Linear optics

Key takeaways

Linear optics belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Linear optics to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Linear optics from memory before moving on to harder problems.

Reference excerpt

Linear optics is a sub-field of optics, consisting of linear systems, and is the opposite of nonlinear optics. Linear optics includes most applications of lenses, mirrors, waveplates, diffraction gratings, and many other common optical components and systems. If an optical system is linear, it has the following properties (among others):

If monochromatic light enters an unchanging linear-optical system, the output will be at the same frequency. For example, if red light enters a lens, it will still be red when it exits the lens. The superposition principle is valid for linear-optical systems. For example, if a mirror transforms light input A into output B, and input C into output D, then an input consisting of A and C simultaneously give an output of B and D simultaneously. Relatedly, if the input light is made more intense, then the output light is made more intense but otherwise unchanged. These properties are violated in nonlinear optics, which frequently involves high-power pulsed lasers. Also, many material interactions including absorption and fluorescence are not part of linear optics.

Linear versus non-linear transformations (examples) As an example, and using the Dirac bracket notations (see bra-ket notations), the transformation | k ⟩ → e i k θ | k ⟩ {\displaystyle |k\rangle \rightarrow e^{ik\theta }|k\rangle }

is linear, while the transformation

α 0 | 0 ⟩ + α 1 | 1 ⟩ + α 2 | 2 ⟩ → α 0 | 0 ⟩ + α 1 | 1 ⟩ − α 2 | 2 ⟩ {\displaystyle \alpha _{0}|0\rangle +\alpha _{1}|1\rangle +\alpha _{2}|2\rangle \rightarrow \alpha _{0}|0\rangle +\alpha _{1}|1\rangle -\alpha _{2}|2\rangle } is non-linear. In the above examples, k = 0 , 1 , … {\displaystyle k=0,1,\ldots } is an integer representing the number of photons. The transformation in the first example is linear in the number of photons, while in the second example it is not. This specific nonlinear transformation plays an important role in optical quantum computing.

Linear versus nonlinear optical devices (examples) Phase shifters and beam splitters are examples of devices commonly used in linear optics. In contrast, frequency-mixing processes, the optical Kerr effect, cross-phase modulation, and Raman amplification, are a few examples of nonlinear effects in optics.

Connections to quantum computing One currently active field of research is the use of linear optics versus the use of nonlinear optics in quantum computing. For example, one model of linear optical quantum computing, the KLM model, is universal for quantum computing, and another model, the boson sampling-based model, is believed to be non-universal (for quantum computing) yet still seems to be able to solve some problems exponentially faster than a classical computer. The specific nonlinear transformation α 0 | 0 ⟩ + α 1 | 1 ⟩ + α 2 | 2 ⟩ → α 0 | 0 ⟩ + α 1 | 1 ⟩ − α 2 | 2 ⟩ {\displaystyle \alpha _{0}|0\rangle +\alpha _{1}|1\rangle +\alpha _{2}|2\rangle \rightarrow \alpha _{0}|0\rangle +\alpha _{1}|1\rangle -\alpha _{2}|2\rangle } , (called "a gate" when using computer science terminology) presented above, plays an important role in optical quantum computing: on the one hand, it is useful for deriving a universal set of gates, and on the other hand, with (only) linear-optical devices and post-selection of specific outcomes plus a feed-forward process, it can be applied with high success probability, and be used for obtaining universal linear-optical quantum computing, as done in the KLM model.

Worked examples

Example 1 — a first encounter with Linear optics

Start with the simplest possible case. Write down what Linear optics claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Linear optics before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Linear optics ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Linear optics

In research: Linear optics appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Linear optics in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Linear optics is common in secondary-school and first-year university syllabi. It links to neighbouring topics Optics, so understanding it makes those chapters shorter.
In everyday life: Look for Linear optics outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Linear optics in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Linear optics means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Linear optics out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Linear optics in simple terms?

Linear optics is a sub-field of optics, consisting of linear systems, and is the opposite of nonlinear optics. Linear optics includes most applications of lenses, mirrors, waveplates, diffraction gratings, and many other common optical components and systems.

Why does Linear optics matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Linear optics?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Linear optics.