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Both Barlow lenses and Powermates act as image amplifiers by increasing the effective focal length of your telescope, thereby increasing the magnification achieved with any eyepiece.
Meade 2x Barlow Lens

Meade 2x Barlow Lens

Typical Barlows have a magnification of 2x and are composed of two or sometimes three elements in a single group, forming a negative lens. This lens produces a diverging ray of light which as well as magnifying the view, moves the eyepiece exit pupil farther outwards thereby increasing the eye relief. This effect is not too noticeable with short to medium focal length eyepieces, but it is more significant in longer focal lengths as it can lead to vignetting if the eyepiece lenses aren't wide enough to let the full width of the altered light cone through. Eyepiece to Barlow lens spacing has a significant effect on the magnification achieved.

A Powermate is a telecentric design with four lenses in two doublet groups - a smaller negative lens and a larger positive lens arranged so that light exits the Powermate in parallel rays. This design ensures that the original eye relief of the eyepiece remains unchanged, making it more suitable for longer length eyepieces and avoiding the risks of vignetting. A Powermate's magnification remains fairly constant irrespective of where the eyepiece is positioned.

2.5x Powermate

2.5x Powermate

From a lunar and planetary imaging point of view, Powermates are the better choice as they can be stacked to produce high levels of magnification.

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