ≤ 0.

The cut-off lines for all other orders fall below those shown on the graph. The green dot illustrates the wavelength-period ratio and angle of incidence associated with the example of a 633 nm laser and 850 lines/mm grating highlighted in Figures 2 and 3. Since this point is below the m = –1, –2, and +1 cut-off lines, but above the m = +2 (and all other) cut-off lines, only these first three orders exist.

Mathematically, the difference between paths AB and A’B’ is a multiple of the wavelength when AB – A’B’ = mλ, where m is an integer and λ is the wavelength of light (typically stated in nm). Since AB = Λsinθm and A’B’ = Λsinθ, where Λ is the grating period and θm and θ are the angles of diffraction and incidence, respectively, relative to the surface normal, the condition for constructive interference is

Successful teams encompass a diversity of skills and thought. This diversity manifests itself in technical strengths, business savvy, complementary personality types, and market knowledge.

The discussion so far has assumed that the diffracted light is reflected off of the grating structure. A grating which predominantly reflects the diffracted light is called, not surprisingly, a reflection grating. A grating which predominantly transmits the diffracted light is called a transmission grating. When analyzing transmission gratings, care should be taken to properly account for refraction of light at the grating and any other interfaces. Refraction of a light ray at a planar interface between two media with different indexes of refraction is described by the well-known Snell’s Law

However, it is important to recognize that there may be orders that do not exist as solutions to (1), yet they exist and propagate inside the substrate. Examples are the +2nd and –3rd orders shown in Figure 6. These orders are totally internally reflected inside the substrate, and therefore do not propagate in the air region outside the substrate. Nevertheless they are real, and can emerge from the edge of the glass substrate.

Free-space optics company AOptix launched with funding in 2000, but folded 16 years later when the market was looking for cheaper telecommunication solutions than its $80,000 links.

Diffraction gratingformula

where nm is the index of refraction of the region into which the diffracted light travels. For the reflected orders, nm = ni, and the Grating Equation becomes

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To be more complete, if a grating is at an interface between two media with indexes ni and nt, the Grating Equation (1) is written as

The Littrow line on the graph in Figure 7 connects the angle of incidence to the wavelength-period ratio. So for example, light with a wavelength exactly equal to the period of a grating (λ/Λ = 1) experiences Littrow diffraction at θ = 30º. For a given wavelength the largest possible period for which only a single diffracted order exists is exactly 1½ wavelengths (λ/Λ = 2/3). Single-order diffraction for such a period occurs at the Littrow angle of θL = arcsin(1/3) ≅ 20º.

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However, if you are differentiating on cost, be cautious. Supply chains change and manufacturing costs are volatile. Your price advantage can be quickly wiped out by a competitor who is able to make a similar product for even less.

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