The magic of Equiangular Spirals

In this series of posts I’ll be writing in more detail about the spiral PCF that I enjoy working on. An introduction and overview on spiral PCF can be found on the My Research page.

So, back to Equiangular Spiral PCF or ES-PCF

schematic of an equiangular spiral curve

Fig. 1: Schematic of an equiangular spiral curve

Equiangular spirals or logarithmic spirals are self similar curves: as the curve grows, the shape remains unchanged. We see them in nature in snails, shells, plants, galaxies and in many other places.

How is the ES design adapted for PCF?

The ES curve grows continuously and is defined by the equation: equiangular spiral equation


definiton of alpha

and θ is the angle between the radius of the ES and tangent at the end point of the radius; rspiral is the distance at any point along the spiral arm from the centre of the structure.

To adapt the design for a PCF, we choose the parameters, ro,, θ , which decide where the air holes will fall (alo

Fig. 2: schematic of the ES-PCF

Fig. 2: schematic of the ES-PCF

ng the ES curve). Several such independent ES patterns of holes can be arranged around a central core to form a cladding with microstructure of air holes. The average refractive index in the cladding  region is lower than the core, similar to standard PCF, and hence Total Internal Reflection (TIR) is the guiding mechanism for light. The cross section of an ES-PCF can be seen in fig. 2.

What makes the ES-PCF so unique and useful?

A number of things:

a) The number of parameters to play around with and optimize performance is bigger than most traditional PCF. We can change the number of ES arms, number of holes per arm, the radius of the arm, angular increment and finally air hole radius.

b) The air holes in the second ring are closer to the core than in a traditional Hexagonal PCF. For the same distance from origin to centre of first air hole (ro), in the ES-PCF the centre of the air holes in the 2nd ring are roughly 0.45 ro away from 1st ring, while this distance is about 0.87 ro  or Hex PCF (see Fig. 3). This means the optical field can be squeezed more tightly into the core by the air

Fig.3: Schematic of the Hexagonal PCF
Fig.3: Schematic of the Hexagonal PCF

holes resulting in properties such as large non-linearity.

What does this give us?

Just be squeezing the field more effectively into the core, we can significantly improve modal properties:

1)      Higher non-linearity (see

2)      Lower bending loss

3)      Flat dispersion

4)      Ability to optimize performance over more than 1 parameter simultaneously

What is the catch?

The catch is that such designs have not been made yet. Feasible fabrication remains the biggest stumbling block for such quasi-crystal designs.

Is there a solution?

Yes. Techniques such as extrusion or drilling can be used to make unconventional PCF designs, so these offer one possibility. Another is to adapt the ‘stack and draw’ technique which is primarily used for hexagonal PCF. In another post I’ll explore how some very elegant mathematics gives us a way to do the latter.


3 comments on “The magic of Equiangular Spirals

  1. hi arti, i am golam kibria from bangladesh and i am interested in spiral
    pcf but i can not understand the term alpha and n in your post and one of your paper i have found that spiral angle = 360/(2n). but in another paper you have mentioned that it has been found possible to tune the modal characteristics of the field in the ES-PCF by choosing the suitable combination of the values for the fiber parameters the number of arms spiral angle spiral radius air hole radius and number of rings. my another question is that in spiral pcf how we define air filling fraction.

    • Hi Golam,
      Thanks for writing in and its nice to know you find the spiral work interesting.
      I will write back to you in detail regarding the connection between alpha and number of arms (on holiday right now and don’t have my papers/notes…)

      By choosing the spiral characteristics the meaning is that for some desired behaviour maybe we need a spiral with 4 rings, 6 arms, in another case it may by 3 rings and 5 arms. Also it is possible to vary the air hole radius.

      In general for a spiral, the air fill fraction : (area of each air hole * number of air holes)/(total area of the finer)

      Hope this helps!

  2. hi arti can you give me any document based on drilling and extrusion. i have found may document on stack and draw method but I want to know about these method of fiber fabrication.

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